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Intrusion detection system evasion techniques are modifications made to attacks in order to prevent detection by an intrusion detection system (IDS). Almost all published evasion techniques modify network attacks. The 1998 paper Insertion, Evasion, and Denial of Service: Eluding Network Intrusion Detection popularized IDS evasion, and discussed both evasion techniques and areas where the correct interpretation was ambiguous depending on the targeted computer system. The 'fragroute' and 'fragrouter' programs implement evasion techniques discussed in the paper. Many web vulnerability scanners, such as 'Nikto', 'whisker' and 'Sandcat', also incorporate IDS evasion techniques.
Most IDSs have been modified to detect or even reverse basic evasion techniques, but IDS evasion (and countering IDS evasion) are still active fields.
An IDS can be evaded by obfuscating or encoding the attack payload in a way that the target computer will reverse but the IDS will not. In this way, an attacker can exploit the end host without alerting the IDS.
Application layer protocols like HTTP allow for multiple encodings of data which are interpreted as the same value. For example, the string "cgi-bin" in a URL can be encoded as "%63%67%69%2d%62%69%6e" (i.e., in hexadecimal). [ 1 ] A web server will view these as the same string and act on them accordingly. An IDS must be aware of all of the possible encodings that its end hosts accept in order to match network traffic to known-malicious signatures. [ 1 ] [ 2 ]
Attacks on encrypted protocols such as HTTPS cannot be read by an IDS unless the IDS has a copy of the private key used by the server to encrypt the communication. [ 3 ] The IDS won't be able to match the encrypted traffic to signatures if it doesn't account for this.
Signature-based IDS often look for common attack patterns to match malicious traffic to signatures. To detect buffer overflow attacks, an IDS might look for the evidence of NOP slides which are used to weaken the protection of address space layout randomization . [ 4 ]
To obfuscate their attacks, attackers can use polymorphic shellcode to create unique attack patterns. This technique typically involves encoding the payload in some fashion (e.g., XOR -ing each byte with 0x95), then placing a decoder in front of the payload before sending it. When the target executes the code, it runs the decoder which rewrites the payload into its original form which the target then executes. [ 1 ] [ 4 ]
Polymorphic attacks don't have a single detectable signature, making them very difficult for signature-based IDS, and even some anomaly-based IDS, to detect. [ 1 ] [ 4 ] Shikata ga nai ("it cannot be helped") is a popular polymorphic encoder in the Metasploit framework used to convert malicious shellcode into difficult-to-detect polymorphic shellcode using XOR additive feedback. [ 5 ]
Attackers can evade IDS by crafting packets in such a way that the end host interprets the attack payload correctly while the IDS either interprets the attack incorrectly or determines that the traffic is benign too quickly. [ 3 ]
One basic technique is to split the attack payload into multiple small packets, so that the IDS must reassemble the packet stream to detect the attack. A simple way of splitting packets is by fragmenting them, but an adversary can also simply craft packets with small payloads. [ 1 ] The 'whisker' evasion tool calls crafting packets with small payloads 'session splicing'.
By itself, small packets will not evade any IDS that reassembles packet streams. However, small packets can be further modified in order to complicate reassembly and detection. One evasion technique is to pause between sending parts of the attack, hoping that the IDS will time out before the target computer does. A second evasion technique is to send the packets out of order,
Another evasion technique is to craft a series of packets with TCP sequence numbers configured to overlap. For example, the first packet will include 80 bytes of payload but the second packet's sequence number will be 76 bytes after the start of the first packet. When the target computer reassembles the TCP stream, they must decide how to handle the four overlapping bytes. Some operating systems will take the older data, and some will take the newer data. [ 3 ] If the IDS doesn't reassemble the TCP in the same way as the target, it can be manipulated into either missing a portion of the attack payload or seeing benign data inserted into the malicious payload, breaking the attack signature. [ 1 ] [ 3 ] This technique can also be used with IP fragmentation in a similar manner.
Some IDS evasion techniques involve deliberately manipulating TCP or IP protocols in a way the target computer will handle differently from the IDS. For example, the TCP urgent pointer is handled differently on different operating systems. If the IDS doesn't handle these protocol violations in a manner consistent with its end hosts, it is vulnerable to insertion and evasion techniques similar to those mentioned earlier. [ 3 ]
Attacks which are spread out across a long period of time or a large number of source IPs, such as nmap's slow scan, can be difficult to pick out of the background of benign traffic. An online password cracker which tests one password for each user every day will look nearly identical to a normal user who mistyped their password.
Due to the fact that passive IDS are inherently fail-open (as opposed to fail-closed ), launching a denial-of-service attack against the IDS on a network is a feasible method of circumventing its protection. [ 3 ] An adversary can accomplish this by exploiting a bug in the IDS, consuming all of the computational resources on the IDS, or deliberately triggering a large number of alerts to disguise the actual attack.
Packets captured by an IDS are stored in a kernel buffer until the CPU is ready to process them. If the CPU is under high load, it can't process the packets quickly enough and this buffer fills up. New (and possibly malicious) packets are then dropped because the buffer is full. [ 3 ]
An attacker can exhaust the IDS's CPU resources in a number of ways. For example, signature-based intrusion detection systems use pattern matching algorithms to match incoming packets against signatures of known attacks. Naturally, some signatures are more computational expensive to match against than others. Exploiting this fact, an attacker can send specially-crafted network traffic to force the IDS to use the maximum amount of CPU time as possible to run its pattern matching algorithm on the traffic. [ 1 ] [ 2 ] This algorithmic complexity attack can overwhelm the IDS with a relatively small amount of bandwidth. [ 1 ]
An IDS that also monitors encrypted traffic can spend a large portion of its CPU resources on decrypting incoming data. [ 3 ]
In order to match certain signatures, an IDS is required to keep state related to the connections it is monitoring. For example, an IDS must maintain "TCP control blocks" (TCBs), chunks of memory which track information such as sequence numbers, window sizes, and connection states (ESTABLISHED, RELATED, CLOSED, etc.), for each TCP connection monitored by the IDS. [ 3 ] Once all of the IDS's random-access memory (RAM) is consumed, it is forced to utilize virtual memory on the hard disk which is much slower than RAM, leading to performance problems and dropped packets similar to the effects of CPU exhaustion. [ 3 ]
If the IDS doesn't garbage collect TCBs correctly and efficiently, an attacker can exhaust the IDS's memory by starting a large number of TCP connections very quickly. [ 3 ] Similar attacks can be made by fragmenting a large number of packets into a larger number of smaller packets, or send a large number of out-of-order TCP segments. [ 3 ]
Alerts generated by an IDS have to be acted upon in order for them to have any value. An attacker can reduce the "availability" of an IDS by overwhelming the human operator with an inordinate number of alerts by sending large amounts of "malicious" traffic intended to generate alerts on the IDS. The attacker can then perform the actual attack using the alert noise as cover. The tools 'stick' and 'snot' were designed for this purpose. They generate a large number of IDS alerts by sending attack signature across the network, but will not trigger alerts in IDS that maintain application protocol context. | https://en.wikipedia.org/wiki/Intrusion_detection_system_evasion_techniques |
Luigi Stipa (30 November 1900 – 9 January 1992) was an Italian aeronautical, hydraulic, and civil engineer and aircraft designer who invented the "intubed propeller" for aircraft, a concept that some aviation historians view as the predecessor of the turbofan engine .
Stipa was born in Appignano del Tronto , Italy on 30 November 1900. He left school to serve in the Italian Army 's Bersaglieri Corps during World War I . After the war, he earned academic degrees in aeronautical engineering , hydraulic engineering and civil engineering . He went to work for the Italian Air Ministry , where he rose to the position of general inspector of the Engineering Division of the Regia Aeronautica (Italian Royal Air Force). [ 1 ]
In the 1920s, Stipa applied his study of hydraulic engineering to develop a theory of how to make aircraft more efficient as they traveled through the air. Noting that in fluid dynamics —in accordance with Bernoulli's principle —a fluid's velocity increases as the diameter of a tube it is passing through decreases, Stipa believed that the same principle could be applied to air flow to make an aircraft's engine more efficient by directing its propeller wash through a Venturi tube in a design he termed an "intubed propeller". In his concept, the fuselage of a single-engined airplane designed around an intubed propeller would be constructed as a tube, with the propeller and engine nacelle inside the tube, and therefore within the fuselage. The propeller would be of the same diameter as the tube, and its slipstream would exit the tube via the opening at the tube's trailing edge at the rear of the fuselage. [ 2 ]
Stipa spent years studying the idea mathematically , eventually determining that the Venturi tube's inner surface needed to be shaped like an airfoil in order to achieve the greatest efficiency. He also determined the optimum shape of the propeller, the most efficient distance between the leading edge of the tube and the propeller, and the best rate of revolution of the propeller. He appears to have intended the intubed propeller for use in large, multi-engine, flying wing aircraft—for which he produced several designs—but saw the construction of an experimental single-engine prototype aircraft as the first step in proving the concept. [ 3 ]
Stipa published his ideas in the Italian aviation journal Rivista Aeronautica ("Aeronautical Review"), then asked the Air Ministry to build a prototype aircraft to prove his concept. Eager for propaganda opportunities to highlight Italian achievements in technology to the world, and particularly interested in aviation advances, the Italian Fascist government approved of the venture, and contracted with the Caproni Aviation Corporation to build the prototype in 1932. [ 4 ]
The prototype, named the Stipa-Caproni , [ 5 ] first flew on October 7, 1932. Remarkably ungainly in appearance, the plane nonetheless proved Stipa's concept in that its intubed propeller increased its engine's efficiency, and the airfoil shape of the tube gave it an improved rate of climb compared to conventional aircraft of similar engine power and wing loading . The Stipa-Caproni also had a very low landing speed and was much quieter than conventional aircraft. Its rudder and elevators were mounted in the propeller's slipstream in the opening at the trailing edge of the tube in order to improve handling, and this configuration gave the aircraft handling characteristics that it made it very stable in flight. [ 6 ]
The Stipa-Caproni's great drawback was that the intubed propeller design created so much aerodynamic drag that most of the design's benefits in efficiency were negated by the drag. However, Stipa viewed the Stipa-Caproni as a mere testbed , and probably did not believe that the intubed propeller's aerodynamic drag problem would be significant in the various large, multi-engine flying wing aircraft he had designed. [ 7 ]
After the Caproni company completed initial testing of the Stipa-Caproni, the Regia Aeronautica took control of it and conducted a brief series of additional tests, but did not develop it further because the Stipa-Caproni offered no performance improvement over aircraft of conventional design. [ 8 ]
Despite the lack of Regia Aeronautica interest in developing the intubed propeller concept further, the Italian government publicized the success of Stipa's idea. Stipa patented the intubed propeller in 1938 in Germany , Italy, and the United States , and his work was published in France , Germany, Italy, the United Kingdom , and the United States, where the National Advisory Committee for Aeronautics studied it. The Regia Aeronautica ' s tests also sparked academic interest in the intubed propeller. [ 9 ]
France in the 1930s based its ANF- Mureaux BN.4 advanced night bomber design on a multi-engine intubed-propeller Stipa design, although the BN.4 was cancelled in 1936 before the first aircraft could be built. In Germany in 1934, Ludwig Kort designed the Kort nozzle , a ducted fan similar to Stipa's intubed propeller and still in use, and the German Heinkel T fighter design bore a similarity to Stipa's concepts. In Italy, none of Stipa's flying wing designs with intubed propellers ever were built, but the Caproni Campini N.1 , an experimental but impractical advanced derivative of the intubed propeller idea powered by a motorjet , appeared in 1940. [ 10 ]
Stipa himself believed that he deserved the credit for inventing the jet engine via his intubed propeller design, and claimed that the pulse jet engine the Germans employed on the V-1 flying bomb of World War II violated his intubed propeller patent in Germany, although the pulse-jet engine was not in fact closely related to his ideas. [ 11 ]
Stipa died on 9 January 1992, embittered over never having received what he viewed as his just recognition for inventing the jet engine. Some aviation historians do at least partially agree with Stipa, noting that the modern turbofan engine has features which show it to be the descendant of his intubed propeller concept. [ 12 ] | https://en.wikipedia.org/wiki/Intubed_propeller |
In the philosophy of mathematics , intuitionism , or neointuitionism (opposed to preintuitionism ), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. [ 1 ] That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition . The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene formally defined intuitionistic truth from a realist position, yet Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill-defined. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything they prove is in fact intuitionistically true. This gives rise to intuitionistic logic .
To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed. Any mathematical object is considered to be a product of a construction of a mind , and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its nonexistence. For the intuitionist, this is not valid; the refutation of the nonexistence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence. As such, intuitionism is a variety of mathematical constructivism ; but it is not the only kind.
The interpretation of negation is different in intuitionist logic than in classical logic. In classical logic, the negation of a statement asserts that the statement is false ; to an intuitionist, it means the statement is refutable . [ 2 ] There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement P is provable, then P certainly cannot be refutable. But even if it can be shown that P cannot be refuted, this does not constitute a proof of P . Thus P is a stronger statement than not-not-P .
Similarly, to assert that A or B holds, to an intuitionist, is to claim that either A or B can be proved . In particular, the law of excluded middle , " A or not A ", is not accepted as a valid principle. For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of " A or not A ". However, the intuitionist will accept that " A and not A " cannot be true. Thus the connectives "and" and "or" of intuitionistic logic do not satisfy de Morgan's laws as they do in classical logic.
Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof of model theory to abstract truth in modern mathematics . The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has been taken as giving philosophical support to several schools of philosophy, most notably the Anti-realism of Michael Dummett . Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines (e.g. Fuzzy Sets and Systems), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which intuitionism attempts to construct/refute/refound are taken as intuitively given. [ citation needed ]
Among the different formulations of intuitionism, there are several different positions on the meaning and reality of infinity.
The term potential infinity refers to a mathematical procedure in which there is an unending series of steps. After each step has been completed, there is always another step to be performed. For example, consider the process of counting: 1 , 2 , . . . {\displaystyle 1,2,...}
The term actual infinity refers to a completed mathematical object which contains an infinite number of elements. An example is the set of natural numbers , N = { 1 , 2 , . . . } {\displaystyle \mathbb {N} =\{1,2,...\}} .
In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers R {\displaystyle \mathbb {R} } is larger than N {\displaystyle \mathbb {N} } , because any attempt to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers "left over". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be "countable" or "denumerable". Infinite sets larger than this are said to be "uncountable". [ 3 ]
Cantor's set theory led to the axiomatic system of Zermelo–Fraenkel set theory (ZFC), now the most common foundation of modern mathematics . Intuitionism was created, in part, as a reaction to Cantor's set theory.
Modern constructive set theory includes the axiom of infinity from ZFC (or a revised version of this axiom) and the set N {\displaystyle \mathbb {N} } of natural numbers. Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example).
Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity.
According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell's vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence.
Intuitionism's history can be traced to two controversies in nineteenth century mathematics.
The first of these was the invention of transfinite arithmetic by Georg Cantor and its subsequent rejection by a number of prominent mathematicians including most famously his teacher Leopold Kronecker —a confirmed finitist .
The second of these was Gottlob Frege 's effort to reduce all of mathematics to a logical formulation via set theory and its derailing by a youthful Bertrand Russell , the discoverer of Russell's paradox . Frege had planned a three-volume definitive work, but just as the second volume was going to press, Russell sent Frege a letter outlining his paradox, which demonstrated that one of Frege's rules of self-reference was self-contradictory. In an appendix to the second volume, Frege acknowledged that one of the axioms of his system did in fact lead to Russell's paradox. [ 4 ]
Frege, the story goes, plunged into depression and did not publish the third volume of his work as he had planned. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and van Heijenoort's commentary.
These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox. Hence how one chooses to resolve Russell's paradox has direct implications on the status accorded to Cantor's transfinite arithmetic.
In the early twentieth century L. E. J. Brouwer represented the intuitionist position and David Hilbert the formalist position—see van Heijenoort. Kurt Gödel offered opinions referred to as Platonist (see various sources re Gödel). Alan Turing considers:
"non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive". [ 5 ] Later, Stephen Cole Kleene brought forth a more rational consideration of intuitionism in his Introduction to metamathematics (1952). [ 6 ]
Nicolas Gisin is adopting intuitionist mathematics to reinterpret quantum indeterminacy , information theory and the physics of time . [ 7 ] | https://en.wikipedia.org/wiki/Intuitionism |
Intuitionistic logic , sometimes more generally called constructive logic , refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof . In particular, systems of intuitionistic logic do not assume the law of excluded middle and double negation elimination , which are fundamental inference rules in classical logic.
Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer 's programme of intuitionism . From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation . [ 1 ]
Several systems of semantics for intuitionistic logic have been studied. One of these semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras . Another semantics uses Kripke models . These, however, are technical means for studying Heyting’s deductive system rather than formalizations of Brouwer’s original informal semantic intuitions. Semantical systems claiming to capture such intuitions, due to offering meaningful concepts of “constructive truth” (rather than merely validity or provability), are Kurt Gödel ’s dialectica interpretation , Stephen Cole Kleene ’s realizability , Yurii Medvedev’s logic of finite problems, [ 2 ] or Giorgi Japaridze ’s computability logic . Yet such semantics persistently induce logics properly stronger than Heyting’s logic. Some authors have argued that this might be an indication of inadequacy of Heyting’s calculus itself, deeming the latter incomplete as a constructive logic. [ 3 ]
In the semantics of classical logic, propositional formulae are assigned truth values from the two-element set { ⊤ , ⊥ } {\displaystyle \{\top ,\bot \}} ("true" and "false" respectively), regardless of whether we have direct evidence for either case. This is referred to as the 'law of excluded middle', because it excludes the possibility of any truth value besides 'true' or 'false'. In contrast, propositional formulae in intuitionistic logic are not assigned a definite truth value and are only considered "true" when we have direct evidence, hence proof . We can also say, instead of the propositional formula being "true" due to direct evidence, that it is inhabited by a proof in the Curry–Howard sense. Operations in intuitionistic logic therefore preserve justification , with respect to evidence and provability, rather than truth-valuation.
Intuitionistic logic is a commonly-used tool in developing approaches to constructivism in mathematics. The use of constructivist logics in general has been a controversial topic among mathematicians and philosophers (see, for example, the Brouwer–Hilbert controversy ). A common objection to their use is the above-cited lack of two central rules of classical logic, the law of excluded middle and double negation elimination. David Hilbert considered them to be so important to the practice of mathematics that he wrote:
Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether.
Intuitionistic logic has found practical use in mathematics despite the challenges presented by the inability to utilize these rules. One reason for this is that its restrictions produce proofs that have the disjunction and existence properties , making it also suitable for other forms of mathematical constructivism . Informally, this means that if there is a constructive proof that an object exists, that constructive proof may be used as an algorithm for generating an example of that object, a principle known as the Curry–Howard correspondence between proofs and algorithms. One reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants . These tools assist their users in the generation and verification of large-scale proofs, whose size usually precludes the usual human-based checking that goes into publishing and reviewing a mathematical proof. As such, the use of proof assistants (such as Agda or Coq ) is enabling modern mathematicians and logicians to develop and prove extremely complex systems, beyond those that are feasible to create and check solely by hand. One example of a proof that was impossible to satisfactorily verify without formal verification is the famous proof of the four color theorem . This theorem stumped mathematicians for more than a hundred years, until a proof was developed that ruled out large classes of possible counterexamples, yet still left open enough possibilities that a computer program was needed to finish the proof. That proof was controversial for some time, but, later, it was verified using Coq.
The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic . However, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic , hence their choice matters. In intuitionistic propositional logic (IPL) it is customary to use →, ∧, ∨, ⊥ as the basic connectives, treating ¬ A as an abbreviation for ( A → ⊥) . In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.
Intuitionistic logic can be defined using the following Hilbert-style calculus . This is similar to a way of axiomatizing classical propositional logic . [ 4 ]
In propositional logic, the inference rule is modus ponens
and the axioms are
To make this a system of first-order predicate logic, the generalization rules
are added, along with the axioms
If one wishes to include a connective ¬ {\displaystyle \neg } for negation rather than consider it an abbreviation for ϕ → ⊥ {\displaystyle \phi \to \bot } , it is enough to add:
There are a number of alternatives available if one wishes to omit the connective ⊥ {\displaystyle \bot } (false). For example, one may replace the three axioms FALSE, NOT-1', and NOT-2' with the two axioms
as at Propositional calculus § Axioms . Alternatives to NOT-1 are ( ϕ → ¬ χ ) → ( χ → ¬ ϕ ) {\displaystyle (\phi \to \neg \chi )\to (\chi \to \neg \phi )} or ( ϕ → ¬ ϕ ) → ¬ ϕ {\displaystyle (\phi \to \neg \phi )\to \neg \phi } .
The connective ↔ {\displaystyle \leftrightarrow } for equivalence may be treated as an abbreviation, with ϕ ↔ χ {\displaystyle \phi \leftrightarrow \chi } standing for ( ϕ → χ ) ∧ ( χ → ϕ ) {\displaystyle (\phi \to \chi )\land (\chi \to \phi )} . Alternatively, one may add the axioms
IFF-1 and IFF-2 can, if desired, be combined into a single axiom ( ϕ ↔ χ ) → ( ( ϕ → χ ) ∧ ( χ → ϕ ) ) {\displaystyle (\phi \leftrightarrow \chi )\to ((\phi \to \chi )\land (\chi \to \phi ))} using conjunction.
Gerhard Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system that is sound and complete with respect to intuitionistic logic. He called this system LJ. In LK any number of formulas is allowed to appear on the conclusion side of a sequent; in contrast LJ allows at most one formula in this position.
Other derivatives of LK are limited to intuitionistic derivations but still allow multiple conclusions in a sequent. LJ' [ 5 ] is one example.
The theorems of the pure logic are the statements provable from the axioms and inference rules. For example, using THEN-1 in THEN-2 reduces it to ( χ → ( ϕ → ψ ) ) → ( ϕ → ( χ → ψ ) ) {\displaystyle {\big (}\chi \to (\phi \to \psi ){\big )}\to {\big (}\phi \to (\chi \to \psi ){\big )}} . A formal proof of the latter using the Hilbert system is given on that page. With ⊥ {\displaystyle \bot } for ψ {\displaystyle \psi } , this in turn implies ( χ → ¬ ϕ ) → ( ϕ → ¬ χ ) {\displaystyle (\chi \to \neg \phi )\to (\phi \to \neg \chi )} . In words: "If χ {\displaystyle \chi } being the case implies that ϕ {\displaystyle \phi } is absurd, then if ϕ {\displaystyle \phi } does hold, one has that χ {\displaystyle \chi } is not the case." Due to the symmetry of the statement, one in fact obtained
When explaining the theorems of intuitionistic logic in terms of classical logic, it can be understood as a weakening thereof: It is more conservative in what it allows a reasoner to infer, while not permitting any new inferences that could not be made under classical logic. Each theorem of intuitionistic logic is a theorem in classical logic, but not conversely. Many tautologies in classical logic are not theorems in intuitionistic logic – in particular, as said above, one of intuitionistic logic's chief aims is to not affirm the law of the excluded middle so as to vitiate the use of non-constructive proof by contradiction , which can be used to furnish existence claims without providing explicit examples of the objects that it proves exist.
A double negation does not affirm the law of the excluded middle ( PEM ); while it is not necessarily the case that PEM is upheld in any context, no counterexample can be given either. Such a counterexample would be an inference (inferring the negation of the law for a certain proposition) disallowed under classical logic and thus PEM is not allowed in a strict weakening like intuitionistic logic. Formally, it is a simple theorem that ( ( ψ ∨ ( ψ → φ ) ) → φ ) ↔ φ {\displaystyle {\big (}(\psi \lor (\psi \to \varphi ))\to \varphi {\big )}\leftrightarrow \varphi } for any two propositions. By considering any φ {\displaystyle \varphi } established to be false this indeed shows that the double negation of the law ¬ ¬ ( ψ ∨ ¬ ψ ) {\displaystyle \neg \neg (\psi \lor \neg \psi )} is retained as a tautology already in minimal logic . This means any ¬ ( ψ ∨ ¬ ψ ) {\displaystyle \neg (\psi \lor \neg \psi )} is established to be inconsistent and the propositional calculus is in turn always compatible with classical logic.
When assuming the law of excluded middle implies a proposition, then by applying contraposition twice and using the double-negated excluded middle, one may prove double-negated variants of various strictly classical tautologies. The situation is more intricate for predicate logic formulas, when some quantified expressions are being negated.
Akin to the above, from modus ponens in the form ψ → ( ( ψ → φ ) → φ ) {\displaystyle \psi \to ((\psi \to \varphi )\to \varphi )} follows ψ → ¬ ¬ ψ {\displaystyle \psi \to \neg \neg \psi } . The relation between them may always be used to obtain new formulas: A weakened premise makes for a strong implication, and vice versa. For example, note that if ( ¬ ¬ ψ ) → ϕ {\displaystyle (\neg \neg \psi )\to \phi } holds, then so does ψ → ϕ {\displaystyle \psi \to \phi } , but the schema in the other direction would imply the double-negation elimination principle. Propositions for which double-negation elimination is possible are also called stable . Intuitionistic logic proves stability only for restricted types of propositions. A formula for which excluded middle holds can be proven stable using the disjunctive syllogism , which is discussed more thoroughly below. The converse does however not hold in general, unless the excluded middle statement at hand is stable itself.
An implication ψ → ¬ ϕ {\displaystyle \psi \to \neg \phi } can be proven to be equivalent to ¬ ¬ ψ → ¬ ϕ {\displaystyle \neg \neg \psi \to \neg \phi } , whatever the propositions. As a special case, it follows that propositions of negated form ( ψ = ¬ ϕ {\displaystyle \psi =\neg \phi } here) are stable, i.e. ¬ ¬ ¬ ϕ → ¬ ϕ {\displaystyle \neg \neg \neg \phi \to \neg \phi } is always valid.
In general, ¬ ¬ ψ → ϕ {\displaystyle \neg \neg \psi \to \phi } is stronger than ψ → ϕ {\displaystyle \psi \to \phi } , which is stronger than ¬ ¬ ( ψ → ϕ ) {\displaystyle \neg \neg (\psi \to \phi )} , which itself implies the three equivalent statements ψ → ( ¬ ¬ ϕ ) {\displaystyle \psi \to (\neg \neg \phi )} , ( ¬ ¬ ψ ) → ( ¬ ¬ ϕ ) {\displaystyle (\neg \neg \psi )\to (\neg \neg \phi )} and ¬ ϕ → ¬ ψ {\displaystyle \neg \phi \to \neg \psi } . Using the disjunctive syllogism, the previous four are indeed equivalent. This also gives an intuitionistically valid derivation of ¬ ¬ ( ¬ ¬ ϕ → ϕ ) {\displaystyle \neg \neg (\neg \neg \phi \to \phi )} , as it is thus equivalent to an identity .
When ψ {\displaystyle \psi } expresses a claim, then its double-negation ¬ ¬ ψ {\displaystyle \neg \neg \psi } merely expresses the claim that a refutation of ψ {\displaystyle \psi } would be inconsistent. Having proven such a mere double-negation also still aids in negating other statements through negation introduction , as then ( ϕ → ¬ ψ ) → ¬ ϕ {\displaystyle (\phi \to \neg \psi )\to \neg \phi } . A double-negated existential statement does not denote existence of an entity with a property, but rather the absurdity of assumed non-existence of any such entity. Also all the principles in the next section involving quantifiers explain use of implications with hypothetical existence as premise.
Weakening statements by adding two negations before existential quantifiers (and atoms) is also the core step in the double-negation translation . It constitutes an embedding of classical first-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if its Gödel–Gentzen translation is provable intuitionistically. For example, any theorem of classical propositional logic of the form ψ → ϕ {\displaystyle \psi \to \phi } has a proof consisting of an intuitionistic proof of ψ → ¬ ¬ ϕ {\displaystyle \psi \to \neg \neg \phi } followed by one application of double-negation elimination. Intuitionistic logic can thus be seen as a means of extending classical logic with constructive semantics.
Already minimal logic easily proves the following theorems, relating conjunction resp. disjunction to the implication using negation . Firstly,
In words: " ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } each imply that it is not the case that both ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } fail to hold together."
And here the logically negative conclusion ¬ ( ¬ ϕ ∧ ¬ ψ ) {\displaystyle \neg (\neg \phi \land \neg \psi )} is in fact equivalent to ¬ ϕ → ¬ ¬ ψ {\displaystyle \neg \phi \to \neg \neg \psi } . The alternative implied theorem, ( ϕ ∨ ψ ) → ( ¬ ϕ → ¬ ¬ ψ ) {\displaystyle (\phi \lor \psi )\to (\neg \phi \to \neg \neg \psi )} , represents a weakened variant of the disjunctive syllogism.
Secondly,
In words: " ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } both together imply that neither ϕ {\displaystyle \phi } nor ψ {\displaystyle \psi } fail to hold."
And here the logically negative conclusion ¬ ( ¬ ϕ ∨ ¬ ψ ) {\displaystyle \neg (\neg \phi \lor \neg \psi )} is in fact equivalent to ¬ ( ϕ → ¬ ψ ) {\displaystyle \neg (\phi \to \neg \psi )} . A variant of the converse of the implied theorem here does also hold, namely
In words: " ϕ {\displaystyle \phi } implying ψ {\displaystyle \psi } implies that it is not the case that ϕ {\displaystyle \phi } holds while ψ {\displaystyle \psi } fails to hold."
And indeed, stronger variants of all of these still do hold - for example the antecedents may be double-negated, as noted, or all ψ {\displaystyle \psi } may be replaced by ¬ ¬ ψ {\displaystyle \neg \neg \psi } on the antecedent sides, as will be discussed.
However, neither of these five implications above can be reversed without immediately implying excluded middle (consider ¬ ψ {\displaystyle \neg \psi } for ϕ {\displaystyle \phi } ) resp. double-negation elimination (consider true ϕ {\displaystyle \phi } ). Hence, the left hand sides do not constitute a possible definition of the right hand sides.
In contrast, in classical propositional logic it is possible to take one of those three connectives plus negation as primitive and define the other two in terms of it, in this way. Such is done, for example, in Łukasiewicz 's three axioms of propositional logic .
It is even possible to define all in terms of a sole sufficient operator such as the Peirce arrow (NOR) or Sheffer stroke (NAND). Similarly, in classical first-order logic, one of the quantifiers can be defined in terms of the other and negation.
These are fundamentally consequences of the law of bivalence , which makes all such connectives merely Boolean functions .
The law of bivalence is not required to hold in intuitionistic logic. As a result, none of the basic connectives can be dispensed with, and the above axioms are all necessary. So most of the classical identities between connectives and quantifiers are only theorems of intuitionistic logic in one direction. Some of the theorems go in both directions, i.e. are equivalences, as subsequently discussed.
Firstly, when x {\displaystyle x} is not free in the proposition φ {\displaystyle \varphi } , then
When the domain of discourse is empty, then by the principle of explosion , an existential statement implies anything. When the domain contains at least one term, then assuming excluded middle for ∀ x ϕ ( x ) {\displaystyle \forall x\,\phi (x)} , the inverse of the above implication becomes provably too, meaning the two sides become equivalent. This inverse direction is equivalent to the drinker's paradox (DP). Moreover, an existential and dual variant of it is given by the independence of premise principle (IP). Classically, the statement above is moreover equivalent to a more disjunctive form discussed further below. Constructively, existence claims are however generally harder to come by.
If the domain of discourse is not empty and ϕ {\displaystyle \phi } is moreover independent of x {\displaystyle x} , such principles are equivalent to formulas in the propositional calculus. Here, the formula then just expresses the identity ( ϕ → φ ) → ( ϕ → φ ) {\displaystyle (\phi \to \varphi )\to (\phi \to \varphi )} . This is the curried form of modus ponens ( ( ϕ → φ ) ∧ ϕ ) → φ {\displaystyle ((\phi \to \varphi )\land \phi )\to \varphi } , which in the special the case with φ {\displaystyle \varphi } as a false proposition results in the law of non-contradiction principle ¬ ( ϕ ∧ ¬ ϕ ) {\displaystyle \neg (\phi \land \neg \phi )} .
Considering a false proposition φ {\displaystyle \varphi } for the original implication results in the important
In words: "If there exists an entity x {\displaystyle x} that does not have the property ϕ {\displaystyle \phi } , then the following is refuted : Each entity has the property ϕ {\displaystyle \phi } ."
The quantifier formula with negations also immediately follows from the non-contradiction principle derived above, each instance of which itself already follows from the more particular ¬ ( ¬ ¬ ϕ ∧ ¬ ϕ ) {\displaystyle \neg (\neg \neg \phi \land \neg \phi )} . To derive a contradiction given ¬ ϕ {\displaystyle \neg \phi } , it suffices to establish its negation ¬ ¬ ϕ {\displaystyle \neg \neg \phi } (as opposed to the stronger ϕ {\displaystyle \phi } ) and this makes proving double-negations valuable also. By the same token, the original quantifier formula in fact still holds with ∀ x ϕ ( x ) {\displaystyle \forall x\ \phi (x)} weakened to ∀ x ( ( ϕ ( x ) → φ ) → φ ) {\displaystyle \forall x{\big (}(\phi (x)\to \varphi )\to \varphi {\big )}} . And so, in fact, a stronger theorem holds:
In words: "If there exists an entity x {\displaystyle x} that does not have the property ϕ {\displaystyle \phi } , then the following is refuted : For each entity, one is not able to prove that it does not have the property ϕ {\displaystyle \phi } ".
Secondly,
where similar considerations apply. Here the existential part is always a hypothesis and this is an equivalence. Considering the special case again,
The proven conversion ( χ → ¬ ϕ ) ↔ ( ϕ → ¬ χ ) {\displaystyle (\chi \to \neg \phi )\leftrightarrow (\phi \to \neg \chi )} can be used to obtain two further implications:
Of course, variants of such formulas can also be derived that have the double-negations in the antecedent.
A special case of the first formula here is ( ∀ x ¬ ϕ ( x ) ) → ¬ ( ∃ x ¬ ¬ ϕ ( x ) ) {\displaystyle (\forall x\,\neg \phi (x))\to \neg (\exists x\,\neg \neg \phi (x))} and this is indeed stronger than the → {\displaystyle \to } -direction of the equivalence bullet point listed above. For simplicity of the discussion here and below, the formulas are generally presented in weakened forms without all possible insertions of double-negations in the antecedents.
More general variants hold. Incorporating the predicate ψ {\displaystyle \psi } and currying, the following generalization also entails the relation between implication and conjunction in the predicate calculus, discussed below.
If the predicate ψ {\displaystyle \psi } is decidedly false for all x {\displaystyle x} , then this equivalence is trivial. If ψ {\displaystyle \psi } is decidedly true for all x {\displaystyle x} , the schema simply reduces to the previously stated equivalence. In the language of classes , A = { x ∣ ϕ ( x ) } {\displaystyle A=\{x\mid \phi (x)\}} and B = { x ∣ ψ ( x ) } {\displaystyle B=\{x\mid \psi (x)\}} , the special case of this equivalence with false φ {\displaystyle \varphi } equates two characterizations of disjointness A ∩ B = ∅ {\displaystyle A\cap B=\emptyset } :
There are finite variations of the quantifier formulas, with just two propositions:
The first principle cannot be reversed: Considering ¬ ψ {\displaystyle \neg \psi } for ϕ {\displaystyle \phi } would imply the weak excluded middle, i.e. the statement ¬ ψ ∨ ¬ ¬ ψ {\displaystyle \neg \psi \lor \neg \neg \psi } . But intuitionistic logic alone does not even prove ¬ ψ ∨ ¬ ¬ ψ ∨ ( ¬ ¬ ψ → ψ ) {\displaystyle \neg \psi \lor \neg \neg \psi \lor (\neg \neg \psi \to \psi )} . So in particular, there is no distributivity principle for negations deriving the claim ¬ ϕ ∨ ¬ ψ {\displaystyle \neg \phi \lor \neg \psi } from ¬ ( ϕ ∧ ψ ) {\displaystyle \neg (\phi \land \psi )} . For an informal example of the constructive reading, consider the following: From conclusive evidence it not to be the case that both Alice and Bob showed up to their date, one cannot derive conclusive evidence, tied to either of the two persons, that this person did not show up. Negated propositions are comparably weak, in that the classically valid De Morgan's law , granting a disjunction from a single negative hypothetical, does not automatically hold constructively. The intuitionistic propositional calculus and some of its extensions exhibit the disjunction property instead, implying one of the disjuncts of any disjunction individually would have to be derivable as well.
The converse variants of those two, and the equivalent variants with double-negated antecedents, had already been mentioned above. Implications towards the negation of a conjunction can often be proven directly from the non-contradiction principle. In this way one may also obtain the mixed form of the implications, e.g. ( ¬ ϕ ∨ ψ ) → ¬ ( ϕ ∧ ¬ ψ ) {\displaystyle (\neg \phi \lor \psi )\to \neg (\phi \land \neg \psi )} . Concatenating the theorems, we also find
The reverse cannot be provable, as it would prove weak excluded middle.
In predicate logic, the constant domain principle is not valid: ∀ x ( φ ∨ ψ ( x ) ) {\displaystyle \forall x{\big (}\varphi \lor \psi (x){\big )}} does not imply the stronger φ ∨ ∀ x ψ ( x ) {\displaystyle \varphi \lor \forall x\,\psi (x)} . The distributive properties does however hold for any finite number of propositions. For a variant of the De Morgan law concerning two existentially closed decidable predicates, see LLPO .
From the general equivalence also follows import-export , expressing incompatibility of two predicates using two different connectives:
Due to the symmetry of the conjunction connective, this again implies the already established ( ϕ → ¬ ψ ) ↔ ( ψ → ¬ ϕ ) {\displaystyle (\phi \to \neg \psi )\leftrightarrow (\psi \to \neg \phi )} .
The equivalence formula for the negated conjunction may be understood as a special case of currying and uncurrying. Many more considerations regarding double-negations again apply. And both non-reversible theorems relating conjunction and implication mentioned in the introduction to non-interdefinability above follow from this equivalence. One is a simply proven variant of a converse, while ( ϕ → ψ ) → ¬ ( ϕ ∧ ¬ ψ ) {\displaystyle (\phi \to \psi )\to \neg (\phi \land \neg \psi )} holds simply because ϕ → ψ {\displaystyle \phi \to \psi } is stronger than ϕ → ¬ ¬ ψ {\displaystyle \phi \to \neg \neg \psi } .
Now when using the principle in the next section, the following variant of the latter, with more negations on the left, also holds:
A consequence is that
Already minimal logic proves excluded middle equivalent to consequentia mirabilis , an instance of Peirce's law .
Now akin to modus ponens, clearly ( ϕ ∨ ψ ) → ( ( ϕ → ψ ) → ψ ) {\displaystyle (\phi \lor \psi )\to ((\phi \to \psi )\to \psi )} already in minimal logic, which is a theorem that does not even involve negations. In classical logic, this implication is in fact an equivalence. With taking ϕ {\displaystyle \phi } to be of the form ψ → φ {\displaystyle \psi \to \varphi } , excluded middle together with explosion is seen to entail Peirce's law.
In intuitionistic logic, one obtains variants of the stated theorem involving ⊥ {\displaystyle \bot } , as follows. Firstly, note that two different formulas for ¬ ( ϕ ∧ ψ ) {\displaystyle \neg (\phi \land \psi )} mentioned above can be used to imply ( ¬ ϕ ∨ ¬ ψ ) → ( ϕ → ¬ ψ ) {\displaystyle (\neg \phi \vee \neg \psi )\to (\phi \to \neg \psi )} . It also followed from direct case-analysis, as do variants where the negations are moved around, such as the theorems ( ¬ ϕ ∨ ψ ) → ( ϕ → ¬ ¬ ψ ) {\displaystyle (\neg \phi \lor \psi )\to (\phi \to \neg \neg \psi )} or ( ϕ ∨ ψ ) → ( ¬ ϕ → ¬ ¬ ψ ) {\displaystyle (\phi \lor \psi )\to (\neg \phi \to \neg \neg \psi )} , the latter being mentioned in the introduction to non-interdefinability. These are forms of the disjunctive syllogism involving negated propositions ¬ ψ {\displaystyle \neg \psi } . Strengthened forms still holds in intuitionistic logic, say
The implication cannot generally be reversed, as that would immediately imply excluded middle. So, intuitionistically, "Either P {\displaystyle P} or Q {\displaystyle Q} " is generally also a stronger propositional formula than "If not P {\displaystyle P} , then Q {\displaystyle Q} ", whereas in classical logic these are interchangeable.
Non-contradiction and explosion together actually also prove the stronger variant ( ¬ ϕ ∨ ψ ) → ( ¬ ¬ ϕ → ψ ) {\displaystyle (\neg \phi \lor \psi )\to (\neg \neg \phi \to \psi )} . And this shows how excluded middle for ψ {\displaystyle \psi } implies double-negation elimination for it. For a fixed ψ {\displaystyle \psi } , this implication also cannot generally be reversed. However, as ¬ ¬ ( ψ ∨ ¬ ψ ) {\displaystyle \neg \neg (\psi \lor \neg \psi )} is always constructively valid, it follows that assuming double-negation elimination for all such disjunctions implies classical logic also.
Of course the formulas established here may be combined to obtain yet more variations. For example, the disjunctive syllogism as presented generalizes to
If some term exists at all, the antecedent here even implies ∃ x ( ϕ ( x ) → φ ) {\displaystyle \exists x{\big (}\phi (x)\to \varphi {\big )}} , which in turn itself also implies the conclusion here (this is again the very first formula mentioned in this section).
The bulk of the discussion in these sections applies just as well to just minimal logic. But as for the disjunctive syllogism with general ψ {\displaystyle \psi } and in its form as a single proposition, minimal logic can at most prove ( ¬ ϕ ∨ ψ ) → ( ¬ ¬ ϕ → ψ ′ ) {\displaystyle (\neg \phi \lor \psi )\to (\neg \neg \phi \to \psi ')} where ψ ′ {\displaystyle \psi '} denotes ¬ ¬ ψ ∧ ( ψ ∨ ¬ ψ ) {\displaystyle \neg \neg \psi \land (\psi \lor \neg \psi )} . The conclusion here can only be simplified to ψ {\displaystyle \psi } using explosion.
The above lists also contain equivalences.
The equivalence involving a conjunction and a disjunction stems from ( P ∨ Q ) → R {\displaystyle (P\lor Q)\to R} actually being stronger than P → R {\displaystyle P\to R} . Both sides of the equivalence can be understood as conjunctions of independent implications. Above, absurdity ⊥ {\displaystyle \bot } is used for R {\displaystyle R} . In functional interpretations, it corresponds to if-clause constructions.
So e.g. "Not ( P {\displaystyle P} or Q {\displaystyle Q} )" is equivalent to "Not P {\displaystyle P} , and also not Q {\displaystyle Q} ".
An equivalence itself is generally defined as, and then equivalent to, a conjunction ( ∧ {\displaystyle \land } ) of implications ( → {\displaystyle \to } ), as follows:
With it, such connectives become in turn definable from it:
In turn, { ∨ , ↔ , ⊥ } {\displaystyle \{\lor ,\leftrightarrow ,\bot \}} and { ∨ , ↔ , ¬ } {\displaystyle \{\lor ,\leftrightarrow ,\neg \}} are complete bases of intuitionistic connectives, for example.
As shown by Alexander V. Kuznetsov , either of the following connectives – the first one ternary, the second one quinary – is by itself functionally complete : either one can serve the role of a sole sufficient operator for intuitionistic propositional logic, thus forming an analog of the Sheffer stroke from classical propositional logic: [ 6 ]
The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics . In 2014, a Tarski-like model theory was proved complete by Bob Constable , but with a different notion of completeness than classically. [ 7 ]
Unproved statements in intuitionistic logic are not given a intermediate or third truth value (as is sometimes mistakenly asserted). One can prove that such statements have no third truth value, a result dating back to Glivenko in 1928. [ 1 ] Instead they remain of unknown truth value, until they are either proved or disproved. Statements are disproved by deducing a contradiction from them.
A consequence of this point of view is that intuitionistic logic has no interpretation as a two-valued logic, nor even as a finite-valued logic, in the familiar sense. Although intuitionistic logic retains the trivial propositions { ⊤ , ⊥ } {\displaystyle \{\top ,\bot \}} from classical logic, each proof of a propositional formula is considered a valid propositional value, thus by Heyting's notion of propositions-as-sets, propositional formulae are (potentially non-finite) sets of their proofs.
In classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra . The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra. Then we have the useful theorem that a formula is a valid proposition of classical logic if and only if its value is 1 for every valuation —that is, for any assignment of values to its variables.
A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from a Heyting algebra , of which Boolean algebras are a special case. A formula is valid in intuitionistic logic if and only if it receives the value of the top element for any valuation on any Heyting algebra.
It can be shown that to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elements are the open subsets of the real line R . [ 8 ] In this algebra we have:
where int( X ) is the interior of X and X ∁ its complement .
The last identity concerning A → B allows us to calculate the value of ¬ A :
With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line. [ 8 ] For example, the formula ¬( A ∧ ¬ A ) is valid, because no matter what set X is chosen as the value of the formula A , the value of ¬( A ∧ ¬ A ) can be shown to be the entire line:
So the valuation of this formula is true, and indeed the formula is valid. But the law of the excluded middle, A ∨ ¬ A , can be shown to be invalid by using a specific value of the set of positive real numbers for A :
The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in the top element, representing true, as the valuation of the formula, regardless of what values from the algebra are assigned to the variables of the formula. [ 8 ] Conversely, for every invalid formula, there is an assignment of values to the variables that yields a valuation that differs from the top element. [ 9 ] [ 10 ] No finite Heyting algebra has the second of these two properties. [ 8 ]
Building upon his work on semantics of modal logic , Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics. [ 11 ] [ 12 ] [ 4 ]
It was discovered that Tarski-like semantics for intuitionistic logic were not possible to prove complete. However, Robert Constable has shown that a weaker notion of completeness still holds for intuitionistic logic under a Tarski-like model. In this notion of completeness we are concerned not with all of the statements that are true of every model, but with the statements that are true in the same way in every model. That is, a single proof that the model judges a formula to be true must be valid for every model. In this case, there is not only a proof of completeness, but one that is valid according to intuitionistic logic. [ 7 ]
In intuitionistic logic or a fixed theory using the logic, the situation can occur that an implication always hold metatheoretically, but not in the language. For example, in the pure propositional calculus, if ( ¬ A ) → ( B ∨ C ) {\displaystyle (\neg A)\to (B\lor C)} is provable, then so is ( ¬ A → B ) ∨ ( ¬ A → C ) {\displaystyle (\neg A\to B)\lor (\neg A\to C)} . Another example is that ( A → B ) → ( A ∨ C ) {\displaystyle (A\to B)\to (A\lor C)} being provable always also means that so is ( ( A → B ) → A ) ∨ ( ( A → B ) → C ) {\displaystyle {\big (}(A\to B)\to A{\big )}\lor {\big (}(A\to B)\to C{\big )}} . One says the system is closed under these implications as rules and they may be adopted.
Theories over constructive logics can exhibit the disjunction property . The pure intuitionistic propositional calculus does so as well.
In particular, it means the excluded middle disjunction for an un-rejectable statement A {\displaystyle A} is provable exactly when A {\displaystyle A} is provable.
This also means, for examples, that the excluded middle disjunction for some the excluded middle disjunctions are not provable also.
Intuitionistic logic is related by duality to a paraconsistent logic known as Brazilian , anti-intuitionistic or dual-intuitionistic logic . [ 13 ]
The subsystem of intuitionistic logic with the FALSE (resp. NOT-2) axiom removed is known as minimal logic and some differences have been elaborated on above.
In 1932, Kurt Gödel defined a system of logics intermediate between classical and intuitionistic logic. Indeed, any finite Heyting algebra that is not equivalent to a Boolean algebra defines (semantically) an intermediate logic . On the other hand, validity of formulae in pure intuitionistic logic is not tied to any individual Heyting algebra but relates to any and all Heyting algebras at the same time.
So for example, for a schema not involving negations, consider the classically valid ( A → B ) ∨ ( B → A ) {\displaystyle (A\to B)\lor (B\to A)} . Adopting this over intuitionistic logic gives the intermediate logic called Gödel-Dummett logic .
The system of classical logic is obtained by adding any one of the following axioms:
Various reformulations, or formulations as schemata in two variables (e.g. Peirce's law), also exist. One notable one is the (reverse) law of contraposition
Such are detailed on the intermediate logics article.
In general, one may take as the extra axiom any classical tautology that is not valid in the two-element Kripke frame ∘ ⟶ ∘ {\displaystyle \circ {\longrightarrow }\circ } (in other words, that is not included in Smetanich's logic ).
Kurt Gödel 's work involving many-valued logic showed in 1932 that intuitionistic logic is not a finite-valued logic . [ 14 ] (See the section titled Heyting algebra semantics above for an infinite-valued logic interpretation of intuitionistic logic.)
Any formula of the intuitionistic propositional logic (IPC) may be translated into the language of the normal modal logic S4 as follows:
and it has been demonstrated that the translated formula is valid in the propositional modal logic S4 if and only if the original formula is valid in IPC. [ 15 ] The above set of formulae are called the Gödel–McKinsey–Tarski translation .
There is also an intuitionistic version of modal logic S4 called Constructive Modal Logic CS4. [ 16 ]
There is an extended Curry–Howard isomorphism between IPC and simply typed lambda calculus . [ 16 ] | https://en.wikipedia.org/wiki/Intuitionistic_logic |
Intuitionistic type theory (also known as constructive type theory , or Martin-Löf type theory ( MLTT )) is a type theory and an alternative foundation of mathematics .
Intuitionistic type theory was created by Per Martin-Löf , a Swedish mathematician and philosopher , who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox , gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types .
Martin-Löf designed the type theory on the principles of mathematical constructivism . Constructivism requires any existence proof to contain a "witness". So, any proof of "there exists a prime greater than 1000" must identify a specific number that is both prime and greater than 1000. Intuitionistic type theory accomplished this design goal by internalizing the BHK interpretation . A useful consequence is that proofs become mathematical objects that can be examined, compared, and manipulated.
Intuitionistic type theory's type constructors were built to follow a one-to-one correspondence with logical connectives. For example, the logical connective called implication ( A ⟹ B {\displaystyle A\implies B} ) corresponds to the type of a function ( A → B {\displaystyle A\to B} ). This correspondence is called the Curry–Howard isomorphism . Prior type theories had also followed this isomorphism, but Martin-Löf's was the first to extend it to predicate logic by introducing dependent types.
A type theory is a kind of mathematical ontology , or foundation , describing the fundamental objects that exist. In the standard foundation, set theory combined with mathematical logic , the fundamental object is the set, which is a container that contains elements. In type theory, the fundamental object is the term, each of which belongs to one and only one type.
Intuitionistic type theory has three finite types, which are then composed using five different type constructors. Unlike set theories , type theories are not built on top of a logic like Frege's . So, each feature of the type theory does double duty as a feature of both math and logic.
There are three finite types: The 0 type contains no terms. The 1 type contains one canonical term. The 2 type contains two canonical terms.
Because the 0 type contains no terms, it is also called the empty type . It is used to represent anything that cannot exist. It is also written ⊥ {\displaystyle \bot } and represents anything unprovable (that is, a proof of it cannot exist). As a result, negation is defined as a function to it: ¬ A := A → ⊥ {\displaystyle \neg A:=A\to \bot } .
Likewise, the 1 type contains one canonical term and represents existence. It also is called the unit type .
Finally, the 2 type contains two canonical terms. It represents a definite choice between two values. It is used for Boolean values but not propositions.
Propositions are instead represented by particular types. For instance, a true proposition can be represented by the 1 type, while a false proposition can be represented by the 0 type. But we cannot assert that these are the only propositions, i.e. the law of excluded middle does not hold for propositions in intuitionistic type theory.
Σ-types contain ordered pairs. As with typical ordered pair (or 2-tuple) types, a Σ-type can describe the Cartesian product , A × B {\displaystyle A\times B} , of two other types, A {\displaystyle A} and B {\displaystyle B} . Logically, such an ordered pair would hold a proof of A {\displaystyle A} and a proof of B {\displaystyle B} , so one may see such a type written as A ∧ B {\displaystyle A\wedge B} .
Σ-types are more powerful than typical ordered pair types because of dependent typing. In the ordered pair, the type of the second term can depend on the value of the first term. For example, the first term of the pair might be a natural number and the second term's type might be a sequence of reals of length equal to the first term. Such a type would be written:
∑ n : N Vec ( R , n ) {\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}\operatorname {Vec} ({\mathbb {R} },n)}
Using set-theory terminology, this is similar to an indexed disjoint union of sets. In the case of the usual cartesian product, the type of the second term does not depend on the value of the first term. Thus the type describing the cartesian product N × R {\displaystyle {\mathbb {N} }\times {\mathbb {R} }} is written:
∑ n : N R {\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}{\mathbb {R} }}
It is important to note here that the value of the first term, n {\displaystyle n} , is not depended on by the type of the second term, R {\displaystyle {\mathbb {R} }} .
Σ-types can be used to build up longer dependently-typed tuples used in mathematics and the records or structs used in most programming languages. An example of a dependently-typed 3-tuple is two integers and a proof that the first integer is smaller than the second integer, described by the type:
∑ m : Z ∑ n : Z ( ( m < n ) = True ) {\displaystyle \sum _{m{\mathbin {:}}{\mathbb {Z} }}{\sum _{n{\mathbin {:}}{\mathbb {Z} }}((m<n)={\text{True}})}}
Dependent typing allows Σ-types to serve the role of existential quantifier . The statement "there exists an n {\displaystyle n} of type N {\displaystyle {\mathbb {N} }} , such that P ( n ) {\displaystyle P(n)} is proven" becomes the type of ordered pairs where the first item is the value n {\displaystyle n} of type N {\displaystyle {\mathbb {N} }} and the second item is a proof of P ( n ) {\displaystyle P(n)} . Notice that the type of the second item (proofs of P ( n ) {\displaystyle P(n)} ) depends on the value in the first part of the ordered pair ( n {\displaystyle n} ). Its type would be:
∑ n : N P ( n ) {\displaystyle \sum _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
Π-types contain functions. As with typical function types, they consist of an input type and an output type. They are more powerful than typical function types however, in that the return type can depend on the input value. Functions in type theory are different from set theory. In set theory, you look up the argument's value in a set of ordered pairs. In type theory, the argument is substituted into a term and then computation ("reduction") is applied to the term.
As an example, the type of a function that, given a natural number n {\displaystyle n} , returns a vector containing n {\displaystyle n} real numbers is written:
∏ n : N Vec ( R , n ) {\displaystyle \prod _{n{\mathbin {:}}{\mathbb {N} }}\operatorname {Vec} ({\mathbb {R} },n)}
When the output type does not depend on the input value, the function type is often simply written with a → {\displaystyle \to } . Thus, N → R {\displaystyle {\mathbb {N} }\to {\mathbb {R} }} is the type of functions from natural numbers to real numbers. Such Π-types correspond to logical implication. The logical proposition A ⟹ B {\displaystyle A\implies B} corresponds to the type A → B {\displaystyle A\to B} , containing functions that take proofs-of-A and return proofs-of-B. This type could be written more consistently as:
∏ a : A B {\displaystyle \prod _{a{\mathbin {:}}A}B}
Π-types are also used in logic for universal quantification . The statement "for every n {\displaystyle n} of type N {\displaystyle {\mathbb {N} }} , P ( n ) {\displaystyle P(n)} is proven" becomes a function from n {\displaystyle n} of type N {\displaystyle {\mathbb {N} }} to proofs of P ( n ) {\displaystyle P(n)} . Thus, given the value for n {\displaystyle n} the function generates a proof that P ( ⋅ ) {\displaystyle P(\,\cdot \,)} holds for that value. The type would be
∏ n : N P ( n ) {\displaystyle \prod _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
=-types are created from two terms. Given two terms like 2 + 2 {\displaystyle 2+2} and 2 ⋅ 2 {\displaystyle 2\cdot 2} , you can create a new type 2 + 2 = 2 ⋅ 2 {\displaystyle 2+2=2\cdot 2} . The terms of that new type represent proofs that the pair reduce to the same canonical term. Thus, since both 2 + 2 {\displaystyle 2+2} and 2 ⋅ 2 {\displaystyle 2\cdot 2} compute to the canonical term 4 {\displaystyle 4} , there will be a term of the type 2 + 2 = 2 ⋅ 2 {\displaystyle 2+2=2\cdot 2} . In intuitionistic type theory, there is a single way to introduce =-types and that is by reflexivity :
refl : ∏ a : A ( a = a ) . {\displaystyle \operatorname {refl} {\mathbin {:}}\prod _{a{\mathbin {:}}A}(a=a).}
It is possible to create =-types such as 1 = 2 {\displaystyle 1=2} where the terms do not reduce to the same canonical term, but you will be unable to create terms of that new type. In fact, if you were able to create a term of 1 = 2 {\displaystyle 1=2} , you could create a term of ⊥ {\displaystyle \bot } . Putting that into a function would generate a function of type 1 = 2 → ⊥ {\displaystyle 1=2\to \bot } . Since … → ⊥ {\displaystyle \ldots \to \bot } is how intuitionistic type theory defines negation, you would have ¬ ( 1 = 2 ) {\displaystyle \neg (1=2)} or, finally, 1 ≠ 2 {\displaystyle 1\neq 2} .
Equality of proofs is an area of active research in proof theory and has led to the development of homotopy type theory and other type theories.
Inductive types allow the creation of complex, self-referential types. For example, a linked list of natural numbers is either an empty list or a pair of a natural number and another linked list. Inductive types can be used to define unbounded mathematical structures like trees , graphs , etc.. In fact, the natural numbers type may be defined as an inductive type, either being 0 {\displaystyle 0} or the successor of another natural number.
Inductive types define new constants, such as zero 0 : N {\displaystyle 0{\mathbin {:}}{\mathbb {N} }} and the successor function S : N → N {\displaystyle S{\mathbin {:}}{\mathbb {N} }\to {\mathbb {N} }} . Since S {\displaystyle S} does not have a definition and cannot be evaluated using substitution, terms like S 0 {\displaystyle S0} and S S S 0 {\displaystyle SSS0} become the canonical terms of the natural numbers.
Proofs on inductive types are made possible by induction . Each new inductive type comes with its own inductive rule. To prove a predicate P ( ⋅ ) {\displaystyle P(\,\cdot \,)} for every natural number, you use the following rule:
N - e l i m : P ( 0 ) → ( ∏ n : N P ( n ) → P ( S ( n ) ) ) → ∏ n : N P ( n ) {\displaystyle {\operatorname {{\mathbb {N} }-elim} }\,{\mathbin {:}}P(0)\,\to \left(\prod _{n{\mathbin {:}}{\mathbb {N} }}P(n)\to P(S(n))\right)\to \prod _{n{\mathbin {:}}{\mathbb {N} }}P(n)}
Inductive types in intuitionistic type theory are defined in terms of W-types, the type of well-founded trees. Later work in type theory generated coinductive types, induction-recursion, and induction-induction for working on types with more obscure kinds of self-referentiality. Higher inductive types allow equality to be defined between terms.
The universe types allow proofs to be written about all the types created with the other type constructors. Every term in the universe type U 0 {\displaystyle {\mathcal {U}}_{0}} can be mapped to a type created with any combination of 0 , 1 , 2 , Σ , Π , = , {\displaystyle 0,1,2,\Sigma ,\Pi ,=,} and the inductive type constructor. However, to avoid paradoxes, there is no term in U n {\displaystyle {\mathcal {U}}_{n}} that maps to U n {\displaystyle {\mathcal {U}}_{n}} for any n ∈ N {\displaystyle {\mathcal {n}}\in \mathbb {N} } . [ 1 ]
To write proofs about all "the small types" and U 0 {\displaystyle {\mathcal {U}}_{0}} , you must use U 1 {\displaystyle {\mathcal {U}}_{1}} , which does contain a term for U 0 {\displaystyle {\mathcal {U}}_{0}} , but not for itself U 1 {\displaystyle {\mathcal {U}}_{1}} . Similarly, for U 2 {\displaystyle {\mathcal {U}}_{2}} . There is a predicative hierarchy of universes, so to quantify a proof over any fixed constant k {\displaystyle k} universes, you can use U k + 1 {\displaystyle {\mathcal {U}}_{k+1}} .
Universe types are a tricky feature of type theories. Martin-Löf's original type theory had to be changed to account for Girard's paradox . Later research covered topics such as "super universes", " Mahlo universes", and impredicative universes.
The formal definition of intuitionistic type theory is written using judgements. For example, in the statement "if A {\displaystyle A} is a type and B {\displaystyle B} is a type then ∑ a : A B {\displaystyle \textstyle \sum _{a:A}B} is a type" there are judgements of "is a type", "and", and "if ... then ...". The expression ∑ a : A B {\displaystyle \textstyle \sum _{a:A}B} is not a judgement; it is the type being defined.
This second level of the type theory can be confusing, particularly where it comes to equality. There is a judgement of term equality, which might say 4 = 2 + 2 {\displaystyle 4=2+2} . It is a statement that two terms reduce to the same canonical term. There is also a judgement of type equality, say that A = B {\displaystyle A=B} , which means every element of A {\displaystyle A} is an element of the type B {\displaystyle B} and vice versa. At the type level, there is a type 4 = 2 + 2 {\displaystyle 4=2+2} and it contains terms if there is a proof that 4 {\displaystyle 4} and 2 + 2 {\displaystyle 2+2} reduce to the same value. (Terms of this type are generated using the term-equality judgement.) Lastly, there is an English-language level of equality, because we use the word "four" and symbol " 4 {\displaystyle 4} " to refer to the canonical term S S S S 0 {\displaystyle SSSS0} . Synonyms like these are called "definitionally equal" by Martin-Löf.
The description of judgements below is based on the discussion in Nordström, Petersson, and Smith.
The formal theory works with types and objects .
A type is declared by:
An object exists and is in a type if:
Objects can be equal
and types can be equal
A type that depends on an object from another type is declared
and removed by substitution
An object that depends on an object from another type can be done two ways.
If the object is "abstracted", then it is written
and removed by substitution
The object-depending-on-object can also be declared as a constant as part of a recursive type. An example of a recursive type is:
Here, S {\displaystyle S} is a constant object-depending-on-object. It is not associated with an abstraction. Constants like S {\displaystyle S} can be removed by defining equality. Here the relationship with addition is defined using equality and using pattern matching to handle the recursive aspect of S {\displaystyle S} :
S {\displaystyle S} is manipulated as an opaque constant - it has no internal structure for substitution.
So, objects and types and these relations are used to express formulae in the theory. The following styles of judgements are used to create new objects, types and relations from existing ones:
By convention, there is a type that represents all other types. It is called U {\displaystyle {\mathcal {U}}} (or Set {\displaystyle \operatorname {Set} } ). Since U {\displaystyle {\mathcal {U}}} is a type, the members of it are objects. There is a dependent type El {\displaystyle \operatorname {El} } that maps each object to its corresponding type. In most texts El {\displaystyle \operatorname {El} } is never written. From the context of the statement, a reader can almost always tell whether A {\displaystyle A} refers to a type, or whether it refers to the object in U {\displaystyle {\mathcal {U}}} that corresponds to the type.
This is the complete foundation of the theory. Everything else is derived.
To implement logic, each proposition is given its own type. The objects in those types represent the different possible ways to prove the proposition. If there is no proof for the proposition, then the type has no objects in it. Operators like "and" and "or" that work on propositions introduce new types and new objects. So A × B {\displaystyle A\times B} is a type that depends on the type A {\displaystyle A} and the type B {\displaystyle B} . The objects in that dependent type are defined to exist for every pair of objects in A {\displaystyle A} and B {\displaystyle B} . If A {\displaystyle A} or B {\displaystyle B} has no proof and is an empty type, then the new type representing A × B {\displaystyle A\times B} is also empty.
This can be done for other types (booleans, natural numbers, etc.) and their operators.
Using the language of category theory , R. A. G. Seely introduced the notion of a locally cartesian closed category (LCCC) as the basic model of type theory. This has been refined by Hofmann and Dybjer to Categories with Families or Categories with Attributes based on earlier work by Cartmell. [ 2 ]
A category with families is a category C of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor T : C op → Fam ( Set ).
Fam ( Set ) is the category of families of Sets, in which objects are pairs ( A , B ) {\displaystyle (A,B)} of an "index set" A and a function B : X → A , and morphisms are pairs of functions f : A → A' and g : X → X' , such that B' ° g = f ° B – in other words, f maps B a to B g ( a ) .
The functor T assigns to a context G a set T y ( G ) {\displaystyle Ty(G)} of types, and for each A : T y ( G ) {\displaystyle A:Ty(G)} , a set T m ( G , A ) {\displaystyle Tm(G,A)} of terms. The axioms for a functor require that these play harmoniously with substitution. Substitution is usually written in the form Af or af , where A is a type in T y ( G ) {\displaystyle Ty(G)} and a is a term in T m ( G , A ) {\displaystyle Tm(G,A)} , and f is a substitution from D to G . Here A f : T y ( D ) {\displaystyle Af:Ty(D)} and a f : T m ( D , A f ) {\displaystyle af:Tm(D,Af)} .
The category C must contain a terminal object (the empty context), and a final object for a form of product called comprehension, or context extension, in which the right element is a type in the context of the left element. If G is a context, and A : T y ( G ) {\displaystyle A:Ty(G)} , then there should be an object ( G , A ) {\displaystyle (G,A)} final among contexts D with mappings p : D → G , q : Tm ( D,Ap ).
A logical framework, such as Martin-Löf's, takes the form of closure conditions on the context-dependent sets of types and terms: that there should be a type called Set, and for each set a type, that the types should be closed under forms of dependent sum and product, and so forth.
A theory such as that of predicative set theory expresses closure conditions on the types of sets and their elements: that they should be closed under operations that reflect dependent sum and product, and under various forms of inductive definition.
A fundamental distinction is extensional vs intensional type theory. In extensional type theory, definitional (i.e., computational) equality is not distinguished from propositional equality, which requires proof. As a consequence type checking becomes undecidable in extensional type theory because programs in the theory might not terminate. For example, such a theory allows one to give a type to the Y-combinator ; a detailed example of this can be found in Nordstöm and Petersson Programming in Martin-Löf's Type Theory . [ 3 ] However, this does not prevent extensional type theory from being a basis for a practical tool; for example, Nuprl is based on extensional type theory.
In contrast, in intensional type theory type checking is decidable , but the representation of standard mathematical concepts is somewhat more cumbersome, since intensional reasoning requires using setoids or similar constructions. There are many common mathematical objects that are hard to work with or cannot be represented without this, for example, integer numbers , rational numbers , and real numbers . Integers and rational numbers can be represented without setoids, but this representation is difficult to work with. Cauchy real numbers cannot be represented without this. [ 4 ]
Homotopy type theory works on resolving this problem. It allows one to define higher inductive types , which not only define first-order constructors ( values or points ), but higher-order constructors, i.e. equalities between elements ( paths ), equalities between equalities ( homotopies ), ad infinitum .
Different forms of type theory have been implemented as the formal systems underlying a number of proof assistants . While many are based on Per Martin-Löf's ideas, many have added features, more axioms, or a different philosophical background. For instance, the Nuprl system is based on computational type theory [ 5 ] and Coq is based on the calculus of (co)inductive constructions . Dependent types also feature in the design of programming languages such as ATS , Cayenne , Epigram , Agda , [ 6 ] and Idris . [ 7 ]
Per Martin-Löf constructed several type theories that were published at various times, some of them much later than when the preprints with their description became accessible to specialists (among others Jean-Yves Girard and Giovanni Sambin). The list below attempts to list all the theories that have been described in a printed form and to sketch the key features that distinguished them from each other. All of these theories had dependent products, dependent sums, disjoint unions, finite types and natural numbers. All the theories had the same reduction rules that did not include η-reduction either for dependent products or for dependent sums, except for MLTT79 where the η-reduction for dependent products is added.
MLTT71 was the first type theory created by Per Martin-Löf. It appeared in a preprint in 1971. It had one universe, but this universe had a name in itself, i.e., it was a type theory with, as it is called today, "Type in Type". Jean-Yves Girard has shown that this system was inconsistent, and the preprint was never published.
MLTT72 was presented in a 1972 preprint that has now been published. [ 8 ] That theory had one universe V and no identity types (=-types). The universe was " predicative " in the sense that the dependent product of a family of objects from V over an object that was not in V such as, for example, V itself, was not assumed to be in V. The universe was à la Russell 's Principia Mathematica , i.e., one would write directly "T∈V" and "t∈T" (Martin-Löf uses the sign "∈" instead of modern ":") without an added constructor such as "El".
MLTT73 was the first definition of a type theory that Per Martin-Löf published (it was presented at the Logic Colloquium '73 and published in 1975 [ 9 ] ). There are identity types, which he describes as "propositions", but since no real distinction between propositions and the rest of the types is introduced the meaning of this is unclear. There is what later acquires the name of J-eliminator but yet without a name (see pp. 94–95). There is in this theory an infinite sequence of universes V 0 , ..., V n , ... . The universes are predicative, à la Russell and non-cumulative . In fact, Corollary 3.10 on p. 115 says that if A∈V m and B∈V n are such that A and B are convertible then m = n . This means, for example, that it would be difficult to formulate univalence axiom in this theory—there are contractible types in each of the V i , but it is unclear how to declare them to be equal since there are no identity types connecting V i and V j for i ≠ j .
MLTT79 was presented in 1979 and published in 1982. [ 10 ] In this paper, Martin-Löf introduced the four basic types of judgement for the dependent type theory that has since become fundamental in the study of the meta-theory of such systems. He also introduced contexts as a separate concept in it (see p. 161). There are identity types with the J-eliminator (which already appeared in MLTT73 but did not have this name there) but also with the rule that makes the theory "extensional" (p. 169). There are W-types. There is an infinite sequence of predicative universes that are cumulative .
Bibliopolis : there is a discussion of a type theory in the Bibliopolis book from 1984, [ 11 ] but it is somewhat open-ended and does not seem to represent a particular set of choices and so there is no specific type theory associated with it. | https://en.wikipedia.org/wiki/Intuitionistic_type_theory |
Intussusceptive angiogenesis also known as splitting angiogenesis , is a type of angiogenesis , the process whereby a new blood vessel is created. By intussusception a new blood vessel is created by splitting of an existing blood vessel in two. [ 1 ] [ 2 ] [ 3 ] Intussusception occurs in normal development as well as in pathologic conditions involving wound healing, [ 4 ] tissue regeneration, inflammation as colitis [ 5 ] [ 6 ] or myocarditis, [ 7 ] lung fibrosis, [ 8 ] and tumors [ 9 ] [ 10 ] amongst others.
Intussusception was first observed in neonatal rats. In this type of vessel formation, the capillary wall extends into the lumen to split a single vessel in two. There are four phases of intussusceptive angiogenesis. First, the two opposing capillary walls establish a zone of contact. Second, the endothelial cell junctions are reorganized and the vessel bilayer is perforated to allow growth factors and cells to penetrate into the lumen. Third, a core is formed between the two new vessels at the zone of contact that is filled with pericytes and myofibroblasts . These cells begin laying collagen fibers into the core to provide an extracellular matrix for growth of the vessel lumen. Finally, the core is fleshed out with no alterations to the basic structure. Intussusception is important because it is a reorganization of existing cells. It allows a vast increase in the number of capillaries without a corresponding increase in the number of endothelial cells . This is especially important in embryonic development as there are not enough resources to create a rich microvasculature with new cells every time a new vessel develops. [ citation needed ]
A process called coalescent angiogenesis [ 11 ] [ 12 ] is considered the opposite of intussusceptive angiogenesis. During coalescent angiogenesis capillaries fuse and form larger vessels to increase blood flow and circulation. Several other modes of angiogenesis have been described, such as sprouting angiogenesis, vessel co-option and vessel elongation. [ 13 ]
In a small study comparing the lungs of patients who had died from COVID-19 to those that had died from influenza A pneumonia (H1N1) to uninfected controls during autopsy; there was a significantly greater density of intussusceptive angiogenic features in the lungs of patients who had died from Covid-19 as compared to influenza A and the control group. The degree of intussusceptive angiogenic features in the lungs from the Covid-19 patients were also found to be greater as the length of hospitalization increased (which was not seen in the influenza or control groups). This suggests that increased or enhanced intussusceptive angiogenesis is seen in Covid-19 and may play a role in pathogenesis. [ 14 ] [ 15 ]
This cardiovascular system article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Intussusceptive_angiogenesis |
Inuit astronomy is centered around the Qilak, the Inuit name for the celestial sphere and the home for souls of departed people. Inuit beliefs about astronomy are shaped by the harsh climate in the Arctic and the resulting difficulties of surviving and hunting in the region. The stars were an important tool to track time, seasons, and location, particularly during winter. [ 1 ]
The Inuit are a group of circumpolar peoples who inhabit the Arctic and subarctic regions of Canada and Alaska (North America), Greenland/KalaallitNunaat (Denmark) and parts of northern Siberia (Russia). There are many similarities between the traditions and beliefs among the indigenous peoples in Arctic regions. For example, the Inuit, Chukchi and Evenks all have a worldview based on their religious beliefs and have related traditions about astronomy. [ 2 ] While differing traditions exist among groups, they overlap in the way the stars, weather, and folk tales assist in hunting, navigation and teaching their young about the world. [ 1 ]
Their astronomy and relationship to the sky is heavily influenced by their spiritual and pragmatic needs, as well as the high northerly latitudes where they reside. For those living above the Arctic Circle , the latitude affects the view of the night sky , especially the fact that during winter polar night may occur for multiple months and the midnight sun during summer. [ 2 ]
The latitudes within the Arctic Circle significantly influence both the behavior of the sun and the ability to see stars. Starting at approximately the end of November to mid-January, at around the 69th parallel north , the Inuit never see the sun . During this time, though dark, the sky is often obscured by weather conditions like blowing snow or cloud cover. Then, for 10 weeks beginning in mid-May, the sun never sets . This also means that in the spring, summer, and early fall, the skies are too bright to visible see stars. These phenomenons and limitations have had a significant influence on Inuit relationships to the Sun and stars. [ 2 ]
The latitude also means that some stars are not visible at all, while those that are visible, but near the horizon , are visibly affected by atmospheric refraction because of the low temperatures . [ 2 ] The appearance of these stars near the horizon changes throughout the day and during "dark days of winter without sunrise, the stars signal the time for villagers to wake up, for children and hunters to begin their days, and for the village to start the routines of the day." [ 2 ]
Refraction also affects the appearance of the Sun, in particular when it first re-appears on the horizon after the long, dark winter. This was a time of great anxiety, so the Inuit observed strict taboos "to ensure the sun's rapid and full return." [ 2 ] The Sun was not believed to be safely and securely back until it reached a height in the sky roughly equivalent to the width of a mitten on an outstretched hand. Only at this point would longer dog-team journeys be taken and the preparations for moving to spring camps begin. [ 2 ]
Inuit use the Moon to keep track of the 'calendar year', counting thirteen "moon months." Each month is named for a predictable seasonal characteristic, mostly related to animal behavior, which coincide with a particular moon. For example, one month is called "the nesting of eider ducks" while another is called "the birth of seal pups." The moon month during the polar night is referred to as tauvijjuaq or the "great darkness." [ 2 ]
Observing the winter solstice was very important, though the equinoxes and summer solstice were not given much attention. Winter solstice marks both the darkest part of winter and the turning point when light begins to increase, marking the promise of the Sun's return. The first appearance of Aagjuuk happens around mid-December and is used across the Arctic to signal winter solstice's arrival. For some tribes, this would also signal the time for a midwinter celebration. [ 2 ]
Some constellations have only seasonal appearances, which help mark the passage of time. For example, Ullakut ( Orion ) and Sakiattiak ( Taurus ) are only visible in the winter. Throughout winter, many stars within Tukturjuit ( Big Dipper ) were used as hour hands to keep track of time during the night or as calendar stars to determine the date. Aagjuuk ( Aquila ) and Kingulliq ( Lyra ) begin to appear near the end of winter, signaling that light will be returning to the region. [ 1 ]
Inuit tradition closely links the Earth and sky, with a spatial understanding of the Earth as a large flat disk ending in cliffs and surrounded by sky. The sky itself is understood as four to five layers of celestial realms. Each layer is a separate land of the dead and its own world. The aurora borealis bears special significance as the place where spirits who died from blood loss, murder or childbirth dwell. Legends warn Inuit against wrongdoing and taboo acts by telling the stories of people being transformed into stars after committing transgressions. For example, "the ubiquitous Inuit epic in which greed, murder, incest, and retribution account for the creation of the sun, moon, and the first stars." [ 2 ]
The Inuit have traditional names for many constellations , asterisms and stars. Inuit astronomy names thirty-three individual stars, two star clusters, and one nebula. The stars are incorporated into 16 or 17 asterisms, though seven stand alone with individual names.
Distinctively, the star Polaris or the North Star is a minor one for the Inuit, possibly because at northern latitudes its location is too high in the sky to be useful for navigation. It is called Nuutuittuq, which means "never moves." It is only used for navigation by the southernmost Inuit. [ 1 ]
Naming practices fall into two main categories: human or animal personification and "intrinsic" designation, drawing from a particular visible feature of the star(s). Intrinsic designation might be based on color, distance to surrounding stars, and movement or progression across the sky. Many stars have two names, an everyday name and "literary" name which would be used when stars personify a mythic character. The stars never collectively make the image of an animal or person because of the belief that each individual star was once an animate being living on Earth. Inanimate objects like the soapstone "lamp-stand" or "collar-bones" are represented by groupings of stars. [ 2 ]
The names of the stars are recalled through myths and legends, which "reflected social ethics and universal concerns about creation, social and cosmic order, nourishment, retribution, and renewal." [ 3 ] These stories are both used as explanations for the way things are or came to be and as a narrative tool to help people remember the location of stars and their relationship to each other, crucial when using the stars for navigation or time telling. [ 2 ]
(alternate spelling: Niqirtsuituq) [ 5 ]
*Asterisks mark names of principle stars, not full constellations or asterisms. | https://en.wikipedia.org/wiki/Inuit_astronomy |
In mathematics , the inverse function of a function f (also called the inverse of f ) is a function that undoes the operation of f . The inverse of f exists if and only if f is bijective , and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.}
For a function f : X → Y {\displaystyle f\colon X\to Y} , its inverse f − 1 : Y → X {\displaystyle f^{-1}\colon Y\to X} admits an explicit description: it sends each element y ∈ Y {\displaystyle y\in Y} to the unique element x ∈ X {\displaystyle x\in X} such that f ( x ) = y .
As an example, consider the real-valued function of a real variable given by f ( x ) = 5 x − 7 . One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function f − 1 : R → R {\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} } defined by f − 1 ( y ) = y + 7 5 . {\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}
Let f be a function whose domain is the set X , and whose codomain is the set Y . Then f is invertible if there exists a function g from Y to X such that g ( f ( x ) ) = x {\displaystyle g(f(x))=x} for all x ∈ X {\displaystyle x\in X} and f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for all y ∈ Y {\displaystyle y\in Y} . [ 1 ]
If f is invertible, then there is exactly one function g satisfying this property. The function g is called the inverse of f , and is usually denoted as f −1 , a notation introduced by John Frederick William Herschel in 1813. [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ nb 1 ]
The function f is invertible if and only if it is bijective. This is because the condition g ( f ( x ) ) = x {\displaystyle g(f(x))=x} for all x ∈ X {\displaystyle x\in X} implies that f is injective , and the condition f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for all y ∈ Y {\displaystyle y\in Y} implies that f is surjective .
The inverse function f −1 to f can be explicitly described as the function
Recall that if f is an invertible function with domain X and codomain Y , then
Using the composition of functions , this statement can be rewritten to the following equations between functions:
where id X is the identity function on the set X ; that is, the function that leaves its argument unchanged. In category theory , this statement is used as the definition of an inverse morphism .
Considering function composition helps to understand the notation f −1 . Repeatedly composing a function f : X → X with itself is called iteration . If f is applied n times, starting with the value x , then this is written as f n ( x ) ; so f 2 ( x ) = f ( f ( x )) , etc. Since f −1 ( f ( x )) = x , composing f −1 and f n yields f n −1 , "undoing" the effect of one application of f .
While the notation f −1 ( x ) might be misunderstood, [ 1 ] ( f ( x )) −1 certainly denotes the multiplicative inverse of f ( x ) and has nothing to do with the inverse function of f . [ 6 ] The notation f ⟨ − 1 ⟩ {\displaystyle f^{\langle -1\rangle }} might be used for the inverse function to avoid ambiguity with the multiplicative inverse . [ 7 ]
In keeping with the general notation, some English authors use expressions like sin −1 ( x ) to denote the inverse of the sine function applied to x (actually a partial inverse ; see below). [ 8 ] [ 6 ] Other authors feel that this may be confused with the notation for the multiplicative inverse of sin ( x ) , which can be denoted as (sin ( x )) −1 . [ 6 ] To avoid any confusion, an inverse trigonometric function is often indicated by the prefix " arc " (for Latin arcus ). [ 9 ] [ 10 ] For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin ( x ) . [ 9 ] [ 10 ] Similarly, the inverse of a hyperbolic function is indicated by the prefix " ar " (for Latin ārea ). [ 10 ] For instance, the inverse of the hyperbolic sine function is typically written as arsinh ( x ) . [ 10 ] The expressions like sin −1 ( x ) can still be useful to distinguish the multivalued inverse from the partial inverse: sin − 1 ( x ) = { ( − 1 ) n arcsin ( x ) + π n : n ∈ Z } {\displaystyle \sin ^{-1}(x)=\{(-1)^{n}\arcsin(x)+\pi n:n\in \mathbb {Z} \}} . Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the f −1 notation should be avoided. [ 11 ] [ 10 ]
The function f : R → [0,∞) given by f ( x ) = x 2 is not injective because ( − x ) 2 = x 2 {\displaystyle (-x)^{2}=x^{2}} for all x ∈ R {\displaystyle x\in \mathbb {R} } . Therefore, f is not invertible.
If the domain of the function is restricted to the nonnegative reals, that is, we take the function f : [ 0 , ∞ ) → [ 0 , ∞ ) ; x ↦ x 2 {\displaystyle f\colon [0,\infty )\to [0,\infty );\ x\mapsto x^{2}} with the same rule as before, then the function is bijective and so, invertible. [ 12 ] The inverse function here is called the (positive) square root function and is denoted by x ↦ x {\displaystyle x\mapsto {\sqrt {x}}} .
The following table shows several standard functions and their inverses:
Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse f − 1 {\displaystyle f^{-1}} of an invertible function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } has an explicit description as
This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if f is the function
then to determine f − 1 ( y ) {\displaystyle f^{-1}(y)} for a real number y , one must find the unique real number x such that (2 x + 8) 3 = y . This equation can be solved:
Thus the inverse function f −1 is given by the formula
Sometimes, the inverse of a function cannot be expressed by a closed-form formula . For example, if f is the function
then f is a bijection, and therefore possesses an inverse function f −1 . The formula for this inverse has an expression as an infinite sum:
Since a function is a special type of binary relation , many of the properties of an inverse function correspond to properties of converse relations .
If an inverse function exists for a given function f , then it is unique. [ 13 ] This follows since the inverse function must be the converse relation, which is completely determined by f .
There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y , then its inverse f −1 has domain Y and image X , and the inverse of f −1 is the original function f . In symbols, for functions f : X → Y and f −1 : Y → X , [ 13 ]
This statement is a consequence of the implication that for f to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by [ 14 ]
The inverse of a composition of functions is given by [ 15 ]
Notice that the order of g and f have been reversed; to undo f followed by g , we must first undo g , and then undo f .
For example, let f ( x ) = 3 x and let g ( x ) = x + 5 . Then the composition g ∘ f is the function that first multiplies by three and then adds five,
To reverse this process, we must first subtract five, and then divide by three,
This is the composition ( f −1 ∘ g −1 )( x ) .
If X is a set, then the identity function on X is its own inverse:
More generally, a function f : X → X is equal to its own inverse, if and only if the composition f ∘ f is equal to id X . Such a function is called an involution .
If f is invertible, then the graph of the function
is the same as the graph of the equation
This is identical to the equation y = f ( x ) that defines the graph of f , except that the roles of x and y have been reversed. Thus the graph of f −1 can be obtained from the graph of f by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line y = x . [ 16 ] [ 1 ]
By the inverse function theorem , a continuous function of a single variable f : A → R {\displaystyle f\colon A\to \mathbb {R} } (where A ⊆ R {\displaystyle A\subseteq \mathbb {R} } ) is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima ). For example, the function
is invertible, since the derivative f′ ( x ) = 3 x 2 + 1 is always positive.
If the function f is differentiable on an interval I and f′ ( x ) ≠ 0 for each x ∈ I , then the inverse f −1 is differentiable on f ( I ) . [ 17 ] If y = f ( x ) , the derivative of the inverse is given by the inverse function theorem,
Using Leibniz's notation the formula above can be written as
This result follows from the chain rule (see the article on inverse functions and differentiation ).
The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable multivariable function f : R n → R n is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible . In this case, the Jacobian of f −1 at f ( p ) is the matrix inverse of the Jacobian of f at p .
Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function
is not one-to-one, since x 2 = (− x ) 2 . However, the function becomes one-to-one if we restrict to the domain x ≥ 0 , in which case
(If we instead restrict to the domain x ≤ 0 , then the inverse is the negative of the square root of y .)
Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function :
Sometimes, this multivalued inverse is called the full inverse of f , and the portions (such as √ x and − √ x ) are called branches . The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch , and its value at y is called the principal value of f −1 ( y ) .
For a continuous function on the real line, one branch is required between each pair of local extrema . For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture).
The above considerations are particularly important for defining the inverses of trigonometric functions . For example, the sine function is not one-to-one, since
for every real x (and more generally sin( x + 2 π n ) = sin( x ) for every integer n ). However, the sine is one-to-one on the interval [− π / 2 , π / 2 ] , and the corresponding partial inverse is called the arcsine . This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between − π / 2 and π / 2 . The following table describes the principal branch of each inverse trigonometric function: [ 19 ]
Function composition on the left and on the right need not coincide. In general, the conditions
imply different properties of f . For example, let f : R → [0, ∞) denote the squaring map, such that f ( x ) = x 2 for all x in R , and let g : [0, ∞) → R denote the square root map, such that g ( x ) = √ x for all x ≥ 0 . Then f ( g ( x )) = x for all x in [0, ∞) ; that is, g is a right inverse to f . However, g is not a left inverse to f , since, e.g., g ( f (−1)) = 1 ≠ −1 .
If f : X → Y , a left inverse for f (or retraction of f ) is a function g : Y → X such that composing f with g from the left gives the identity function [ 20 ] g ∘ f = id X . {\displaystyle g\circ f=\operatorname {id} _{X}{\text{.}}} That is, the function g satisfies the rule
The function g must equal the inverse of f on the image of f , but may take any values for elements of Y not in the image.
A function f with nonempty domain is injective if and only if it has a left inverse. [ 21 ] An elementary proof runs as follows:
If nonempty f : X → Y is injective, construct a left inverse g : Y → X as follows: for all y ∈ Y , if y is in the image of f , then there exists x ∈ X such that f ( x ) = y . Let g ( y ) = x ; this definition is unique because f is injective. Otherwise, let g ( y ) be an arbitrary element of X .
For all x ∈ X , f ( x ) is in the image of f . By construction, g ( f ( x )) = x , the condition for a left inverse.
In classical mathematics, every injective function f with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics . For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1} . [ 22 ]
A right inverse for f (or section of f ) is a function h : Y → X such that
That is, the function h satisfies the rule
Thus, h ( y ) may be any of the elements of X that map to y under f .
A function f has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice ).
An inverse that is both a left and right inverse (a two-sided inverse ), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse .
A function has a two-sided inverse if and only if it is bijective.
If f : X → Y is any function (not necessarily invertible), the preimage (or inverse image ) of an element y ∈ Y is defined to be the set of all elements of X that map to y :
The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f .
The notion can be generalized to subsets of the range. Specifically, if S is any subset of Y , the preimage of S , denoted by f − 1 ( S ) {\displaystyle f^{-1}(S)} , is the set of all elements of X that map to S :
For example, take the function f : R → R ; x ↦ x 2 . This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.
The original notion and its generalization are related by the identity f − 1 ( y ) = f − 1 ( { y } ) , {\displaystyle f^{-1}(y)=f^{-1}(\{y\}),} The preimage of a single element y ∈ Y – a singleton set { y } – is sometimes called the fiber of y . When Y is the set of real numbers, it is common to refer to f −1 ({ y }) as a level set . | https://en.wikipedia.org/wiki/Inv_(function_prefix) |
In ecology , invader potential is the qualitative and quantitative measures of a given invasive species probability to invade a given ecosystem . This is often seen through climate matching. There are many reasons why a species may invade a new area. The term invader potential may also be interchangeable with invasiveness. Invader potential is a large threat to global biodiversity . It has been shown that there is an ecosystem function loss due to the introduction of species in areas they are not native to.
Invaders are species that, through biomass, abundance, and strong interactions with native species, have significantly altered the structure and composition of the established community. This differs greatly from the term "introduced", which merely refers to species that have been introduced to an environment, disregarding whether or not they have created a successful establishment. [ 1 ] They are simply organisms that have been accidentally, or deliberately, placed into an unfamiliar area. [ 2 ] Many times, in fact, species do not have a strong impact on the introduced habitat. This can be for a variety of reasons; either the newcomers are not abundant or because they are small and unobtrusive. [ 1 ]
Understanding the mechanisms of invader potential is important to understanding why species relocate and to predict future invasions. There are three predicted reasons as to why species invade an area. They are as follows: adaptation to physical environment, resource competition and/or utilization, and enemy release. Some of these reasons as to why species move seem relatively simple to understand. For example, species may adapt to the new physical environment through having great phenotypic plasticity and environmental tolerance. Species with high rates of these find it easier to adapt to new environments. In terms of resources, those with low resource requirements thrive in unknown areas more than those with complex resource needs. This is shown directly through Tilman's R* rule. Those with less needs can competitively exclude those with more complex needs and take over an area. And finally, species with high reproduction rate and low defense to natural enemies have a better chance of invading other areas. All of these are reasons why species may thrive in places they are non-native to, due to having desirable flexibility within their species' needs. [ 3 ]
Climate matching is a technique used to identify extralimital destinations that invasive species may like to overtake, based on its similarities to the species previous native range. Species are more likely to invade areas that match their origin for ease of use, and abundance of resources. Climate matching assesses the invasion risk and heavily prioritizes destination-specific action. [ 4 ]
The Bioga irregularis , the brown tree snake, is a great example of a species that climate matches. This species is native to northern and eastern Australia , eastern Indonesia , Papua New Guinea and most of the Solomon Islands . The brown tree snake was accidentally translocated by means of ship cargo to Guam, where it is responsible for replacing the majority of the native bird species. [ 4 ]
Humans play a significant role in the ways species invade an area. By changing the habitat, an invasion is made easier or more advantageous for an invasive species. As previously mentioned, species are more likely to invade areas they feel they can competitively win in. [ 5 ]
As an example, human led shoreline development, specifically in New England, was found to explain over 90% of intermarsh variation. This has boosted nitrogen availability, which can draw in new species. This human made change, among others, was the reason that Phragmites australis invaded the New England salt marshes. [ 5 ]
In a study by Sillman and Bertness, 22 salt marshes were surveyed for changes following this invasion. This study specifically looked at how human habitat alteration led to the invasion success of this species. Shoreline development, nutrient enrichment, and salinity reduction were all human made changes that contributed to the species ability to invade. [ 5 ]
It is critical, especially in conservation biology, to have the ability to foresee impacts on ecosystems. For example, the predictions of the identities and ecological impacts of invasive alien species assists in risk assessment. Currently, scientists are lacking the universal and standardized metrics that are reliable enough to predict the likelihood and degree of impact of the specific invaders. Data on the measurable changes in populations of the affected species, for instance, would be especially beneficial. [ 6 ]
Invader potential is a tool to aid in this dilemma. By understanding the qualitative and quantitative measures of a given invasive species probability to invade a given ecosystem, researchers can hypothesize which species will impact which environments. The addition, or removal, of a species from an ecosystem can cause drastic changes to environmental factors as well as the community's food web. Predicting these inevitable situations can aid in both maintenance and conservation. This is especially advised for emerging and potential future invaders that have no invasion history. [ 6 ]
Although the focus is typically on the invading species' adverse impacts on native species, they are also often negatively impacted, as well. The new colonization of a foreign species has proven to lead to introduced species being subject to genetic bottlenecks, random genetic drift, and increased levels of inbreeding. [ 7 ] Genetic changes, such as these, can pose a potential threat to allelic diversity. This could lead to genetic differentiation of the introduced population. In addition, invasive organisms face new biotic and abiotic factors.
Invasion potential has a great impact on whether or not the invasive organism will survive these biotic or abiotic factors. The species' ability to adapt to the new conditions will contribute to the success of the particular invasion. In the majority of cases, a small subset of introduced species become invaders as a result of rapid changes in the new habitat. In other cases, the species fails to thrive symbiotically with the ecosystem. [ 7 ] | https://en.wikipedia.org/wiki/Invader_potential |
Invagination is the process of a surface folding in on itself to form a cavity, pouch or tube. In developmental biology , invagination of epithelial sheets occurs in many contexts during embryonic development . Invagination is critical for making the primitive gut during gastrulation in many organisms, forming the neural tube in vertebrates , and in the morphogenesis of countless organs and sensory structures. Models of invagination that have been most thoroughly studied include the ventral furrow in Drosophila melanogaster , neural tube formation, and gastrulation in many marine organisms. The cellular mechanisms of invagination vary from one context to another but at their core they involve changing the mechanics of one side of a sheet of cells such that this pressure induces a bend in the tissue.
The term, originally used in embryology , has been adopted in other disciplines as well.
The process of tissue invagination has fascinated scientists for over a century and a half. Since the beginning, scientists have tried to understand the process of invagination as a mechanical process resulting from forces acting in the embryo . [ 1 ] For example, the Swiss biologist Wilhelm His , observing the invagination of the chick neural tube, experimented with modeling this process using sheets of different materials and suggested that pushing forces from the lateral edges of the neural plate might drive its invagination. [ 2 ] Scientists throughout the next century have speculated on the mechanisms of invagination, often making models of this process using either physical analogs, [ 3 ] or, especially in recent years, mathematical and computational modeling .
Invagination can be driven by a number of mechanisms at the cellular level. Regardless of the force-generating mechanism that causes the bending of the epithelium , most instances of invagination result in a stereotypical cell shape change. At the side of the epithelium exposed to the environment (the apical side), the surface of cells shrinks, and at the side of the cell in contact with the basement membrane (the basal side), the cell surfaces expand. Thus, cells become wedge-shaped. As these cells change shape, the tissue bends in the direction of the apical surface. In many–– though not all––cases, this process involves active constriction of the apical surface by the actin - myosin cytoskeleton . Furthermore, while most invagination processes involve shrinking of the apical surface, there have been cases observed where the opposite happens - the basal surface constricts and the apical surface expands, such as in optic cup morphogenesis and formation of the midbrain-hindbrain boundary in zebrafish . [ 4 ] [ 5 ] [ 6 ]
Apical constriction is an active process that results in the shrinkage of the apical side of the cell. This causes the cell shape to change from a column or cube-shaped cell to become wedge-shaped. Apical constriction is powered by the activity of the proteins actin and myosin interacting in a complex network known as the actin-myosin cytoskeleton. Myosin, a motor protein, generates force by pulling filaments of actin together. Myosin activity is regulated by the phosphorylation of one of its subunits , myosin regulatory light chain . Thus, kinases such as Rho-associated coiled-coil kinase (ROCK), which phosphorylate myosin, as well as phosphatases , which dephosphorylate myosin, are regulators of actomyosin contraction in cells. [ 7 ]
The arrangement of actin and myosin in the cell cortex and the way they generate force can vary across contexts. Classical models of apical constriction in embryos and epithelia in cell culture showed that actin-myosin bundles are assembled around the circumference of the cell in association with adherens junctions between cells. Contraction of the actin-myosin bundles thus results in a constriction of the apical surface in a process that has been likened to the tightening of a purse string. [ 7 ] More recently, in the context of a cultured epithelium derived from the mouse organ of Corti , it has also been shown that the arrangement of the actin and myosin around the cell circumerence is similar to a muscle sarcomere , where there are a repeating units of myosin connected to antiparallel actin bundles. [ 8 ] In other cells, a network of myosin and actin in the middle of the apical surface can also generate apical constriction. For example, in cells of the Drosophila ventral furrow, the organization of actin and myosin is analogous to a muscle sarcomere arranged radially. [ 9 ] [ 10 ] In some contexts, a less clearly organized “cortical flow” of actin and myosin can also generate contraction of the apical surface. [ 8 ]
To maintain a constant cell volume during apical constriction, cells must either change their height or expand the basal surface of their cells. While the process of basal relaxation has been less thoroughly studied, in some cases it has been directly observed that the process of apical constriction occurs alongside an active disassembly of the actin-myosin network at the basal surface of the cell, allowing the basal side of the cell to expand. For example, this has been observed in the Drosophila ventral furrow invagination [ 11 ] [ 12 ] and the formation of the otic placode in the chicken. [ 13 ] [ 14 ]
Invagination also often involves, and can be driven by, changes in cell height. When apical constriction occurs, this can lead to elongation of cells to maintain constant cell volume, and consequently a thickening of the epithelium. However, shortening of cells along the apical-basal axis can also help deepen the pit formed during invagination. [ 15 ] Active changes in cell shape to cause cell shortening have been shown to contribute to invagination in a few cases. For example, in the Drosophila leg epithelium, apoptotic cells shrink and pull on the apical surface of the epithelium via an apical-basal cable made up of actin and myosin. [ 16 ] In the invagination that occurs in ascidian gastrulation, cells first undergo apical constriction and then change their shape to become rounder ––and thus shorter along the apical-basal axis––which is responsible for the completion of the invagination movement. [ 17 ] During cell division , cells also naturally take on a rounded morphology. The rapid drop in cell height caused by rounding of cells during mitosis has also been implicated in invagination of the Drosophila tracheal placode. [ 18 ]
Supracellular actomyosin cables are structures of actin and myosin that align between cells next to each other and are connected by cell junctions. [ 12 ] These cables play many roles in morphogenesis during embryonic development, including invagination. [ 19 ] Rather than solely relying on apical constriction of individual cells, invagination can be driven by compressive forces from this cable contracting around the site of invagination, such as in the case of salivary gland invagination in Drosophila . [ 20 ] [ 21 ] In neural tube formation in the chick embryo, rows of supracellular cables stretching across the site of invagination help pull the tissue together to facilitate bending into a tube. [ 19 ] [ 22 ] [ 23 ]
One of the most well studied models of invagination is the ventral furrow in Drosophila melanogaster . The formation of this structure is one of the first major cell movements in Drosophila gastrulation. In this process, the prospective mesoderm ––the region of cells along the ventral midline of the embryo––folds inwards to form the ventral furrow. This furrow eventually pinches off and becomes a tube inside the embryo and ultimately flattens to form a layer of tissue underneath the ventral surface. [ 24 ]
Ventral furrow formation is driven by apical constriction of the future mesoderm cells, which first flatten along the apical surface and then contract their apical membranes. The classical models for how apical constriction worked in this context were based on the “purse-string” mechanism where an actin-myosin band around the circumference of the apical cell surface contracts. [ 25 ] However, more recent investigations have revealed that, while there is a circumferential band of actin associated with cell junctions on the side of cells, it is actually an actin-myosin network arranged radially across the apical surface that powers apical constriction. [ 26 ] This structure acts like a radial version of a muscle sarcomere. [ 10 ] Force generated by myosin results in contraction towards the center of the cell. The cells do not contract continuously but rather have pulsed contractions. In between contractions, the actin network around the circumference of the cell helps stabilize the reduced size of the cell, allowing for a progressive decrease in size of the apical surface. [ 26 ] In addition to apical constriction, adhesion between cells through adherens junctions is critical for transforming these individual cell-level contractions into a deformation of a whole tissue.
Genetically, formation of the ventral furrow relies on the activity of the transcription factors twist and snail , which are expressed in the prospective ventral mesoderm before furrow formation. [ 25 ] Downstream of twist is the Fog signaling pathway, which controls the changes that occur in the apical domain of cells. [ 27 ]
Scientists have studied the process of neural tube formation in vertebrate embryos since the late 1800s. [ 2 ] Across vertebrate groups including amphibians , reptiles , birds , and mammals , the neural tube (the embryonic precursor of the spinal cord ) forms through the invagination of the neural plate into a tube, known as primary neurulation. In fish (and in some contexts in other vertebrates), the neural tube can also be formed by a non-invagination-mediated process known as secondary neurulation. [ 24 ] While some differences exist in the mechanism of primary neurulation between vertebrate species, the general process is similar. Neurulation involves the formation of a medial hinge point at the middle of the neural plate, which is where tissue bending is initiated. The cells at the medial hinge point become wedge shaped. In some contexts, such as in Xenopus frog embryos, this cell shape change appears to be due to apical constriction. [ 28 ] [ 29 ] However, in chickens and mice, bending at this hinge point is mediated by a process called basal wedging, rather than apical constriction. [ 12 ] [ 30 ] [ 31 ] In this case, the cells are so thin that the movement of the nucleus to the basal side of the cell causes a bulge in the basal part of the cell. This process may be regulated by how the cell divisions take place. Contractions of actin-myosin cables are also important for the invagination of the neural plate. Supracellular actin cables stretching across the neural plate help pull the tissue together (see § Supracellular cables ). Furthermore, forces pushing into the neural plate from the adjacent tissue also may play a role in the folding of the neural plate. [ 32 ] [ 33 ] [ 34 ]
Sea urchin gastrulation is another classic model for invagination in embryology. One of the early gastrulation movements in sea urchins is the invagination of a region of cells at the vegetal side of the embryo (vegetal plate) to become the archenteron , or future gut tube. There are multiple stages of archenteron invagination: a first stage where the initial folding in of tissue occurs, a second stage where the archenteron elongates, and in some species a third stage where the archenteron contacts the other side of the cell cavity and finishes its elongation. [ 24 ]
Apical constriction occurs in archenteron invagination, with a ring of cells called “bottle cells” in the center of the vegetal plate becoming wedge-shaped. [ 35 ] However, invagination does not seem to be solely driven by the apical constriction of bottle cells, as inhibiting actin polymerization [ 36 ] or removing bottle cells does not fully block invagination. [ 35 ] Several other mechanisms have been proposed to be involved in the process, including a role for extraembryonic extracellular matrix . [ 37 ] In this model, there are two layers of extracellular matrix at the apical surface of cells made of different proteins. When cells from the vegetal plate secrete a molecule ( chondroitin sulfate proteoglycan ) that is highly water absorbent into the inner layer, this causes the layer to swell, making the tissue buckle inwards. [ 36 ] Several genetic pathways have been implicated in this process. Wnt signaling through the non-canonical planar cell polarity pathway has been shown to be important, with one of its downstream targets being the small GTPase RhoA . FGF signaling also plays a role in invagination. [ 38 ]
The invagination in amphioxus is the first cell movement of gastrulation. This process was first described by Conklin . During gastrulation, the blastula will be transformed by the invagination. The endoderm folds towards the inner part and thus the blastocoel transforms into a cup-shaped structure with a double wall. The inner wall is now called the archenteron ; the primitive gut. The archenteron will open to the exterior through the blastopore . The outer wall will become the ectoderm , later forming the epidermis and nervous system . [ 39 ]
In tunicates , invagination is the first mechanism that takes place during gastrulation. The four largest endoderm cells induce the invagination process in the tunicates. Invagination consists of the internal movements of a sheet of cells (the endoderm) based on changes in their shape. The blastula of the tunicates is a little flattened in the vegetal pole making a change of shape from a columnar to a wedge shape. Once the endoderm cells were invaginated, the cells will keep moving beneath the ectoderm. Later, the blastopore will be formed and with this, the invagination process is complete. The blastopore will be surrounded by the mesoderm by all sides. [ 40 ]
In geology, invagination is used to describe a deep depression of strata. Used by Donald L. Baars in "The Colorado Plateau". | https://en.wikipedia.org/wiki/Invagination |
The invariable plane of a planetary system , also called Laplace's invariable plane , is the plane passing through its barycenter (center of mass) perpendicular to its angular momentum vector .
In the Solar System , about 98% of this effect is contributed by the orbital angular momenta of the four giant planets ( Jupiter , Saturn , Uranus , and Neptune ). The invariable plane is within 0.5° of the orbital plane of Jupiter, [ 1 ] and may be regarded as the weighted average of all planetary orbital and rotational planes.
This plane is sometimes called the "Laplacian" or "Laplace plane" or the "invariable plane of Laplace", though it should not be confused with the Laplace plane , which is the plane about which the individual orbital planes of planetary satellites precess . [ 4 ] Both derive from the work of (and are at least sometimes named for) the French astronomer Pierre-Simon Laplace . [ 5 ] The two are equivalent only in the case where all perturbers and resonances are far from the precessing body. The invariable plane is derived from the sum of angular momenta, and is "invariable" over the entire system, while the Laplace plane for different orbiting objects within a system may be different. Laplace called the invariable plane the plane of maximum areas , where the "area" in this case is the product of the radius R and its time rate of change d R / d t , that is, its radial velocity, multiplied by the mass.
The magnitude of the orbital angular momentum vector of a planet is L = R 2 M θ ˙ {\displaystyle L=R^{2}M{\dot {\theta }}} , where R {\displaystyle R} is the orbital radius of the planet (from the barycenter ), M {\displaystyle M} is the mass of the planet, and θ ˙ {\displaystyle {\dot {\theta }}} is its orbital angular velocity. That of Jupiter contributes the bulk of the Solar System's angular momentum, 60.3%. Then comes Saturn at 24.5%, Neptune at 7.9%, and Uranus at 5.3%. The Sun forms a counterbalance to all of the planets, so it is near the barycenter when Jupiter is on one side and the other three jovian planets are diametrically opposite on the other side, but the Sun moves to 2.17 R ☉ away from the barycenter when all jovian planets are in line on the other side. The orbital angular momenta of the Sun and all non-jovian planets, moons, and small Solar System bodies , as well as the axial rotation momenta of all bodies, including the Sun, total only about 2%.
If all Solar System bodies were point masses, or were rigid bodies having spherically symmetric mass distributions, and further if there were no external effects due to the uneven gravitation of the Milky Way Galaxy , then an invariable plane defined on orbits alone would be truly invariable and would constitute an inertial frame of reference. But almost all are not, allowing the transfer of a very small amount of momenta from axial rotations to orbital revolutions due to tidal friction and to bodies being non-spherical. This causes a change in the magnitude of the orbital angular momentum, as well as a change in its direction (precession) because the rotational axes are not parallel to the orbital axes.
Nevertheless, these changes are exceedingly small compared to the total angular momentum of the system, which is very nearly conserved despite these effects. For almost all purposes, the plane defined from the giant planets' orbits alone can be considered invariable when working in Newtonian dynamics , by also ignoring the even tinier amounts of angular momentum ejected in material and gravitational waves leaving the Solar System, and the extremely small torques exerted on the Solar System by other stars passing nearby, Milky Way galactic tides, etc. | https://en.wikipedia.org/wiki/Invariable_plane |
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space R n {\displaystyle \mathbb {R} ^{n}} .
It states:
The theorem and its proof are due to L. E. J. Brouwer , published in 1912. [ 1 ] The proof uses tools of algebraic topology , notably the Brouwer fixed point theorem .
The conclusion of the theorem can equivalently be formulated as: " f {\displaystyle f} is an open map ".
Normally, to check that f {\displaystyle f} is a homeomorphism, one would have to verify that both f {\displaystyle f} and its inverse function f − 1 {\displaystyle f^{-1}} are continuous;
the theorem says that if the domain is an open subset of R n {\displaystyle \mathbb {R} ^{n}} and the image is also in R n , {\displaystyle \mathbb {R} ^{n},} then continuity of f − 1 {\displaystyle f^{-1}} is automatic.
Furthermore, the theorem says that if two subsets U {\displaystyle U} and V {\displaystyle V} of R n {\displaystyle \mathbb {R} ^{n}} are homeomorphic, and U {\displaystyle U} is open, then V {\displaystyle V} must be open as well.
(Note that V {\displaystyle V} is open as a subset of R n , {\displaystyle \mathbb {R} ^{n},} and not just in the subspace topology.
Openness of V {\displaystyle V} in the subspace topology is automatic.)
Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space.
It is of crucial importance that both domain and image of f {\displaystyle f} are contained in Euclidean space of the same dimension .
Consider for instance the map f : ( 0 , 1 ) → R 2 {\displaystyle f:(0,1)\to \mathbb {R} ^{2}} defined by f ( t ) = ( t , 0 ) . {\displaystyle f(t)=(t,0).} This map is injective and continuous, the domain is an open subset of R {\displaystyle \mathbb {R} } , but the image is not open in R 2 . {\displaystyle \mathbb {R} ^{2}.} A more extreme example is the map g : ( − 1.1 , 1 ) → R 2 {\displaystyle g:(-1.1,1)\to \mathbb {R} ^{2}} defined by g ( t ) = ( t 2 − 1 , t 3 − t ) {\displaystyle g(t)=\left(t^{2}-1,t^{3}-t\right)} because here g {\displaystyle g} is injective and continuous but does not even yield a homeomorphism onto its image.
The theorem is also not generally true in infinitely many dimensions. Consider for instance the Banach L p space ℓ ∞ {\displaystyle \ell ^{\infty }} of all bounded real sequences .
Define f : ℓ ∞ → ℓ ∞ {\displaystyle f:\ell ^{\infty }\to \ell ^{\infty }} as the shift f ( x 1 , x 2 , … ) = ( 0 , x 1 , x 2 , … ) . {\displaystyle f\left(x_{1},x_{2},\ldots \right)=\left(0,x_{1},x_{2},\ldots \right).} Then f {\displaystyle f} is injective and continuous, the domain is open in ℓ ∞ {\displaystyle \ell ^{\infty }} , but the image is not.
If n > m {\displaystyle n>m} , there exists no continuous injective map f : U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} for a nonempty open set U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} . To see this, suppose there exists such a map f . {\displaystyle f.} Composing f {\displaystyle f} with the standard inclusion of R m {\displaystyle \mathbb {R} ^{m}} into R n {\displaystyle \mathbb {R} ^{n}} would give a continuous injection from R n {\displaystyle \mathbb {R} ^{n}} to itself, but with an image with empty interior in R n {\displaystyle \mathbb {R} ^{n}} . This would contradict invariance of domain.
In particular, if n ≠ m {\displaystyle n\neq m} , no nonempty open subset of R n {\displaystyle \mathbb {R} ^{n}} can be homeomorphic to an open subset of R m {\displaystyle \mathbb {R} ^{m}} .
And R n {\displaystyle \mathbb {R} ^{n}} is not homeomorphic to R m {\displaystyle \mathbb {R} ^{m}} if n ≠ m . {\displaystyle n\neq m.}
The domain invariance theorem may be generalized to manifolds : if M {\displaystyle M} and N {\displaystyle N} are topological n -manifolds without boundary and f : M → N {\displaystyle f:M\to N} is a continuous map which is locally one-to-one (meaning that every point in M {\displaystyle M} has a neighborhood such that f {\displaystyle f} restricted to this neighborhood is injective), then f {\displaystyle f} is an open map (meaning that f ( U ) {\displaystyle f(U)} is open in N {\displaystyle N} whenever U {\displaystyle U} is an open subset of M {\displaystyle M} ) and a local homeomorphism .
There are also generalizations to certain types of continuous maps from a Banach space to itself. [ 2 ] | https://en.wikipedia.org/wiki/Invariance_of_domain |
In mathematics , an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. [ 1 ] [ 2 ] The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane . The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class . [ 3 ]
Invariants are used in diverse areas of mathematics such as geometry , topology , algebra and discrete mathematics . Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles . The discovery of invariants is an important step in the process of classifying mathematical objects. [ 2 ] [ 3 ]
A simple example of invariance is expressed in our ability to count . For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in which we count the objects in the set . The quantity—a cardinal number —is associated with the set, and is invariant under the process of counting.
An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change.
The distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand, multiplication does not have this same property, as distance is not invariant under multiplication.
Angles and ratios of distances are invariant under scalings , rotations , translations and reflections . These transformations produce similar shapes, which is the basis of trigonometry . In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. As another example, all circles are similar: they can be transformed into each other and the ratio of the circumference to the diameter is invariant (denoted by the Greek letter π ( pi )).
Some more complicated examples:
The MU puzzle [ 7 ] is a good example of a logical problem where determining an invariant is of use for an impossibility proof . The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:
An example derivation (with superscripts indicating the applied rules) is
In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a property that is invariant to all rules (that is, not changed by any of them), and that demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider:
This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules:
The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either.
Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).
A subset S of the domain U of a mapping T : U → U is an invariant set under the mapping when x ∈ S ⟺ T ( x ) ∈ S . {\displaystyle x\in S\iff T(x)\in S.} The elements of S are not necessarily fixed , even though the set S is fixed in the power set of U . (Some authors use the terminology setwise invariant, [ 8 ] vs. pointwise invariant, [ 9 ] to distinguish between these cases.)
For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under a homothety of space.
An invariant set of an operation T is also said to be stable under T . For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group . [ 10 ] [ 11 ] [ 12 ] In linear algebra , if a linear transformation T has an eigenvector v , then the line through 0 and v is an invariant set under T , in which case the eigenvectors span an invariant subspace which is stable under T .
When T is a screw displacement , the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points.
In probability theory and ergodic theory , invariant sets are usually defined via the stronger property x ∈ S ⇔ T ( x ) ∈ S . {\displaystyle x\in S\Leftrightarrow T(x)\in S.} [ 13 ] [ 14 ] [ 15 ] When the map T {\displaystyle T} is measurable, invariant sets form a sigma-algebra , the invariant sigma-algebra .
The notion of invariance is formalized in three different ways in mathematics: via group actions , presentations, and deformation.
Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group.
Frequently one will have a group acting on a set X , which leaves one to determine which objects in an associated set F ( X ) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L ( P ); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.
More importantly, one may define a function on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions.
Dual to the notion of invariants are coinvariants , also known as orbits, which formalizes the notion of congruence : objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the perimeter of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.
These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In classification problems , one might seek to find a complete set of invariants , such that if two objects have the same values for this set of invariants, then they are congruent.
For example, triangles such that all three sides are equal are congruent under rigid motions, via SSS congruence , and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the AAA similarity criterion shows that this is a complete set of invariants.
Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying topological space (the manifold ) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.
The most common examples are:
Thirdly, if one is studying an object which varies in a family, as is common in algebraic geometry and differential geometry , one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric).
In computer science , an invariant is a logical assertion that is always held to be true during a certain phase of execution of a computer program . For example, a loop invariant is a condition that is true at the beginning and the end of every iteration of a loop.
Invariants are especially useful when reasoning about the correctness of a computer program . The theory of optimizing compilers , the methodology of design by contract , and formal methods for determining program correctness , all rely heavily on invariants.
Programmers often use assertions in their code to make invariants explicit. Some object oriented programming languages have a special syntax for specifying class invariants .
Abstract interpretation tools can compute simple invariants of given imperative computer programs. The kind of properties that can be found depend on the abstract domains used. Typical example properties are single integer variable ranges like 0<=x<1024 , relations between several variables like 0<=i-j<2*n-1 , and modulus information like y%4==0 . Academic research prototypes also consider simple properties of pointer structures. [ 16 ]
More sophisticated invariants generally have to be provided manually.
In particular, when verifying an imperative program using the Hoare calculus , [ 17 ] a loop invariant has to be provided manually for each loop in the program, which is one of the reasons that this approach is generally impractical for most programs.
In the context of the above MU puzzle example, there is currently no general automated tool that can detect that a derivation from MI to MU is impossible using only the rules 1–4. However, once the abstraction from the string to the number of its "I"s has been made by hand, leading, for example, to the following C program, an abstract interpretation tool will be able to detect that ICount%3 cannot be 0, and hence the "while"-loop will never terminate. | https://en.wikipedia.org/wiki/Invariant_(mathematics) |
In theoretical physics , an invariant is an observable of a physical system which remains unchanged under some transformation . Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition . Invariants of a system are deeply tied to the symmetries imposed by its environment.
Invariance is an important concept in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants.
In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the conservation of momentum , whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy . In general, by Noether's theorem , any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law .
In crystals , the electron density is periodic and invariant with respect to discrete translations by unit cell vectors. In very few materials, this symmetry can be broken due to enhanced electron correlations .
Another examples of physical invariants are the speed of light , and charge and mass of a particle observed from two reference frames moving with respect to one another (invariance under a spacetime Lorentz transformation [ 1 ] ), and invariance of time and acceleration under a Galilean transformation between two such frames moving at low velocities.
Quantities can be invariant under some common transformations but not under others. For example, the velocity of a particle is invariant when switching coordinate representations from rectangular to curvilinear coordinates, but is not invariant when transforming between frames of reference that are moving with respect to each other. Other quantities, like the speed of light, are always invariant.
Physical laws are said to be invariant under transformations when their predictions remain unchanged. This generally means that the form of the law (e.g. the type of differential equations used to describe the law) is unchanged in transformations so that no additional or different solutions are obtained.
For example the rule describing Newton's force of gravity between two chunks of matter is the same whether they are in this galaxy or another ( translational invariance in space). It is also the same today as it was a million years ago (translational invariance in time). The law does not work differently depending on whether one chunk is east or north of the other one ( rotational invariance ). Nor does the law have to be changed depending on whether you measure the force between the two chunks in a railroad station, or do the same experiment with the two chunks on a uniformly moving train ( principle of relativity ).
Covariance and contravariance generalize the mathematical properties of invariance in tensor mathematics , and are frequently used in electromagnetism , special relativity , and general relativity .
In the field of physics , the adjective covariant (as in covariance and contravariance of vectors ) is often used informally as a synonym for "invariant". For example, the Schrödinger equation does not keep its written form under the coordinate transformations of special relativity . Thus, a physicist might say that the Schrödinger equation is not covariant . In contrast, the Klein–Gordon equation and the Dirac equation do keep their written form under these coordinate transformations. Thus, a physicist might say that these equations are covariant .
Despite this usage of "covariant", it is more accurate to say that the Klein–Gordon and Dirac equations are invariant, and that the Schrödinger equation is not invariant. Additionally, to remove ambiguity, the transformation by which the invariance is evaluated should be indicated. | https://en.wikipedia.org/wiki/Invariant_(physics) |
The factorization of a linear partial differential operator (LPDO) is an important issue in the theory of integrability, due to the Laplace-Darboux transformations, [ 1 ] which allow construction of integrable LPDEs. Laplace solved the factorization problem for a bivariate hyperbolic operator of the second order (see Hyperbolic partial differential equation ), constructing two Laplace invariants. Each Laplace invariant is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO. The polynomial conditions of factorization are called invariants because they have the same form for equivalent (i.e. self-adjoint) operators.
Beals-Kartashova-factorization (also called BK-factorization) is a constructive procedure to factorize a bivariate operator of the arbitrary order and arbitrary form . Correspondingly, the factorization conditions in this case also have polynomial form, are invariants and coincide with Laplace invariants for bivariate hyperbolic operators of the second order. The factorization procedure is purely algebraic, the number of possible factorizations depending on the number of simple roots of the Characteristic polynomial (also called symbol) of the initial LPDO and reduced LPDOs appearing at each factorization step. Below the factorization procedure is described for a bivariate operator of arbitrary form, of order 2 and 3. Explicit factorization formulas for an operator of the order n {\displaystyle n} can be found in [ 2 ] General invariants are defined in [ 3 ] and invariant formulation of the Beals-Kartashova factorization is given in [ 4 ]
Consider an operator
with smooth coefficients and look for a factorization
Let us write down the equations on p i {\displaystyle p_{i}} explicitly, keeping in
mind the rule of left composition, i.e. that
Then in all cases
where the notation L = p 1 ∂ x + p 2 ∂ y {\displaystyle {\mathcal {L}}=p_{1}\partial _{x}+p_{2}\partial _{y}} is used.
Without loss of generality, a 20 ≠ 0 , {\displaystyle a_{20}\neq 0,} i.e. p 1 ≠ 0 , {\displaystyle p_{1}\neq 0,} and it can be taken as 1, p 1 = 1. {\displaystyle p_{1}=1.} Now solution of the system of 6 equations on the variables
can be found in three steps .
At the first step , the roots of a quadratic polynomial have to be found.
At the second step , a linear system of two algebraic equations has to be solved.
At the third step , one algebraic condition has to be checked.
Step 1. Variables
can be found from the first three equations,
The (possible) solutions are then the functions of the roots of a quadratic polynomial:
Let ω {\displaystyle \omega } be a root of the polynomial P 2 , {\displaystyle {\mathcal {P}}_{2},} then
Step 2. Substitution of the results obtained at the first step, into the next two equations
yields linear system of two algebraic equations:
In particularly , if the root ω {\displaystyle \omega } is simple,
i.e.
equations have the unique solution:
At this step, for each
root of the polynomial P 2 {\displaystyle {\mathcal {P}}_{2}} a corresponding set of coefficients p j {\displaystyle p_{j}} is computed.
Step 3. Check factorization condition (which is the last of the initial 6 equations)
written in the known variables p j {\displaystyle p_{j}} and ω {\displaystyle \omega } ):
If
the operator A 2 {\displaystyle {\mathcal {A}}_{2}} is factorizable and explicit form for the factorization coefficients p j {\displaystyle p_{j}} is given above.
Consider an operator
with smooth coefficients and look for a factorization
Similar to the case of the operator A 2 , {\displaystyle {\mathcal {A}}_{2},} the conditions of factorization are described by the following system:
with L = p 1 ∂ x + p 2 ∂ y , {\displaystyle {\mathcal {L}}=p_{1}\partial _{x}+p_{2}\partial _{y},} and again a 30 ≠ 0 , {\displaystyle a_{30}\neq 0,} i.e. p 1 = 1 , {\displaystyle p_{1}=1,} and three-step procedure yields:
At the first step , the roots of a cubic polynomial
have to be found. Again ω {\displaystyle \omega } denotes a root and first four coefficients are
At the second step , a linear system of three algebraic equations has to be solved:
At the third step , two algebraic conditions have to be checked.
Definition The operators A {\displaystyle {\mathcal {A}}} , A ~ {\displaystyle {\tilde {\mathcal {A}}}} are called
equivalent if there is a gauge transformation that takes one to the
other:
BK-factorization is then pure algebraic procedure which allows to
construct explicitly a factorization of an arbitrary order LPDO A ~ {\displaystyle {\tilde {\mathcal {A}}}} in the form
with first-order operator L = ∂ x − ω ∂ y + p {\displaystyle {\mathcal {L}}=\partial _{x}-\omega \partial _{y}+p} where ω {\displaystyle \omega } is an arbitrary simple root of the characteristic polynomial
Factorization is possible then for each simple root ω ~ {\displaystyle {\tilde {\omega }}} iff
for n = 2 → l 2 = 0 , {\displaystyle n=2\ \ \rightarrow l_{2}=0,}
for n = 3 → l 3 = 0 , l 31 = 0 , {\displaystyle n=3\ \ \rightarrow l_{3}=0,l_{31}=0,}
for n = 4 → l 4 = 0 , l 41 = 0 , l 42 = 0 , {\displaystyle n=4\ \ \rightarrow l_{4}=0,l_{41}=0,l_{42}=0,}
and so on. All functions l 2 , l 3 , l 31 , l 4 , l 41 , l 42 , . . . {\displaystyle l_{2},l_{3},l_{31},l_{4},l_{41},\ \ l_{42},...} are known functions, for instance,
and so on.
Theorem All functions
are invariants under gauge transformations.
Definition Invariants l 2 = a 00 − L ( p 6 ) + p 3 p 6 , l 3 = a 00 − L ( p 9 ) + p 3 p 9 , l 31 , . . . . . {\displaystyle l_{2}=a_{00}-{\mathcal {L}}(p_{6})+p_{3}p_{6},l_{3}=a_{00}-{\mathcal {L}}(p_{9})+p_{3}p_{9},l_{31},.....} are
called generalized invariants of a bivariate operator of arbitrary
order.
In particular case of the bivariate hyperbolic operator its generalized
invariants coincide with Laplace invariants (see Laplace invariant ).
Corollary If an operator A ~ {\displaystyle {\tilde {\mathcal {A}}}} is factorizable, then all
operators equivalent to it, are also factorizable.
Equivalent operators are easy to compute:
and so on. Some example are given below:
Factorization of an operator is the first step on the way of solving corresponding equation. But for solution we need right factors and BK-factorization constructs left factors which are easy to construct. On the other hand, the existence of a certain right factor of a LPDO is equivalent to the existence of a corresponding left factor of the transpose of that operator.
Definition The transpose A t {\displaystyle {\mathcal {A}}^{t}} of an operator A = ∑ a α ∂ α , ∂ α = ∂ 1 α 1 ⋯ ∂ n α n . {\displaystyle {\mathcal {A}}=\sum a_{\alpha }\partial ^{\alpha },\qquad \partial ^{\alpha }=\partial _{1}^{\alpha _{1}}\cdots \partial _{n}^{\alpha _{n}}.} is defined as A t u = ∑ ( − 1 ) | α | ∂ α ( a α u ) . {\displaystyle {\mathcal {A}}^{t}u=\sum (-1)^{|\alpha |}\partial ^{\alpha }(a_{\alpha }u).} and the identity ∂ γ ( u v ) = ∑ ( γ α ) ∂ α u , ∂ γ − α v {\displaystyle \partial ^{\gamma }(uv)=\sum {\binom {\gamma }{\alpha }}\partial ^{\alpha }u,\partial ^{\gamma -\alpha }v} implies that A t = ∑ ( − 1 ) | α + β | ( α + β α ) ( ∂ β a α + β ) ∂ α . {\displaystyle {\mathcal {A}}^{t}=\sum (-1)^{|\alpha +\beta |}{\binom {\alpha +\beta }{\alpha }}(\partial ^{\beta }a_{\alpha +\beta })\partial ^{\alpha }.}
Now the coefficients are
A t = ∑ a ~ α ∂ α , {\displaystyle {\mathcal {A}}^{t}=\sum {\tilde {a}}_{\alpha }\partial ^{\alpha },} a ~ α = ∑ ( − 1 ) | α + β | ( α + β α ) ∂ β ( a α + β ) . {\displaystyle {\tilde {a}}_{\alpha }=\sum (-1)^{|\alpha +\beta |}{\binom {\alpha +\beta }{\alpha }}\partial ^{\beta }(a_{\alpha +\beta }).}
with a standard convention for binomial coefficients in several
variables (see Binomial coefficient ), e.g. in two variables
In particular, for the operator A 2 {\displaystyle {\mathcal {A}}_{2}} the coefficients are a ~ j k = a j k , j + k = 2 ; a ~ 10 = − a 10 + 2 ∂ x a 20 + ∂ y a 11 , a ~ 01 = − a 01 + ∂ x a 11 + 2 ∂ y a 02 , {\displaystyle {\tilde {a}}_{jk}=a_{jk},\quad j+k=2;{\tilde {a}}_{10}=-a_{10}+2\partial _{x}a_{20}+\partial _{y}a_{11},{\tilde {a}}_{01}=-a_{01}+\partial _{x}a_{11}+2\partial _{y}a_{02},}
For instance, the operator
is factorizable as
and its transpose A 1 t {\displaystyle {\mathcal {A}}_{1}^{t}} is factorizable then as [ . . . ] [ ∂ x − ∂ y + 1 2 ( y + x ) ] . {\displaystyle {\big [}...{\big ]}\,{\big [}\partial _{x}-\partial _{y}+{\tfrac {1}{2}}(y+x){\big ]}.} | https://en.wikipedia.org/wiki/Invariant_factorization_of_LPDOs |
In dynamical systems , a branch of mathematics , an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. [ 1 ] Examples include the slow manifold , center manifold , stable manifold , unstable manifold , subcenter manifold and inertial manifold .
Typically, although by no means always, invariant manifolds are constructed as a 'perturbation' of an invariant subspace about an equilibrium.
In dissipative systems, an invariant manifold based upon the gravest, longest lasting modes forms an effective low-dimensional, reduced, model of the dynamics. [ 2 ]
Consider the differential equation d x / d t = f ( x ) , x ∈ R n , {\displaystyle dx/dt=f(x),\ x\in \mathbb {R} ^{n},} with flow x ( t ) = ϕ t ( x 0 ) {\displaystyle x(t)=\phi _{t}(x_{0})} being the solution of the differential equation with x ( 0 ) = x 0 {\displaystyle x(0)=x_{0}} .
A set S ⊂ R n {\displaystyle S\subset \mathbb {R} ^{n}} is called an invariant set for the differential equation if, for each x 0 ∈ S {\displaystyle x_{0}\in S} , the solution t ↦ ϕ t ( x 0 ) {\displaystyle t\mapsto \phi _{t}(x_{0})} , defined on its maximal interval of existence, has its image in S {\displaystyle S} . Alternatively, the orbit
passing through each x 0 ∈ S {\displaystyle x_{0}\in S} lies in S {\displaystyle S} . In addition, S {\displaystyle S} is called an invariant manifold if S {\displaystyle S} is a manifold . [ 3 ]
For any fixed parameter a {\displaystyle a} , consider the variables x ( t ) , y ( t ) {\displaystyle x(t),y(t)} governed by the pair of coupled differential equations
The origin is an equilibrium. This system has two invariant manifolds of interest through the origin.
A differential equation
represents a non-autonomous dynamical system , whose solutions are of the form x ( t ; t 0 , x 0 ) = ϕ t 0 t ( x 0 ) {\displaystyle x(t;t_{0},x_{0})=\phi _{t_{0}}^{t}(x_{0})} with x ( t 0 ; t 0 , x 0 ) = x 0 {\displaystyle x(t_{0};t_{0},x_{0})=x_{0}} . In the extended phase space R n × R {\displaystyle \mathbb {R} ^{n}\times \mathbb {R} } of such a system, any initial surface M 0 ⊂ R n {\displaystyle M_{0}\subset \mathbb {R} ^{n}} generates an invariant manifold
A fundamental question is then how one can locate, out of this large family of invariant manifolds, the ones that have the highest influence on the overall system dynamics. These most influential invariant manifolds in the extended phase space of a non-autonomous dynamical systems are known as Lagrangian Coherent Structures . [ 4 ] | https://en.wikipedia.org/wiki/Invariant_manifold |
The invariant mass , rest mass , intrinsic mass , proper mass , or in the case of bound systems simply mass , is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total energy and momentum that is the same in all frames of reference related by Lorentz transformations . [ 1 ] If a center-of-momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is non-zero, the total mass (a.k.a. relativistic mass ) of the system is greater than the invariant mass, but the invariant mass remains unchanged.
Because of mass–energy equivalence , the rest energy of the system is simply the invariant mass times the speed of light squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.
Systems whose four-momentum is a null vector , a light-like vector within the context of Minkowski space (for example, a single photon or many photons moving in exactly the same direction) have zero invariant mass and are referred to as massless . A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized tachyon ), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass.
If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by c 2 . See mass–energy equivalence for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy.
For an isolated massive system, the center of mass of the system moves in a straight line with a steady subluminal velocity (with a velocity depending on the reference frame used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by c 2 . This total energy in the center of momentum frame, is the minimum energy which the system may be observed to have, when seen by various observers from various inertial frames.
Note that for reasons above, such a rest frame does not exist for single photons , or rays of light moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon.
The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).
Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is not equal to the sum of the rest masses of the particles within it.
The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound).
They will often also interact through one or more of the fundamental forces , giving them a potential energy of interaction, possibly negative .
In particle physics , the invariant mass m 0 is equal to the mass in the rest frame of the particle, and can be calculated by the particle's energy E and its momentum p as measured in any frame, by the energy–momentum relation : m 0 2 c 2 = ( E c ) 2 − ‖ p ‖ 2 {\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}} or in natural units where c = 1 , m 0 2 = E 2 − ‖ p ‖ 2 . {\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.}
This invariant mass is the same in all frames of reference (see also special relativity ). This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector ( E , p ) , calculated using the relativistic version of the Pythagorean theorem which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic Dirac equation for an elementary particle. The Dirac quantum operator corresponds to the particle four-momentum vector.
Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed.
The mass of a system of particles can be calculated from the general formula: ( W c 2 ) 2 = ( ∑ E ) 2 − ‖ ∑ p c ‖ 2 , {\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},} where
The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows (in natural units): W 2 = ( ∑ E in − ∑ E out ) 2 − ‖ ∑ p in − ∑ p out ‖ 2 . {\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.}
If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.
In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the missing energy ) the transverse mass is used.
In a two-particle collision (or a two-particle decay) the square of the invariant mass (in natural units ) is M 2 = ( E 1 + E 2 ) 2 − ‖ p 1 + p 2 ‖ 2 = m 1 2 + m 2 2 + 2 ( E 1 E 2 − p 1 ⋅ p 2 ) . {\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|\mathbf {p} _{1}+\mathbf {p} _{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\mathbf {p} _{1}\cdot \mathbf {p} _{2}\right).\end{aligned}}}
The invariant mass of a system made of two massless particles whose momenta form an angle θ {\displaystyle \theta } has a convenient expression: M 2 = ( E 1 + E 2 ) 2 − ‖ p 1 + p 2 ‖ 2 = [ ( p 1 , 0 , 0 , p 1 ) + ( p 2 , 0 , p 2 sin θ , p 2 cos θ ) ] 2 = ( p 1 + p 2 ) 2 − p 2 2 sin 2 θ − ( p 1 + p 2 cos θ ) 2 = 2 p 1 p 2 ( 1 − cos θ ) . {\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin \theta ,p_{2}\cos \theta )]^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}}
In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle ϕ {\displaystyle \phi } and pseudorapidity η {\displaystyle \eta } . Additionally the transverse momentum, p T {\displaystyle p_{T}} , is usually measured. In this case if the particles are massless, or highly relativistic ( E ≫ m {\displaystyle E\gg m} ) then the invariant mass becomes: M 2 = 2 p T 1 p T 2 ( cosh ( η 1 − η 2 ) − cos ( ϕ 1 − ϕ 2 ) ) . {\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).}
Rest energy (also called rest mass energy ) is the energy associated with a particle's invariant mass. [ 2 ] [ 3 ]
The rest energy E 0 {\displaystyle E_{0}} of a particle is defined as: E 0 = m 0 c 2 , {\displaystyle E_{0}=m_{0}c^{2},} where c {\displaystyle c} is the speed of light in vacuum . [ 2 ] [ 3 ] [ 4 ] In general, only differences in energy have physical significance. [ 5 ]
The concept of rest energy follows from the special theory of relativity that leads to Einstein's famous conclusion about equivalence of energy and mass. See Special relativity § Relativistic dynamics and invariance . | https://en.wikipedia.org/wiki/Invariant_mass |
In mathematics , an invariant measure is a measure that is preserved by some function . The function may be a geometric transformation . For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping , and a difference of slopes is invariant under shear mapping . [ 1 ]
Ergodic theory is the study of invariant measures in dynamical systems . The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
Let ( X , Σ ) {\displaystyle (X,\Sigma )} be a measurable space and let f : X → X {\displaystyle f:X\to X} be a measurable function from X {\displaystyle X} to itself. A measure μ {\displaystyle \mu } on ( X , Σ ) {\displaystyle (X,\Sigma )} is said to be invariant under f {\displaystyle f} if, for every measurable set A {\displaystyle A} in Σ , {\displaystyle \Sigma ,} μ ( f − 1 ( A ) ) = μ ( A ) . {\displaystyle \mu \left(f^{-1}(A)\right)=\mu (A).}
In terms of the pushforward measure , this states that f ∗ ( μ ) = μ . {\displaystyle f_{*}(\mu )=\mu .}
The collection of measures (usually probability measures ) on X {\displaystyle X} that are invariant under f {\displaystyle f} is sometimes denoted M f ( X ) . {\displaystyle M_{f}(X).} The collection of ergodic measures , E f ( X ) , {\displaystyle E_{f}(X),} is a subset of M f ( X ) . {\displaystyle M_{f}(X).} Moreover, any convex combination of two invariant measures is also invariant, so M f ( X ) {\displaystyle M_{f}(X)} is a convex set ; E f ( X ) {\displaystyle E_{f}(X)} consists precisely of the extreme points of M f ( X ) . {\displaystyle M_{f}(X).}
In the case of a dynamical system ( X , T , φ ) , {\displaystyle (X,T,\varphi ),} where ( X , Σ ) {\displaystyle (X,\Sigma )} is a measurable space as before, T {\displaystyle T} is a monoid and φ : T × X → X {\displaystyle \varphi :T\times X\to X} is the flow map, a measure μ {\displaystyle \mu } on ( X , Σ ) {\displaystyle (X,\Sigma )} is said to be an invariant measure if it is an invariant measure for each map φ t : X → X . {\displaystyle \varphi _{t}:X\to X.} Explicitly, μ {\displaystyle \mu } is invariant if and only if μ ( φ t − 1 ( A ) ) = μ ( A ) for all t ∈ T , A ∈ Σ . {\displaystyle \mu \left(\varphi _{t}^{-1}(A)\right)=\mu (A)\qquad {\text{ for all }}t\in T,A\in \Sigma .}
Put another way, μ {\displaystyle \mu } is an invariant measure for a sequence of random variables ( Z t ) t ≥ 0 {\displaystyle \left(Z_{t}\right)_{t\geq 0}} (perhaps a Markov chain or the solution to a stochastic differential equation ) if, whenever the initial condition Z 0 {\displaystyle Z_{0}} is distributed according to μ , {\displaystyle \mu ,} so is Z t {\displaystyle Z_{t}} for any later time t . {\displaystyle t.}
When the dynamical system can be described by a transfer operator , then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of 1 , {\displaystyle 1,} this being the largest eigenvalue as given by the Frobenius–Perron theorem . | https://en.wikipedia.org/wiki/Invariant_measure |
In mathematics , an invariant polynomial is a polynomial P {\displaystyle P} that is invariant under a group Γ {\displaystyle \Gamma } acting on a vector space V {\displaystyle V} . Therefore, P {\displaystyle P} is a Γ {\displaystyle \Gamma } -invariant polynomial if
for all γ ∈ Γ {\displaystyle \gamma \in \Gamma } and x ∈ V {\displaystyle x\in V} . [ 1 ]
Cases of particular importance are for Γ a finite group (in the theory of Molien series , in particular), a compact group , a Lie group or algebraic group . For a basis-independent definition of 'polynomial' nothing is lost by referring to the symmetric powers of the given linear representation of Γ. [ 2 ]
This article incorporates material from Invariant polynomial on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
This commutative algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Invariant_polynomial |
The invariant set postulate concerns the possible relationship between fractal geometry and quantum mechanics and in particular the hypothesis that the former can assist in resolving some of the challenges posed by the latter. It is underpinned by nonlinear dynamical systems theory and black hole thermodynamics . [ 1 ]
The proposer of the postulate is climate scientist and physicist Tim Palmer . Palmer completed a PhD at the University of Oxford under Dennis Sciama , the same supervisor that Stephen Hawking had and then worked with Hawking himself at the University of Cambridge on supergravity theory . He later switched to meteorology and has established a reputation pioneering ensemble forecasting . [ 2 ] He now works at the European Centre for Medium-Range Weather Forecasts in Reading , England. [ 3 ]
Palmer argues that the postulate may help to resolve some of the paradoxes of quantum mechanics that have been discussed since the Bohr–Einstein debates of the 1920s and 30s and which remain unresolved. The idea backs Einstein's view that quantum theory is incomplete, but also agrees with Bohr's contention that quantum systems are not independent of the observer.
The key idea involved is that there exists a state space for the Universe, and that the state of the entire Universe can be expressed as a point in this state space. This state space can then be divided into "real" and "unreal" sets (parts), where, for example, the states where the Nazis lost WW2 are in the "real" set, and the states where the Nazis won WW2 are in the "unreal" set of points. The partition of state space into these two sets is unchanging, making the sets invariant.
If the Universe is a complex system affected by chaos then its invariant set (a fixed state of rest) is likely to be a fractal. According to Palmer this could resolve problems posed by the Kochen–Specker theorem , which appears to indicate that physics may have to abandon the idea of any kind of objective reality, and the apparent paradox of action at a distance . In a paper submitted to the Proceedings of the Royal Society he indicates how the idea can account for quantum uncertainty and problems of "contextuality". [ 3 ] For example, exploring the quantum problem of wave-particle duality , one of the central mysteries of quantum theory, the author claims that "in terms of the Invariant Set Postulate, the paradox is easily resolved, in principle at least". [ 1 ] The paper and related talks given at the Perimeter Institute and University of Oxford also explores the role of gravity in quantum physics. [ 1 ] [ 4 ] [ 5 ]
New Scientist quotes Bob Coeke of Oxford University as stating "What makes this really interesting is that it gets away from the usual debates over multiple universes and hidden variables and so on. It suggests there might be an underlying physical geometry that physics has just missed, which is radical and very positive". He added that "Palmer manages to explain some quantum phenomena, but he hasn't yet derived the whole rigid structure of the theory. This is really necessary." [ 3 ]
Robert Spekkens has said: "I think his approach is really interesting and novel. Other physicists have shown how you can find a way out of the Kochen–Specker theorem , but this work actually provides a mechanism to explain the theorem." [ 3 ]
According to Todd Brun , it is a tall order, to make a serious rival to quantum mechanics, a really predictive theory, out of Palmer's ideas. This goal has not been achieved yet. [ 6 ] [ irrelevant citation ] | https://en.wikipedia.org/wiki/Invariant_set_postulate |
The invariant speed or observer invariant speed is a speed which is measured to be the same in all reference frames by all observers. The invariance of the speed of light is one of the postulates of special relativity , and the terms speed of light and invariant speed are often considered synonymous. In non-relativistic classical mechanics , or Newtonian mechanics, finite invariant speed does not exist (the only invariant speed predicted by Newtonian mechanics is infinity). [ 1 ] [ 2 ]
This relativity -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Invariant_speed |
In the field of mathematics known as functional analysis , the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces (in other words, each example, found so far, of an operator with no non-trivial invariant subspaces is an operator that acts on a Banach space that is not isomorphic to a separable Hilbert space).
The problem seems to have been stated in the mid-20th century after work by Beurling and von Neumann , [ 1 ] who found (but never published) a positive solution for the case of compact operators . It was then posed by Paul Halmos for the case of operators T {\displaystyle T} such that T 2 {\displaystyle T^{2}} is compact. This was resolved affirmatively, for the more general class of polynomially compact operators (operators T {\displaystyle T} such that p ( T ) {\displaystyle p(T)} is a compact operator for a suitably chosen nonzero polynomial p {\displaystyle p} ), by Allen R. Bernstein and Abraham Robinson in 1966 (see Non-standard analysis § Invariant subspace problem for a summary of the proof).
For Banach spaces , the first example of an operator without an invariant subspace was constructed by Per Enflo . He proposed a counterexample to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987. [ 2 ] Enflo's long "manuscript had a world-wide circulation among mathematicians" [ 1 ] and some of its ideas were described in publications besides Enflo (1976). [ 3 ] Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Bernard Beauzamy , who acknowledged Enflo's ideas. [ 2 ]
In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces. [ 4 ]
In May 2023, a preprint of Enflo appeared on arXiv, [ 5 ] which, if correct, solves the problem for Hilbert spaces and completes the picture.
In July 2023, a second and independent preprint of Neville appeared on arXiv, [ 6 ] claiming the solution of the problem for separable Hilbert spaces.
In September 2024, a peer-reviewed article published in Axioms by a team of four Jordanian academic researchers announced that they had solved the invariant subspace problem. [ 7 ] However, basic mistakes in the proof were pointed out. [ 8 ] [ 9 ]
Formally, the invariant subspace problem for a complex Banach space H {\displaystyle H} of dimension > 1 is the question whether every bounded linear operator T : H → H {\displaystyle T:H\to H} has a non-trivial closed T {\displaystyle T} -invariant subspace : a closed linear subspace W {\displaystyle W} of H {\displaystyle H} , which is different from { 0 } {\displaystyle \{0\}} and from H {\displaystyle H} , such that T ( W ) ⊂ W {\displaystyle T(W)\subset W} .
A negative answer to the problem is closely related to properties of the orbits T {\displaystyle T} . If x {\displaystyle x} is an element of the Banach space H {\displaystyle H} , the orbit of x {\displaystyle x} under the action of T {\displaystyle T} , denoted by [ x ] {\displaystyle [x]} , is the subspace generated by the sequence { T n ( x ) : n ≥ 0 } {\displaystyle \{T^{n}(x)\,:\,n\geq 0\}} . This is also called the T {\displaystyle T} -cyclic subspace generated by x {\displaystyle x} . From the definition it follows that [ x ] {\displaystyle [x]} is a T {\displaystyle T} -invariant subspace. Moreover, it is the minimal T {\displaystyle T} -invariant subspace containing x {\displaystyle x} : if W {\displaystyle W} is another invariant subspace containing x {\displaystyle x} , then necessarily T n ( x ) ∈ W {\displaystyle T^{n}(x)\in W} for all n ≥ 0 {\displaystyle n\geq 0} (since W {\displaystyle W} is T {\displaystyle T} -invariant), and so [ x ] ⊂ W {\displaystyle [x]\subset W} . If x {\displaystyle x} is non-zero, then [ x ] {\displaystyle [x]} is not equal to { 0 } {\displaystyle \{0\}} , so its closure is either the whole space H {\displaystyle H} (in which case x {\displaystyle x} is said to be a cyclic vector for T {\displaystyle T} ) or it is a non-trivial T {\displaystyle T} -invariant subspace. Therefore, a counterexample to the invariant subspace problem would be a Banach space H {\displaystyle H} and a bounded operator T : H → H {\displaystyle T:H\to H} for which every non-zero vector x ∈ H {\displaystyle x\in H} is a cyclic vector for T {\displaystyle T} . (Where a "cyclic vector" x {\displaystyle x} for an operator T {\displaystyle T} on a Banach space H {\displaystyle H} means one for which the orbit [ x ] {\displaystyle [x]} of x {\displaystyle x} is dense in H {\displaystyle H} .)
While the case of the invariant subspace problem for separable Hilbert spaces is still open, several other cases have been settled for topological vector spaces (over the field of complex numbers): | https://en.wikipedia.org/wiki/Invariant_subspace_problem |
In mathematics , in the fields of multilinear algebra and representation theory , the principal invariants of the second rank tensor A {\displaystyle \mathbf {A} } are the coefficients of the characteristic polynomial [ 1 ]
where I {\displaystyle \mathbf {I} } is the identity operator and λ i ∈ C {\displaystyle \lambda _{i}\in \mathbb {C} } are the roots of the polynomial p {\displaystyle \ p} and the eigenvalues of A {\displaystyle \mathbf {A} } .
More broadly, any scalar-valued function f ( A ) {\displaystyle f(\mathbf {A} )} is an invariant of A {\displaystyle \mathbf {A} } if and only if f ( Q A Q T ) = f ( A ) {\displaystyle f(\mathbf {Q} \mathbf {A} \mathbf {Q} ^{T})=f(\mathbf {A} )} for all orthogonal Q {\displaystyle \mathbf {Q} } . This means that a formula expressing an invariant in terms of components, A i j {\displaystyle A_{ij}} , will give the same result for all Cartesian bases. For example, even though individual diagonal components of A {\displaystyle \mathbf {A} } will change with a change in basis, the sum of diagonal components will not change.
The principal invariants do not change with rotations of the coordinate system (they are objective, or in more modern terminology, satisfy the principle of material frame-indifference ) and any function of the principal invariants is also objective.
In a majority of engineering applications , the principal invariants of (rank two) tensors of dimension three are sought, such as those for the right Cauchy-Green deformation tensor C {\displaystyle \mathbf {C} } which has the eigenvalues λ 1 2 {\displaystyle \lambda _{1}^{2}} , λ 2 2 {\displaystyle \lambda _{2}^{2}} , and λ 3 2 {\displaystyle \lambda _{3}^{2}} . Where λ 1 {\displaystyle \lambda _{1}} , λ 2 {\displaystyle \lambda _{2}} , and λ 3 {\displaystyle \lambda _{3}} are the principal stretches, i.e. the eigenvalues of U = C {\displaystyle \mathbf {U} ={\sqrt {\mathbf {C} }}} .
For such tensors, the principal invariants are given by:
For symmetric tensors, these definitions are reduced. [ 2 ]
The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the Cayley–Hamilton theorem reveals that
where I {\displaystyle \mathbf {I} } is the second-order identity tensor.
In addition to the principal invariants listed above, it is also possible to introduce the notion of main invariants [ 3 ] [ 4 ]
which are functions of the principal invariants above. These are the coefficients of the characteristic polynomial of the deviator A − ( t r ( A ) / 3 ) I {\displaystyle \mathbf {A} -(\mathrm {tr} (\mathbf {A} )/3)\mathbf {I} } , such that it is traceless. The separation of a tensor into a component that is a multiple of the identity and a traceless component is standard in hydrodynamics, where the former is called isotropic, providing the modified pressure, and the latter is called deviatoric, providing shear effects.
Furthermore, mixed invariants between pairs of rank two tensors may also be defined. [ 4 ]
These may be extracted by evaluating the characteristic polynomial directly, using the Faddeev-LeVerrier algorithm for example.
The invariants of rank three, four, and higher order tensors may also be determined. [ 5 ]
A scalar function f {\displaystyle f} that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density , or Helmholtz free energy , of a nonlinear material possessing isotropic symmetry. [ 6 ]
This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940 where he was able to derive Kármán–Howarth equation from the invariant principle. [ 7 ] George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence. [ 8 ] [ 9 ] [ 10 ]
A real tensor A {\displaystyle \mathbf {A} } in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part. For example, if the Cartesian components of A {\displaystyle \mathbf {A} } are
the first step would be to evaluate the axial vector w {\displaystyle \mathbf {w} } associated with the skew-symmetric part. Specifically, the axial vector has components
The next step finds the principal values of the symmetric part of A {\displaystyle \mathbf {A} } . Even though the eigenvalues of a real non-symmetric tensor might be complex, the eigenvalues of its symmetric part will always be real and therefore can be ordered from largest to smallest. The corresponding orthonormal principal basis directions can be assigned senses to ensure that the axial vector w {\displaystyle \mathbf {w} } points within the first octant. With respect to that special basis, the components of A {\displaystyle \mathbf {A} } are
The first three invariants of A {\displaystyle \mathbf {A} } are the diagonal components of this matrix: a 1 = A 11 ′ = 1875 , a 2 = A 22 ′ = 1250 , a 3 = A 33 ′ = 625 {\displaystyle a_{1}=A'_{11}=1875,a_{2}=A'_{22}=1250,a_{3}=A'_{33}=625} (equal to the ordered principal values of the tensor's symmetric part). The remaining three invariants are the axial vector's components in this basis: w 1 ′ = A 32 ′ = 3750 , w 2 ′ = A 13 ′ = 3125 , w 3 ′ = A 21 ′ = 2500 {\displaystyle w'_{1}=A'_{32}=3750,w'_{2}=A'_{13}=3125,w'_{3}=A'_{21}=2500} . Note: the magnitude of the axial vector, w ⋅ w {\displaystyle {\sqrt {\mathbf {w} \cdot \mathbf {w} }}} , is the sole invariant of the skew part of A {\displaystyle \mathbf {A} } , whereas these distinct three invariants characterize (in a sense) "alignment" between the symmetric and skew parts of A {\displaystyle \mathbf {A} } . Incidentally, it is a myth that a tensor is positive definite if its eigenvalues are positive. Instead, it is positive definite if and only if the eigenvalues of its symmetric part are positive. | https://en.wikipedia.org/wiki/Invariants_of_tensors |
Invasion genetics is the area of study within biology that examines evolutionary processes in the context of biological invasions . Invasion genetics considers how genetic and demographic factors affect the success of a species introduced outside of its native range , and how the mechanisms of evolution , such as natural selection , mutation , and genetic drift , operate in these populations. Researchers exploring these questions draw upon theory and approaches from a range of biological disciplines, including population genetics , evolutionary ecology , population biology , and phylogeography .
Invasion genetics, due to its focus on the biology of introduced species, is useful for identifying potential invasive species and developing practices for managing biological invasions. It is distinguished from the broader study of invasive species because it is less directly concerned with the impacts of biological invasions, such as environmental or economic harm. In addition to applications for invasive species management, insights gained from invasion genetics also contribute to a broader understanding of evolutionary processes such as genetic drift and adaptive evolution .
Charles Elton formed the basis for examining biological invasions as a unified issue in his 1958 monograph, The Ecology of Invasions by Animals and Plants , drawing together case studies of species introductions. Other important events in the study of invasive species include a series of issues published by the Scientific Committee on Problems of the Environment in the 1980s and the founding of the journal Biological Invasions in 1999. [ 1 ] Much of the research motivated by Elton's monograph is generally identified with invasion ecology , and focuses on the ecological causes and impacts of biological invasions. [ 2 ]
The evolutionary modern synthesis in the early 20th century brought together Charles Darwin 's theory of evolution by natural selection and classical genetics through the development of population genetics , which provided the conceptual basis for studying how evolutionary processes shape variation in populations. This development was crucial to the emergence of invasion genetics, which is concerned with the evolution of populations of introduced species. [ 3 ] The beginning of invasion genetics as a distinct study has been identified with a symposium held at Asilomar in 1964 which included a number of major contributors to the modern synthesis, including Theodosius Dobzhansky , Ernst Mayr , and G. Ledyard Stebbins , as well as scientists with experience working in areas of weed and pest control. [ 1 ]
Stebbins, working with another botanist, Herbert G. Baker , collected a series of articles which emerged from the Asilomar symposium and published a volume titled The Genetics of Colonizing Species in 1965. This volume introduced many of the questions which continue to motivate research in invasion genetics today, including questions about the characteristics of successful invaders, the importance of a species' mating system in colonization success, the relative importance of genetic variation and phenotypic plasticity in adaptation to new environments, and the effect of population bottlenecks on genetic variation. [ 1 ]
Since its publication in 1965, The Genetics of Colonizing Species helped to motivate research which would provide a theoretical and empirical foundation for invasion genetics. [ 1 ] However, the term invasion genetics only first appeared in the literature in 1998, [ 4 ] and the first published definition appeared in 2005. [ 5 ] The success of introduced species is quite variable, consequently researchers have sought to develop terminology which allows distinguishing different levels of success. These approaches rely on describing invasion as a biological process. [ 6 ]
Researchers have proposed a number of different methods for describing biological invasions. In 1992, the ecologists Mark Williamson and Alastair Fitter divided the process of biological invasion into three stages : escaping, establishing, and becoming a pest. [ 7 ] Since then, there has been an expanding effort to develop a framework for categorizing biological invasions in terms that are neutral with respect to a species' environmental and economic impacts. This approach has allowed biologists to focus on the processes which facilitate or inhibit the spread of introduced species.
David M. Richardson and colleagues describe how introduced species must pass a series of barriers prior to becoming naturalized or invasive in a new range. [ 8 ] Alternatively, the stages of an invasion may be separated by filters , as described by Robert I. Colautti and Hugh MacIsaac , so that invasion success would depend on the rate of introduction ( propagule pressure ) as well as the traits possessed by the organism. [ 9 ]
The most recent systematic effort to describe the steps of a biological invasion was made by Tim Blackburn and colleagues in 2011, which combined the concepts of barriers and stages. According to this framework, there are four stages of an invasion: transport, introduction, establishment, and spread. Each of these stages is accompanied by one or more barriers. [ 6 ]
Invasion genetics can be used to understand the processes involved at each stage of a biological invasion. Many of the foundational questions of invasion genetics focused on processes involved during establishment and spread. As early as 1955, Herbert G. Baker proposed that self-fertilization would be a favourable trait for colonizing species because successful establishment would not require the simultaneous introduction of two individuals of opposite sexes. [ 10 ] Baker subsequently elaborated a series of "ideal weed characteristics" in an article in The Genetics of Colonizing Species , which included traits such as the ability to tolerate environmental variation, dispersal ability , and the ability to tolerate generalist herbivores and pathogens . While some of the traits, such as ease of germination, may aid a species in transport or introduction, most of the traits Baker identified were primarily conducive to establishment and spread. [ 11 ]
Advances in the study of molecular evolution may help biologists to understand better the processes of transport and introduction. Genomicist Melania Cristescu and her colleagues examined mitochondrial DNA of the fishhook waterflea introduced into the Great Lakes , tracing the source of the invasive populations to the Baltic Sea . [ 12 ] More recently, Cristescu has argued for expanding the use of phylogenetic and phylogenomic approaches, as well as applying metabarcoding and population genomics , to understand how species are introduced and identify "failed invasions" where introduction does not lead to establishment. [ 13 ]
Propagule pressure describes the number of individuals introduced into an area in which they are not native, and can strongly affect the ability of species to reach a later stage of invasion. Factors which may influence the rate of transport and introduction into a novel environment include the species' abundance in its native range, as well as its tendency to co-occur with or be deliberately moved by humans.
The likelihood of reaching establishment is also highly dependent on the number of individuals introduced. Small populations can be limited by Allee effects , as individuals may have difficulty finding suitable mates and populations are vulnerable to demographic stochasticity . Small populations may also suffer from inbreeding depression . Species that are introduced in larger numbers are more likely to establish in different environments, and high propagule pressure will introduce more genetic diversity into a population. These factors can help a species adapt to different environmental conditions during establishment as well as during subsequent spread in a new range. [ 14 ]
Herbert G. Baker 's list of 14 "ideal weed characteristics", published in the 1965 volume The Genetics of Colonizing Species , has been the basis for investigation into characteristics which could contribute to invasion success of plants. Since Baker first proposed this list, researchers have debated whether or not particular traits could be linked to the "invasiveness" of a species. Mark van Kleunen , in revisiting the question, proposed examining the traits of candidate invaders in the context of the process of biological invasion. According to this approach, particular traits might be useful for introduced species because they would allow them to pass through a filter associated with a particular stage of an invasion. [ 11 ]
A population of introduced species exhibiting higher genetic variation could be more successful during establishment and spread, due to the higher likelihood of possessing a suitable genotype for the novel environment. However, populations of a species in an introduced range are likely to exhibit lower genetic variation compared to populations in the native range due to population bottlenecks and founder effects experienced during introduction. A classic study on population bottlenecks, conducted by Masatoshi Nei , described a genetic signature of bottlenecks on introduced populations of Drosophila pseudoobscura in Colombia. The ecological success of many invaders despite these apparent genetic limitations suggests a "genetic paradox of invasion", for which a number of answers have been proposed. [ 15 ]
One of the possible resolutions for the genetic paradox of invasion is that most bottlenecks experienced by introduced species are typically not severe enough to have a strong effect on genetic variation. As well, a species may be introduced multiple times from multiple sources, resulting in genetic admixture which could compensate for lost genetic variation. The evolutionary ecologist Katrina Dlugosch has noted that the relationship between genetic variation and capacity for adaptation is nonlinear and may depend on factors such as the effect size of adaptive loci (in quantitative genetics , effect size refers to the magnitude of change in a phenotypic trait value associated with a particular locus) and the presence of cryptic variation . [ 15 ]
Phenotypic plasticity is the expression of different traits (or phenotypes ), such as morphology or behaviour , in response to different environments. Plasticity allows organisms to cope with environmental variation without necessitating genetic evolution.
Herbert G. Baker proposed that the possession of "general purpose" genotypes which were tolerant of a range of environments could be advantageous for species introduced into new areas. [ 16 ] General purpose genotypes could help introduced species encountering environmental variation during establishment and spread, in part because introduced species should have less genetic variation than native species. [ 17 ] However, it remains disputed whether or not invasive species exhibit higher plasticity than native and non-invasive species. [ 18 ]
Range expansion is the process by which an organism spreads and establishes new populations across a geographic scale, so it is part of a biological invasion. During a range expansion, there exists an expanding wave front , where rapidly-growing populations are established by a relatively small number of individuals. Under these demographic conditions, the phenomenon of gene surfing can lead to the accumulation of deleterious mutations . This reduces the fitness of individuals at the wave front, and is described as an expansion load (see also: mutation load ). [ 19 ] These mutations can limit the rate of range expansion and, in the absence of effective recombination and natural selection which would remove such mutations, can have severe and persisting negative effects on populations. [ 20 ]
Invasive species may encounter environments which differ either from those experienced in their natural range or where they are introduced. In these environments natural selection can act on these introduced populations, provided that there is sufficient genetic variation present in the population, which may lead to local adaptation . Such adaptation can facilitate both the establishment and spread of an introduced species.
Local adaptation can, however, be inhibited by genetic admixture between populations. Admixture can result in hybrid breakdown by breaking up beneficial gene linkages and introducing maladapted alleles. [ 22 ]
Admixture can also facilitation species introductions by increasing genetic variation, thereby limiting the cost of inbreeding in small populations. Through heterosis , the increased quality of hybrid offspring, admixture has also been shown to increase the vigour of introduced populations of common yellow monkeyflower . [ 23 ]
Hybridization broadly refers to breeding between individuals from genetically isolated populations, and may therefore be within a species (intraspecific) or between species (interspecific). When offspring are distinct from either parent, hybridization can be a source of evolutionary novelty. Hybridization can also lead to gene flow between populations or species through the mechanism of introgression .
Hybridization and its contribution to evolution was a subject of interest for G. Ledyard Stebbins , [ 24 ] [ 25 ] who noted in a 1959 review that the introduction of European species of the genus Tragopogon to North America had led to hybrid speciation ; [ 26 ] this example was also discussed by Herbert G. Baker in The Genetics of Colonizing Species . [ 27 ] The first systematic review of the role of invasive plant species in interspecific hybridization appeared in 1992, [ 28 ] and the phenomenon has also been explored in fish and aquatic invertebrates. [ 29 ] Hybridization may increase the invasiveness of introduced species, either by introducing genetic variation, heterosis , or by creating novel genotypes which perform better in a given environment. [ 30 ] Gene flow between introduced and native species can also result in the loss of biodiversity through genetic pollution .
Because biological invasions can have a profound impact on the invaded environment, it is expected that the arrival of invasive species creates new selective pressures on native organisms, typically through competitive or predatory interactions. Through adaptive evolution, species in affected ecological communities could evolve to tolerate invasive species. This means that biological invasions potentially have both ecological and evolutionary consequences for native species. However, many studies have failed to detect an adaptive response of native species to ecological disruptions. The ecologists Jennifer Lau and Casey terHorst have pointed to this absence of an evolutionary response as an important consideration for understanding how invasive species disrupt ecological communities and the multiple challenges faced by native populations. [ 31 ] | https://en.wikipedia.org/wiki/Invasion_genetics |
The Invasive Species Forecasting System , or simply ISFS , was a proposed [ 1 ] modeling environment for creating predictive habitat suitability maps for invasive species . It was developed by the National Aeronautics and Space Administration (NASA) in cooperation with various Department of Interior bureaus, including the United States Geological Survey (USGS), the Bureau of Land Management (BLM), and the National Park Service (NPS). [ 2 ]
As of 26 October 2020 [update] this ISFS website is dead. [ 3 ] The development of such a system is still proposed however. [ 4 ] | https://en.wikipedia.org/wiki/Invasive_Species_Forecasting_System |
InvenSense Inc. is an American consumer electronics company, founded in 2003 in San Jose, California by Steve Nasiri. [ 1 ] They are the provider of the MotionTracking sensor system on chip (SoC) which functions as a gyroscope for consumer electronic devices such as smartphones , tablets, wearables , gaming devices, optical image stabilization , and remote controls for Smart TVs. InvenSense provides the motion controller in the Nintendo Wii game controller and the Oculus Rift DK1 . [ 2 ] Its motion controllers are found in the Samsung Galaxy smartphones and most recently in the Apple iPhone 6 . [ 3 ]
Founded in 2003, InvenSense is headquartered in San Jose, California with offices in Wilmington, Massachusetts , China, Taiwan, Korea, Japan, France, Canada, Slovakia and Italy. [ 4 ] [ 5 ]
In December 2016, the company was acquired by electronics company TDK for US$1.3 billion. [ 6 ] InvenSense became part of the MEMS Sensors Business Group in 2017. In February 2018, Chirp Microsystems joined InvenSense through its acquisition by TDK. [ 7 ]
InvenSense MotionTracking tracks complex user motions with the use of motion sensors such as microelectromechanical gyroscopes , (including 3-axis gyroscopes), [ 3 ] accelerometers , compasses, and pressure sensors. the system then calibrates data, and creates a single data stream. [ 8 ] With complex movement tracking comes a drain on battery life. In June, 2014, the company announced a low power gyroscope chip that used just under six milliwatts of power in a chip and was just 0.75 millimeters thick. [ 4 ]
InvenSense also provides Optical Image Stabilisation for smartphone cameras, which are important to detect hand movements and reduce shake in photographs. InvenSense's compact gyroscope was designed to provide antishake features on the smallest camera phones. [ 9 ] | https://en.wikipedia.org/wiki/InvenSense |
The inventor's paradox is a phenomenon that occurs in seeking a solution to a given problem. Instead of solving a specific type of problem, which would seem intuitively easier, it can be easier to solve a more general problem, which covers the specifics of the sought-after solution. The inventor's paradox has been used to describe phenomena in mathematics , programming , and logic , as well as other areas that involve critical thinking .
In the book How to Solve It , Hungarian mathematician George Pólya introduces what he defines as the inventor's paradox:
The more ambitious plan may have more chances of success […] provided it is not based on a mere pretension but on some vision of the things beyond those immediately present. [ 1 ]
Or, in other words, to solve what one desires to solve, one may have to solve more than that in order to get a properly working flow of information. [ 2 ]
When solving a problem, the natural inclination typically is to remove as much excessive variability and produce limitations on the subject at hand as possible. Doing this can create unforeseen and intrinsically awkward parameters. [ 3 ] The goal is to find elegant and relatively simple solutions to broader problems, allowing for the ability to focus on the specific portion that was originally of concern. [ 4 ] There lies the inventor's paradox , that it is often significantly easier to find a general solution than a more specific one, since the general solution may naturally have a simpler algorithm and cleaner design, and typically can take less time to solve in comparison with a particular problem. [ 3 ]
The sum of numbers sequentially from 1-99:
This process, although not impossible to do in one's head, can prove to be difficult for most. However, the ability to generalize the problem exists, in this case by reordering the sequence to:
In this form, the example can be solved by most without the use of a calculator. [ 3 ] If one notices the problem's lowest and highest numbers (1 + 99) sum to 100, and that the next pair of lowest and highest numbers (2 + 98) also sum to 100, they'll also realize that all 49 numbers are matching pairs that each sum to 100, except for the single number in the middle, 50. The inventive mathematician will reformulate the problem in their mind as (49 * 100) + 50. Since 49 * 100 is easy to calculate by adding 2 zeros to the digit places of 49, they think: 4900 + 50. This is easy to add, because 50's maximum ordinal placement of the most significant digit (number 5 in the 2nd position "10s" place) is less than the minimum ordinal position of 4900's smallest significant digit (number 9 in the 3rd position "100s" place). So the solver simply replaces the last two 0s in 4900 with 50 to add them together, yielding the answer 4950. While the text description of this process seems complicated, each of the steps performed in the mind is simple and fast.
Although appearing in several applications, it can be easiest to explain through inspection of a relatively simple mathematical sequence. [ 5 ]
and further along in the sequence:
In allowing the sequence to expand to a point where the sum cannot be found quickly, we can simplify by finding that the sum of consecutive odd numbers follows: [ 2 ]
As an example in applying the same logic, it may be harder to solve a 25-case problem than it would be to solve an n-case problem, and then apply it to the case where n=25. [ 6 ] [ further explanation needed ]
This paradox has applications in writing efficient computer programs. It is intuitive to write programs that are specialized, but in practice it can become easier to develop more generalized procedures. [ 7 ] According to Bruce Tate , some of the most successful frameworks are simple generalizations of complex problems, and he says that Visual Basic , the Internet, and Apache web servers plug-ins are primary examples of such practice. [ 4 ] In the investigation of the semantics of language, many logicians find themselves facing this paradox. An example of application can be seen in the inherent concern of logicians with the conditions of truth within a sentence, and not, in fact, with the conditions under which a sentence can be truly asserted. [ 2 ] Additionally, the paradox has been shown to have applications in industry. [ 3 ] | https://en.wikipedia.org/wiki/Inventor's_paradox |
Inventor Labs is a CD-ROM software from Houghton Mifflin Interactive and Red Hill Studios.
The CD has a virtual tour through three of the most famous science labs ever: the workshops of Thomas Edison , Alexander Graham Bell , and James Watt . [ 3 ] [ 4 ]
Inventor Labs was developed by Red Hill Studios, a company founded in 1991. [ 1 ]
CNET said "Combining an interactive look at science history with an eye to the future, this virtual tour will be as much fun for kids as their first magnifying glass". [ 3 ] New York Daily News gave Inventor Labs a score of 2 out of 4. [ 5 ]
Publishers Weekly said "Though the science here is solid, kids will likely seek out something more entertaining". [ 6 ]
The CD-ROM won a Gold Invision Award for Best Young Adult Title. [ 7 ] [ 8 ]
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Inventor_Labs |
Inventory ( British English ) or stock ( American English ) is a quantity of the goods and materials that a business holds for the ultimate goal of resale, production or utilisation. [ nb 1 ]
Inventory management is a discipline primarily about specifying the shape and placement of stocked goods. It is required at different locations within a facility or within many locations of a supply network to precede the regular and planned course of production and stock of materials.
The concept of inventory, stock or work in process (or work in progress) has been extended from manufacturing systems to service businesses [ 1 ] [ 2 ] [ 3 ] and projects, [ 4 ] by generalizing the definition to be "all work within the process of production—all work that is or has occurred prior to the completion of production". In the context of a manufacturing production system, inventory refers to all work that has occurred—raw materials, partially finished products, finished products prior to sale and departure from the manufacturing system. In the context of services, inventory refers to all work done prior to sale, including partially process information.
There are five basic reasons for keeping an inventory:
All these stock reasons can apply to any owner or product.
While accountants often discuss inventory in terms of goods for sale, organizations— manufacturers , service-providers and not-for-profits —also have inventories (fixtures, equipment, furniture, supplies, parts, etc.) that they do not intend to sell. Manufacturers', distributors ', and wholesalers' inventory tends to cluster in warehouses . Retailers ' inventory may exist in a warehouse or in a shop or store accessible to customers . Inventories not intended for sale to customers or to clients may be held in any premises an organization uses. Stock ties up cash and, if uncontrolled, it will be impossible to know the actual level of stocks and therefore difficult to keep the costs associated with holding too much or too little inventory under control.
While the reasons for holding stock were covered earlier, most manufacturing organizations usually divide their "goods for sale" inventory into:
For example:
A canned food manufacturer's materials inventory includes the ingredients to form the foods to be canned, empty cans and their lids (or coils of steel or aluminum for constructing those components), labels, and anything else (solder, glue, etc.) that will form part of a finished can. The firm's work in process includes those materials from the time of release to the work floor until they become complete and ready for sale to wholesale or retail customers. This may be vats of prepared food, filled cans not yet labeled or sub-assemblies of food components. It may also include finished cans that are not yet packaged into cartons or pallets. Its finished good inventory consists of all the filled and labeled cans of food in its warehouse that it has manufactured and wishes to sell to food distributors (wholesalers), to grocery stores (retailers), and even perhaps to consumers through arrangements like factory stores and outlet centers.
The partially completed work (or work in process) is a measure of inventory built during the work execution of a capital project, [ 9 ] [ 10 ] [ 11 ] such as encountered in civilian infrastructure construction or oil and gas. Inventory may not only reflect physical items (such as materials, parts, partially-finished sub-assemblies) but also knowledge work-in-process (such as partially completed engineering designs of components and assemblies to be fabricated).
A "virtual inventory" (also known as a "bank inventory") enables a group of users to share common parts, especially where their availability at short notice may be critical but they are unlikely to required by more than a few bank members at any one time. [ 12 ] Virtual inventory also allows distributors and fulfilment houses to ship goods to retailers direct from stock, regardless of whether the stock is held in a retail store, stock room or warehouse. [ 13 ] Virtual inventories allow participants to access a wider mix of products and to reduce the risks involved in carrying inventory for which expected demand does not materialise. [ 14 ]
There are several costs associated with inventory:
Inventory proportionality is the goal of demand-driven inventory management. The primary optimal outcome is to have the same number of days' (or hours', etc.) worth of inventory on hand across all products so that the time of runout of all products would be simultaneous. In such a case, there is no "excess inventory", that is, inventory that would be left over of another product when the first product runs out. Holding excess inventory is sub-optimal because the money spent to obtain and the cost of holding it could have been utilized better elsewhere, i.e. to the product that just ran out.
The secondary goal of inventory proportionality is inventory minimization. By integrating accurate demand forecasting with inventory management, rather than only looking at past averages, a much more accurate and optimal outcome is expected. Integrating demand forecasting into inventory management in this way also allows for the prediction of the "can fit" point when inventory storage is limited on a per-product basis.
The technique of inventory proportionality is most appropriate for inventories that remain unseen by the consumer, as opposed to "keep full" systems where a retail consumer would like to see full shelves of the product they are buying so as not to think they are buying something old, unwanted or stale; and differentiated from the "trigger point" systems where product is reordered when it hits a certain level; inventory proportionality is used effectively by just-in-time manufacturing processes and retail applications where the product is hidden from view.
One early example of inventory proportionality used in a retail application in the United States was for motor fuel. Motor fuel (e.g. gasoline) is generally stored in underground storage tanks. The motorists do not know whether they are buying gasoline off the top or bottom of the tank, nor need they care. Additionally, these storage tanks have a maximum capacity and cannot be overfilled. Finally, the product is expensive. Inventory proportionality is used to balance the inventories of the different grades of motor fuel, each stored in dedicated tanks, in proportion to the sales of each grade. Excess inventory is not seen or valued by the consumer, so it is simply cash sunk (literally) into the ground. Inventory proportionality minimizes the amount of excess inventory carried in underground storage tanks. This application for motor fuel was first developed and implemented by Petrolsoft Corporation in 1990 for Chevron Products Company. Most major oil companies use such systems today. [ 16 ]
The use of inventory proportionality in the United States is thought to have been inspired by Japanese just-in-time parts inventory management made famous by Toyota Motors in the 1980s. [ 17 ]
It seems that around 1880 [ 18 ] there was a change in manufacturing practice from companies with relatively homogeneous lines of products to horizontally integrated companies with unprecedented diversity in processes and products. Those companies (especially in metalworking) attempted to achieve success through economies of scope—the gains of jointly producing two or more products in one facility. The managers now needed information on the effect of product-mix decisions on overall profits and therefore needed accurate product-cost information. A variety of attempts to achieve this were unsuccessful due to the huge overhead of the information processing of the time. However, the burgeoning need for financial reporting after 1900 created unavoidable pressure for financial accounting of stock and the management need to cost manage products became overshadowed. In particular, it was the need for audited accounts that sealed the fate of managerial cost accounting. The dominance of financial reporting accounting over management accounting remains to this day with few exceptions, and the financial reporting definitions of 'cost' have distorted effective management 'cost' accounting since that time. This is particularly true of inventory.
Hence, high-level financial inventory has these two basic formulas, which relate to the accounting period:
The benefit of these formulas is that the first absorbs all overheads of production and raw material costs into a value of inventory for reporting. The second formula then creates the new start point for the next period and gives a figure to be subtracted from the sales price to determine some form of sales-margin figure.
Manufacturing management is more interested in inventory turnover ratio or average days to sell inventory since it tells them something about relative inventory levels.
and its inverse
This ratio estimates how many times the inventory turns over a year. This number tells how much cash/goods are tied up waiting for the process and is a critical measure of process reliability and effectiveness. So a factory with two inventory turns has six months stock on hand, which is generally not a good figure (depending upon the industry), whereas a factory that moves from six turns to twelve turns has probably improved effectiveness by 100%. This improvement will have some negative results in the financial reporting, since the 'value' now stored in the factory as inventory is reduced.
While these accounting measures of inventory are very useful because of their simplicity, they are also fraught with the danger of their own assumptions. There are, in fact, so many things that can vary hidden under this appearance of simplicity that a variety of 'adjusting' assumptions may be used. These include:
Inventory Turn is a financial accounting tool for evaluating inventory and it is not necessarily a management tool. Inventory management should be forward looking. The methodology applied is based on historical cost of goods sold. The ratio may not be able to reflect the usability of future production demand, as well as customer demand.
Business models, including Just in Time (JIT) Inventory, Vendor Managed Inventory (VMI) and Customer Managed Inventory (CMI), attempt to minimize on-hand inventory and increase inventory turns. VMI and CMI have gained considerable attention due to the success of third-party vendors who offer added expertise and knowledge that organizations may not possess.
Inventory management also involves risk which varies depending upon a firm's position in the distribution channel. Some typical measures of inventory exposure [ definition needed ] are width of commitment [ definition needed ] , time of duration [ definition needed ] and depth [ definition needed ] . [ 20 ]
Inventory management in modern days is online oriented and more viable in digital. This type of dynamics order management will require end-to-end visibility, collaboration across fulfillment processes, real-time data automation among different companies, and integration among multiple systems. [ 21 ]
Each country has its own rules about accounting for inventory that fit with their financial-reporting rules.
For example, organizations in the U.S. define inventory to suit their needs within US Generally Accepted Accounting Practices (GAAP), the rules defined by the Financial Accounting Standards Board (FASB) (and others) and enforced by the U.S. Securities and Exchange Commission (SEC) and other federal and state agencies. Other countries often have similar arrangements but with their own accounting standards and national agencies instead.
It is intentional that financial accounting uses standards that allow the public to compare firms' performance, cost accounting functions internally to an organization and potentially with much greater flexibility. A discussion of inventory from standard and Theory of Constraints -based ( throughput ) cost accounting perspective follows some examples and a discussion of inventory from a financial accounting perspective.
The internal costing/valuation of inventory can be complex. Whereas in the past most enterprises ran simple, one-process factories, such enterprises are quite probably in the minority in the 21st century. Where 'one process' factories exist, there is a market for the goods created, which establishes an independent market value for the good. Today, with multistage-process companies, there is much inventory that would once have been finished goods which is now held as 'work in process' (WIP). This needs to be valued in the accounts, but the valuation is a management decision since there is no market for the partially finished product. This somewhat arbitrary 'valuation' of WIP combined with the allocation of overheads to it has led to some unintended and undesirable results. [ example needed ]
An organization's inventory can appear a mixed blessing, since it counts as an asset on the balance sheet , but it also ties up money that could serve for other purposes and requires additional expense for its protection. Inventory may also cause significant tax expenses, depending on particular countries' laws regarding depreciation of inventory, as in Thor Power Tool Company v. Commissioner .
Inventory appears as a current asset on an organization's balance sheet because the organization can, in principle, turn it into cash by selling it. Some organizations hold larger inventories than their operations require in order to inflate their apparent asset value and their perceived profitability.
In addition to the money tied up by acquiring inventory, inventory also brings associated costs for warehouse space, for utilities, and for insurance to cover staff to handle and protect it from fire and other disasters, obsolescence, shrinkage (theft and errors), and others. Such holding costs can mount up: between a third and a half of its acquisition value per year.
Businesses that stock too little inventory cannot take advantage of large orders from customers if they cannot deliver. The conflicting objectives of cost control and customer service often put an organization's financial and operating managers against its sales and marketing departments. Salespeople, in particular, often receive sales-commission payments, so unavailable goods may reduce their potential personal income. This conflict can be minimised by reducing production time to being near or less than customers' expected delivery time. This effort, known as " Lean production " will significantly reduce working capital tied up in inventory and reduce manufacturing costs (See the Toyota Production System ).
By helping the organization to make better decisions, the accountants can help the public sector to change in a very positive way that delivers increased value for the taxpayer's investment. It can also help to incentive's progress and to ensure that reforms are sustainable and effective in the long term, by ensuring that success is appropriately recognized in both the formal and informal reward systems of the organization.
To say that they have a key role to play is an understatement. Finance is connected to most, if not all, of the key business processes within the organization. It should be steering the stewardship and accountability systems that ensure that the organization is conducting its business in an appropriate, ethical manner. It is critical that these foundations are firmly laid. So often they are the litmus test by which public confidence in the institution is either won or lost.
Finance should also be providing the information, analysis and advice to enable the organizations' service managers to operate effectively. This goes beyond the traditional preoccupation with budgets—how much have we spent so far, how much do we have left to spend? It is about helping the organization to better understand its own performance. That means making the connections and understanding the relationships between given inputs—the resources brought to bear—and the outputs and outcomes that they achieve. It is also about understanding and actively managing risks within the organization and its activities.
When a merchant buys goods from inventory, the value of the inventory account is reduced by the cost of goods sold (COGS). This is simple where the cost has not varied across those held in stock; but where it has, then an agreed method must be derived to evaluate it. For inventory items that one cannot track individually, accountants must choose a method that fits the nature of the sale. Two popular methods in use are: FIFO (first in, first out) and LIFO (last in, first out).
FIFO treats the first unit that arrived in inventory as the first one sold. LIFO considers the last unit arriving in inventory as the first one sold. Which method an accountant selects can have a significant effect on net income and book value and, in turn, on taxation. Using LIFO accounting for inventory, a company generally reports lower net income and lower book value, due to the effects of inflation. This generally results in lower taxation. Due to LIFO's potential to skew inventory value, UK GAAP and IAS have effectively banned LIFO inventory accounting. LIFO accounting is permitted in the United States subject to section 472 of the Internal Revenue Code . [ 22 ]
Standard cost accounting uses ratios called efficiencies that compare the labour and materials actually used to produce a good with those that the same goods would have required under "standard" conditions. As long as actual and standard conditions are similar, few problems arise. Unfortunately, standard cost accounting methods developed about 100 years ago, when labor comprised the most important cost in manufactured goods. Standard methods continue to emphasize labor efficiency even though that resource now constitutes a (very) small part of cost in most cases.
Standard cost accounting can hurt managers, workers, and firms in several ways. For example, a policy decision to increase inventory can harm a manufacturing manager's performance evaluation . Increasing inventory requires increased production, which means that processes must operate at higher rates. When (not if) something goes wrong, the process takes longer and uses more than the standard labor time. The manager appears responsible for the excess, even though s/he has no control over the production requirement or the problem.
In adverse economic times, firms use the same efficiencies to downsize, rightsize, or otherwise reduce their labor force. Workers laid off under those circumstances have even less control over excess inventory and cost efficiencies than their managers.
Many financial and cost accountants have agreed for many years on the desirability of replacing standard cost accounting. They have not, however, found a successor.
Eliyahu M. Goldratt developed the Theory of Constraints in part to address the cost-accounting problems in what he calls the "cost world." He offers a substitute, called throughput accounting , that uses throughput (money for goods sold to customers) in place of output (goods produced that may sell or may boost inventory) and considers labor as a fixed rather than as a variable cost. He defines inventory simply as everything the organization owns that it plans to sell, including buildings, machinery, and many other things in addition to the categories listed here. Throughput accounting recognizes only one class of variable costs: the truly variable costs, like materials and components, which vary directly with the quantity produced
Finished goods inventories remain balance-sheet assets, but labor-efficiency ratios no longer evaluate managers and workers. Instead of an incentive to reduce labor cost, throughput accounting focuses attention on the relationships between throughput (revenue or income) on one hand and controllable operating expenses and changes in inventory on the other.
Inventories also play an important role in national accounts and the analysis of the business cycle . Some short-term macroeconomic fluctuations are attributed to the inventory cycle .
Also known as distressed or expired stock, distressed inventory is inventory whose potential to be sold at a normal cost has passed or will soon pass. In certain industries it could also mean that the stock is or will soon be impossible to sell. Examples of distressed inventory include products which have reached their expiry date , or have reached a date in advance of expiry at which the planned market will no longer purchase them (e.g. 3 months left to expiry), clothing which is out of fashion , music which is no longer popular and old newspapers or magazines. It also includes computer or consumer-electronic equipment which is obsolete or discontinued and whose manufacturer is unable to support it, along with products which use that type of equipment e.g. VHS format equipment and videos. [ 23 ]
In 2001, Cisco wrote off inventory worth US$2.25 billion due to duplicate orders. [ 24 ] This is considered one of the biggest inventory write-offs in business history. [ citation needed ]
Stock rotation is the practice of changing the way inventory is displayed on a regular basis. This is most commonly used in hospitality and retail - particularity where food products are sold. For example, in the case of supermarkets that a customer frequents on a regular basis, the customer may know exactly what they want and where it is. This results in many customers going straight to the product they seek and do not look at other items on sale. To discourage this practice, stores will rotate the location of stock to encourage customers to look through the entire store. This is in hopes the customer will pick up items they would not normally see. [ 25 ]
Inventory credit refers to the use of stock, or inventory, as collateral to raise finance. Where banks may be reluctant to accept traditional collateral, for example in developing countries where land title may be lacking, inventory credit is a potentially important way of overcoming financing constraints. [ 26 ] This is not a new concept; archaeological evidence suggests that it was practiced in Ancient Rome. Obtaining finance against stocks of a wide range of products held in a bonded warehouse is common in much of the world. It is, for example, used with Parmesan cheese in Italy. [ 27 ] Inventory credit on the basis of stored agricultural produce is widely used in Latin American countries and in some Asian countries. [ 28 ] A precondition for such credit is that banks must be confident that the stored product will be available if they need to call on the collateral; this implies the existence of a reliable network of certified warehouses. [ 29 ] Banks also face problems in valuing the inventory. The possibility of sudden falls in commodity prices means that they are usually reluctant to lend more than about 60% of the value of the inventory at the time of the loan. | https://en.wikipedia.org/wiki/Inventory |
Inventory control or stock control is the process of managing stock held within a warehouse, store or other storage location, including auditing actions concerned with "checking a shop's stock". [ 1 ] These processes ensure that the right amount of supply is available within a business. [ 2 ] However, a more focused definition takes into account the more science-based, methodical practice of not only verifying a business's inventory but also maximising the amount of profit from the least amount of inventory investment without affecting customer satisfaction. [ 3 ] Other facets of inventory control include forecasting future demand, supply chain management , production control , financial flexibility, purchasing data, loss prevention and turnover, and customer satisfaction. [ 4 ]
An extension of inventory control is the inventory control system. This may come in the form of a technological system and its programmed software used for managing various aspects of inventory problems, [ 5 ] or it may refer to a methodology (which may include the use of technological barriers) for handling loss prevention in a business. [ 6 ] [ 7 ] The inventory control system allows for companies to assess their current state concerning assets, account balances, and financial reports. [ 2 ]
An inventory control system is used to keep inventories in a desired state while continuing to adequately supply customers, [ 8 ] [ 9 ] and its success depends on maintaining clear records on a periodic or perpetual basis. [ 9 ] [ 10 ]
Inventory management software often plays an important role in the modern inventory control system, providing timely and accurate analytical, optimization, and forecasting techniques for complex inventory management problems. [ 11 ] [ 12 ] Typical features of this type of software include: [ 9 ] [ 12 ]
Through this functionality, a business may better detail what has sold, how quickly, and at what price , for example. Reports could be used to predict when to stock up on extra products around a holiday or to make decisions about special offers , discontinuing products, and so on.
Inventory control techniques often rely upon barcodes and radio-frequency identification (RFID) tags to provide automatic identification of inventory objects—including but not limited to merchandise , consumables , fixed assets , circulating tools, library books, and capital equipment —which in turn can be processed with inventory management software. [ 13 ] A new trend in inventory management is to label inventory and assets with a QR Code , which can then be read with smart-phones to keep track of inventory count and movement. [ 14 ] These new systems are especially useful for field service operations, where an employee needs to record inventory transaction or look up inventory stock in the field, away from the computers and hand-held scanners.
The control of inventory involves managing the physical quantities as well as the costing of the goods as it flows through the supply chain. In managing the cost prices of the goods throughout the supply chain, several costing methods are employed:
The calculation can be done for different periods. If the calculation is done on a monthly basis, then it is referred to the periodic method. In this method, the available stock is calculated by:
ADD Stock at beginning of period ADD Stock purchased during the period AVERAGE total cost by total qty to arrive at the Average Cost of Goods for the period.
This Average Cost Price is applied to all movements and adjustments in that period. Ending stock in qty is arrived at by Applying all the changes in qty to the Available balance. Multiplying the stock balance in qty by the Average cost gives the Stock cost at the end of the period.
Using the perpetual method, the calculation is done upon every purchase transaction.
Thus, the calculation is the same based on the periodic calculation whether by period (periodic) or by transaction (perpetual).
The only difference is the 'periodicity' or scope of the calculation.
In practice, the daily averaging has been used to closely approximate the perpetual method.
6. Bottle neck method (depends on proper planning support)
Inventory control systems have advantages and disadvantages, based on what style of system is being run. A purely periodic (physical) inventory control system takes "an actual physical count and valuation of all inventory on hand ... at the close of an accounting period," [ 15 ] whereas a perpetual inventory control system takes an initial count of an entire inventory and then closely monitors any additions and deletions as they occur. [ 15 ] [ 10 ] Various advantages and disadvantages, in comparison, include:
While these terms are sometimes used interchangeably, inventory management and inventory control deal with different aspects of inventory:
Just-in-time inventory (JIT), vendor managed inventory (VMI) and customer managed inventory (CMI) are a few of the popular models being employed by organizations looking to have greater stock management control.
JIT is a model that attempts to replenish inventory for organizations when the inventory is required. The model attempts to avoid excess inventory and its associated costs. As a result, companies receive inventory only when the need for more stock is approaching.
VMI (vendor managed inventory) and (co-managed inventory) are two business models that adhere to the JIT inventory principles. VMI gives the vendor in a vendor/customer relationship the ability to monitor, plan and control inventory for their customers. Customers relinquish the order making responsibilities in exchange for timely inventory replenishment that increases organizational efficiency.
CMI allows the customer to order and control their inventory from their vendors/suppliers. Both VMI and CMI benefit the vendor as well as the customer. Vendors see a significant increase in sales due to increased inventory turns and cost savings realized by their customers, while customers realize similar benefits. | https://en.wikipedia.org/wiki/Inventory_control |
In science , an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
Radar energy expands during both the signal transmission and the reflected return, so the inverse square for both paths means that the radar will receive energy according to the inverse fourth power of the range.
To prevent dilution of energy while propagating a signal, certain methods can be used such as a waveguide , which acts like a canal does for water, or how a gun barrel restricts hot gas expansion to one dimension in order to prevent loss of energy transfer to a bullet .
In mathematical notation the inverse square law can be expressed as an intensity (I) varying as a function of distance (d) from some centre. The intensity is proportional (see ∝ ) to the reciprocal of the square of the distance thus: intensity ∝ 1 distance 2 {\displaystyle {\text{intensity}}\ \propto \ {\frac {1}{{\text{distance}}^{2}}}\,}
It can also be mathematically expressed as : intensity 1 intensity 2 = distance 2 2 distance 1 2 {\displaystyle {\frac {{\text{intensity}}_{1}}{{\text{intensity}}_{2}}}={\frac {{\text{distance}}_{2}^{2}}{{\text{distance}}_{1}^{2}}}}
or as the formulation of a constant quantity: intensity 1 × distance 1 2 = intensity 2 × distance 2 2 {\displaystyle {\text{intensity}}_{1}\times {\text{distance}}_{1}^{2}={\text{intensity}}_{2}\times {\text{distance}}_{2}^{2}}
The divergence of a vector field which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources. Newton's law of universal gravitation follows an inverse-square law, as do the effects of electric , light , sound , and radiation phenomena.
The inverse-square law generally applies when some force, energy, or other conserved quantity is evenly radiated outward from a point source in three-dimensional space . Since the surface area of a sphere (which is 4π r 2 ) is proportional to the square of the radius, as the emitted radiation gets farther from the source, it is spread out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of radiation passing through any unit area (directly facing the point source) is inversely proportional to the square of the distance from the point source. Gauss's law for gravity is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.
Gravitation is the attraction between objects that have mass. Newton's law states:
The gravitational attraction force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation distance. The force is always attractive and acts along the line joining them. [ 1 ]
F = G m 1 m 2 r 2 {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}}
If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the shell theorem . Otherwise, if we want to calculate the attraction between massive bodies, we need to add all the point-point attraction forces vectorially and the net attraction might not be exact inverse square. However, if the separation between the massive bodies is much larger compared to their sizes, then to a good approximation, it is reasonable to treat the masses as a point mass located at the object's center of mass while calculating the gravitational force.
As the law of gravitation, this law was suggested in 1645 by Ismaël Bullialdus . But Bullialdus did not accept Kepler's second and third laws , nor did he appreciate Christiaan Huygens 's solution for circular motion (motion in a straight line pulled aside by the central force). Indeed, Bullialdus maintained the sun's force was attractive at aphelion and repulsive at perihelion. Robert Hooke and Giovanni Alfonso Borelli both expounded gravitation in 1666 as an attractive force. [ 2 ] Hooke's lecture "On gravity" was at the Royal Society, in London, on 21 March. [ 3 ] Borelli's "Theory of the Planets" was published later in 1666. [ 4 ] Hooke's 1670 Gresham lecture explained that gravitation applied to "all celestiall bodys" and added the principles that the gravitating power decreases with distance and that in the absence of any such power bodies move in straight lines. By 1679, Hooke thought gravitation had inverse square dependence and communicated this in a letter to Isaac Newton : [ 5 ] my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall . [ 6 ]
Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686 Principia acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the Solar System , [ 7 ] as well as giving some credit to Bullialdus. [ 8 ]
The force of attraction or repulsion between two electrically charged particles, in addition to being directly proportional to the product of the electric charges, is inversely proportional to the square of the distance between them; this is known as Coulomb's law . The deviation of the exponent from 2 is less than one part in 10 15 . [ 9 ]
F = k e q 1 q 2 r 2 {\displaystyle F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}}
The intensity (or illuminance or irradiance ) of light or other linear waves radiating from a point source (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source, so an object (of the same size) twice as far away receives only one-quarter the energy (in the same time period).
More generally, the irradiance, i.e., the intensity (or power per unit area in the direction of propagation ), of a spherical wavefront varies inversely with the square of the distance from the source (assuming there are no losses caused by absorption or scattering ).
For example, the intensity of radiation from the Sun is 9126 watts per square meter at the distance of Mercury (0.387 AU ); but only 1367 watts per square meter at the distance of Earth (1 AU)—an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.
For non- isotropic radiators such as parabolic antennas , headlights, and lasers , the effective origin is located far behind the beam aperture. If you are close to the origin, you don't have to go far to double the radius, so the signal drops quickly. When you are far from the origin and still have a strong signal, like with a laser, you have to travel very far to double the radius and reduce the signal. This means you have a stronger signal or have antenna gain in the direction of the narrow beam relative to a wide beam in all directions of an isotropic antenna .
In photography and stage lighting , the inverse-square law is used to determine the “fall off” or the difference in illumination on a subject as it moves closer to or further from the light source. For quick approximations, it is enough to remember that doubling the distance reduces illumination to one quarter; [ 10 ] or similarly, to halve the illumination increase the distance by a factor of 1.4 (the square root of 2 ), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%. [ 11 ]
The fractional reduction in electromagnetic fluence (Φ) for indirectly ionizing radiation with increasing distance from a point source can be calculated using the inverse-square law. Since emissions from a point source have radial directions, they intercept at a perpendicular incidence. The area of such a shell is 4π r 2 where r is the radial distance from the center. The law is particularly important in diagnostic radiography and radiotherapy treatment planning, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance. As stated in Fourier theory of heat “as the point source is magnification by distances, its radiation is dilute proportional to the sin of the angle, of the increasing circumference arc from the point of origin”.
Let P be the total power radiated from a point source (for example, an omnidirectional isotropic radiator ). At large distances from the source (compared to the size of the source), this power is distributed over larger and larger spherical surfaces as the distance from the source increases. Since the surface area of a sphere of radius r is A = 4 πr 2 , the intensity I (power per unit area) of radiation at distance r is I = P A = P 4 π r 2 . {\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
The energy or intensity decreases (divided by 4) as the distance r is doubled; if measured in dB would decrease by 6.02 dB per doubling of distance. When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference value.
In acoustics , the sound pressure of a spherical wavefront radiating from a point source decreases by 50% as the distance r is doubled; measured in dB , the decrease is still 6.02 dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):
p ∝ 1 r {\displaystyle p\ \propto \ {\frac {1}{r}}\,}
The same is true for the component of particle velocity v {\displaystyle v\,} that is in-phase with the instantaneous sound pressure p {\displaystyle p\,} :
v ∝ 1 r {\displaystyle v\ \propto {\frac {1}{r}}\ \,}
In the near field is a quadrature component of the particle velocity that is 90° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound. The sound intensity is the product of the RMS sound pressure and the in-phase component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:
I = p v ∝ 1 r 2 . {\displaystyle I\ =\ pv\ \propto \ {\frac {1}{r^{2}}}.\,}
For an irrotational vector field in three-dimensional space, the inverse-square law corresponds to the property that the divergence is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in n -dimensional Euclidean space , the intensity "I" of the vector field falls off with the distance "r" following the inverse ( n − 1) th power law
I ∝ 1 r n − 1 , {\displaystyle I\propto {\frac {1}{r^{n-1}}},}
given that the space outside the source is divergence free. [ citation needed ]
The inverse-square law, fundamental in Euclidean spaces, also applies to non-Euclidean geometries , including hyperbolic space . The curvature present in these spaces alters physical laws, influencing a variety of fields such as cosmology , general relativity , and string theory . [ 12 ]
John D. Barrow , in his 2020 paper "Non-Euclidean Newtonian Cosmology," expands on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He explains that F and Φ obey the relationships F ∝ 1 / R² sinh²(r/R) and Φ ∝ coth(r/R), where R represents the curvature radius and r represents the distance from the focal point.
The concept of spatial dimensionality, first proposed by Immanuel Kant, remains a topic of debate concerning the inverse-square law. [ 13 ] Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, contend that the inverse-square law is more closely related to force distribution symmetry than to the dimensionality of space.
In the context of non-Euclidean geometries and general relativity, deviations from the inverse-square law do not arise from the law itself but rather from the assumption that the force between two bodies is instantaneous, which contradicts special relativity . General relativity reinterprets gravity as the curvature of spacetime, leading particles to move along geodesics in this curved spacetime. [ 14 ]
John Dumbleton of the 14th-century Oxford Calculators , was one of the first to express functional relationships in graphical form. He gave a proof of the mean speed theorem stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitative decrease in intensity of illumination in his Summa logicæ et philosophiæ naturalis (ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law. [ 15 ]
In proposition 9 of Book 1 in his book Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur (1604), the astronomer Johannes Kepler argued that the spreading of light from a point source obeys an inverse square law: [ 16 ] [ 17 ]
Sicut se habent spharicae superificies, quibus origo lucis pro centro est, amplior ad angustiorem: ita se habet fortitudo seu densitas lucis radiorum in angustiori, ad illamin in laxiori sphaerica, hoc est, conversim. Nam per 6. 7. tantundem lucis est in angustiori sphaerica superficie, quantum in fusiore, tanto ergo illie stipatior & densior quam hic.
Just as [the ratio of] spherical surfaces, for which the source of light is the center, [is] from the wider to the narrower, so the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely. For according to [propositions] 6 & 7, there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there.
In 1645, in his book Astronomia Philolaica ..., the French astronomer Ismaël Bullialdus (1605–1694) refuted Johannes Kepler's suggestion that "gravity" [ 18 ] weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance: [ 19 ] [ 20 ]
Virtus autem illa, qua Sol prehendit seu harpagat planetas, corporalis quae ipsi pro manibus est, lineis rectis in omnem mundi amplitudinem emissa quasi species solis cum illius corpore rotatur: cum ergo sit corporalis imminuitur, & extenuatur in maiori spatio & intervallo, ratio autem huius imminutionis eadem est, ac luminus, in ratione nempe dupla intervallorum, sed eversa.
As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances [that is, 1/d²].
In England, the Anglican bishop Seth Ward (1617–1689) publicized the ideas of Bullialdus in his critique In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis (1653) and publicized the planetary astronomy of Kepler in his book Astronomia geometrica (1656).
In 1663–1664, the English scientist Robert Hooke was writing his book Micrographia (1666) in which he discussed, among other things, the relation between the height of the atmosphere and the barometric pressure at the surface. Since the atmosphere surrounds the Earth, which itself is a sphere, the volume of atmosphere bearing on any unit area of the Earth's surface is a truncated cone (which extends from the Earth's center to the vacuum of space; obviously only the section of the cone from the Earth's surface to space bears on the Earth's surface). Although the volume of a cone is proportional to the cube of its height, Hooke argued that the air's pressure at the Earth's surface is instead proportional to the height of the atmosphere because gravity diminishes with altitude. Although Hooke did not explicitly state so, the relation that he proposed would be true only if gravity decreases as the inverse square of the distance from the Earth's center. [ 21 ] [ 22 ]
This article incorporates public domain material from Federal Standard 1037C . General Services Administration . Archived from the original on 22 January 2022. | https://en.wikipedia.org/wiki/Inverse-square_law |
In geometry , the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem [ 1 ] or the upside down Pythagorean theorem [ 2 ] ) is as follows: [ 3 ]
This theorem should not be confused with proposition 48 in book 1 of Euclid 's Elements , the converse of the Pythagorean theorem, which states that if the square on one side of a triangle is equal to the sum of the squares on the other two sides then the other two sides contain a right angle.
The area of triangle △ ABC can be expressed in terms of either AC and BC , or AB and CD :
given CD > 0 , AC > 0 and BC > 0 .
Using the Pythagorean theorem ,
as above.
Note in particular:
The cruciform curve or cross curve is a quartic plane curve given by the equation
where the two parameters determining the shape of the curve, a and b are each CD .
Substituting x with AC and y with BC gives
Inverse-Pythagorean triples can be generated using integer parameters t and u as follows. [ 4 ]
If two identical lamps are placed at A and B , the theorem and the inverse-square law imply that the light intensity at C is the same as when a single lamp is placed at D .
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Inverse_Pythagorean_theorem |
The Inverse Symbolic Calculator is an online number checker established July 18, 1995 by Peter Benjamin Borwein , Jonathan Michael Borwein and Simon Plouffe of the Canadian Centre for Experimental and Constructive Mathematics (Burnaby, Canada). A user will input a number and the Calculator will use an algorithm to search for and calculate closed-form expressions or suitable functions that have roots near this number. Hence, the calculator is of great importance for those working in numerical areas of experimental mathematics .
The ISC contains 54 million mathematical constants. Plouffe's Inverter (opened in 1998) contains 214 million. A newer version of the tables with 3.702 billion entries (as of June 19, 2010) exists.
In 2016, Plouffe released a portable version of Plouffe's Inverter containing 3 billion entries. [ 1 ]
This applied mathematics –related article is a stub . You can help Wikipedia by expanding it .
This article about a search engine website is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Inverse_Symbolic_Calculator |
In nuclear and particle physics , inverse beta decay , commonly abbreviated to IBD , [ 1 ] is a nuclear reaction involving an electron antineutrino scattering off a proton , creating a positron and a neutron . This process is commonly used in the detection of electron antineutrinos in neutrino detectors , such as the first detection of antineutrinos in the Cowan–Reines neutrino experiment , or in neutrino experiments such as KamLAND and Borexino . It is an essential process to experiments involving low-energy neutrinos (< 60 MeV ) [ 2 ] such as those studying neutrino oscillation , [ 2 ] reactor neutrinos , sterile neutrinos , and geoneutrinos. [ 3 ]
Inverse beta decay proceeds as [ 2 ] [ 3 ] [ 4 ]
where an electron antineutrino ( ν e ) interacts with a proton ( p ) to produce a positron ( e + ) and a neutron ( n ). The IBD reaction can only be initiated when the antineutrino possesses at least 1.806 MeV [ 3 ] [ 4 ] of kinetic energy (called the threshold energy ). This threshold energy is due to a difference in mass between the products ( e + and n ) and the reactants ( ν e and p ) and also slightly due to a relativistic mass effect on the antineutrino. Most of the antineutrino energy is distributed to the positron due to its small mass relative to the neutron. The positron promptly [ 4 ] undergoes matter–antimatter annihilation after creation and yields a flash of light with energy calculated as [ 5 ]
E vis = 511 keV + 511 keV + E ν ¯ e − 1806 keV = E ν ¯ e − 784 keV {\displaystyle {\begin{aligned}E_{\text{vis}}&=511{\text{ keV}}+511{\text{ keV}}+E_{\rm {\,{\overline {\nu }}_{e}}}-1806{\text{ keV}}\\[2pt]&=E_{\rm {\,{\overline {\nu }}_{e}}}-784{\text{ keV}}\end{aligned}}}
where 511 keV is the electron and positron rest energy , E vis is the visible energy from the reaction, and E ν ¯ e {\displaystyle E_{\rm {\,{\overline {\nu }}_{e}}}} is the antineutrino kinetic energy . After the prompt positron annihilation , the neutron undergoes neutron capture on an element in the detector, producing a delayed flash of 2.22 MeV if captured on a proton. [ 4 ] The timing of the delayed capture is 200–300 microseconds after IBD initiation ( ≈256 μs in the Borexino detector [ 4 ] ). The timing and spatial coincidence between the prompt positron annihilation and delayed neutron capture provides a clear IBD signature in neutrino detectors , allowing for discrimination from background. [ 4 ] The IBD cross section is dependent on antineutrino energy and capturing element, although is generally on the order of 10 −44 cm 2 (~ attobarns ). [ 6 ]
Another kind of inverse beta decay is the reaction
The Homestake experiment used the reaction
to detect solar neutrinos.
During the formation of neutron stars , or in radioactive isotopes capable of electron capture , neutrons are created by electron capture:
This is similar to the inverse beta reaction in that a proton is changed to a neutron, but is induced by the capture of an electron instead of an antineutrino. | https://en.wikipedia.org/wiki/Inverse_beta_decay |
A chord (from the Latin chorda , meaning " bowstring ") of a circle is a straight line segment whose endpoints both lie on a circular arc . If a chord were to be extended infinitely on both directions into a line , the object is a secant line . The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow").
More generally, a chord is a line segment joining two points on any curve , for instance, on an ellipse . A chord that passes through a circle's center point is the circle's diameter .
Among properties of chords of a circle are the following:
The midpoints of a set of parallel chords of a conic are collinear ( midpoint theorem for conics ). [ 1 ]
Chords were used extensively in the early development of trigonometry . The first known trigonometric table, compiled by Hipparchus in the 2nd century BC, is no longer extant but tabulated the value of the chord function for every 7 + 1 / 2 degrees . In the 2nd century AD, Ptolemy compiled a more extensive table of chords in his book on astronomy , giving the value of the chord for angles ranging from 1 / 2 to 180 degrees by increments of 1 / 2 degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part. [ 2 ]
The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle . The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be ( cos θ , sin θ ), and then using the Pythagorean theorem to calculate the chord length: [ 2 ]
The last step uses the half-angle formula . Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably, a great deal was known about them. In the table below (where c is the chord length, and D the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones:
The inverse function exists as well: [ 4 ] | https://en.wikipedia.org/wiki/Inverse_chord |
The external secant function (abbreviated exsecant , symbolized exsec ) is a trigonometric function defined in terms of the secant function:
exsec θ = sec θ − 1 = 1 cos θ − 1. {\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}
It was introduced in 1855 by American civil engineer Charles Haslett , who used it in conjunction with the existing versine function, vers θ = 1 − cos θ , {\displaystyle \operatorname {vers} \theta =1-\cos \theta ,} for designing and measuring circular sections of railroad track. [ 3 ] It was adopted by surveyors and civil engineers in the United States for railroad and road design , and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. [ 4 ] For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc ), coexsec θ = {\displaystyle \operatorname {coexsec} \theta ={}} csc θ − 1 , {\displaystyle \csc \theta -1,} the exsecant of the complementary angle , [ 5 ] [ 6 ] though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest. [ 7 ]
As a line segment , an external secant of a circle has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of the central angle between the segment's inner endpoint and the point of tangency for a line through the outer endpoint and tangent to the circle.
The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements , as used e.g. in the intersecting secants theorem . 18th century sources in Latin called any non- tangential line segment external to a circle with one endpoint on the circumference a secans exterior . [ 8 ]
The trigonometric secant , named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under the name secant . [ 9 ]
In the 19th century, most railroad tracks were constructed out of arcs of circles , called simple curves . [ 10 ] Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups. [ 11 ]
The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, R exsec 1 2 Δ . {\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .} [ 12 ] By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to the track [ 13 ] – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, R vers 1 2 Δ . {\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .} These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. [ 3 ] The same idea was adopted by other authors, such as Searles (1880). [ 14 ] By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants". [ 15 ]
In the late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines. [ 15 ] [ 16 ]
Solving the same types of problems is required when surveying circular sections of canals [ 17 ] and roads, and the exsecant was still used in mid-20th century books about road surveying. [ 18 ]
The exsecant has sometimes been used for other applications, such as beam theory [ 19 ] and depth sounding with a wire. [ 20 ]
In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. [ 21 ] Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries ), [ 22 ] and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.
Naïvely evaluating the expressions 1 − cos θ {\displaystyle 1-\cos \theta } (versine) and sec θ − 1 {\displaystyle \sec \theta -1} (exsecant) is problematic for small angles where sec θ ≈ cos θ ≈ 1. {\displaystyle \sec \theta \approx \cos \theta \approx 1.} Computing the difference between two approximately equal quantities results in catastrophic cancellation : because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.
For example, the secant of 1° is approximately 1.000 152 , with the leading several digits wasted on zeros, while the common logarithm of the exsecant of 1° is approximately −3.817 220 , [ 23 ] all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place trigonometric table and then subtracting 1 , the difference sec 1° − 1 ≈ 0.000 152 has only 3 significant digits , and after computing the logarithm only three digits are correct, log(sec 1° − 1) ≈ −3.81 8 156 . [ 24 ] For even smaller angles loss of precision is worse.
If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as exsec θ = tan θ tan 1 2 θ | , {\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},} or using versine, exsec θ = vers θ sec θ , {\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,} which can itself be computed as vers θ = 2 ( sin 1 2 θ ) ) 2 | = {\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}} sin θ tan 1 2 θ | {\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}} ; Haslett used these identities to compute his 1855 exsecant and versine tables. [ 25 ] [ 26 ]
For a sufficiently small angle, a circular arc is approximately shaped like a parabola , and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength. [ 27 ]
The inverse of the exsecant function, which might be symbolized arcexsec , [ 6 ] is well defined if its argument y ≥ 0 {\displaystyle y\geq 0} or y ≤ − 2 {\displaystyle y\leq -2} and can be expressed in terms of other inverse trigonometric functions (using radians for the angle):
arcexsec y = arcsec ( y + 1 ) = { arctan ( y 2 + 2 y ) if y ≥ 0 , undefined if − 2 < y < 0 , π − arctan ( y 2 + 2 y ) if y ≤ − 2 ; . {\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\[6mu]{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\[4mu]\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}
the arctangent expression is well behaved for small angles. [ 28 ]
While historical uses of the exsecant did not explicitly involve calculus , its derivative and antiderivative (for x in radians) are: [ 29 ]
d d x exsec x = tan x sec x , ∫ exsec x d x = ln | sec x + tan x | − x + C , ∫ | {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\[10mu]\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}}
where ln is the natural logarithm . See also Integral of the secant function .
The exsecant of twice an angle is: [ 6 ]
exsec 2 θ = 2 sin 2 θ 1 − 2 sin 2 θ . {\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.}
Van Sickle, Jenna (2011). "The history of one definition: Teaching trigonometry in the US before 1900" . International Journal for the History of Mathematics Education . 6 (2): 55– 70.
Review: Poor, Henry Varnum , ed. (1856-03-22). " Practical Book of Reference, and Engineer's Field Book . By Charles Haslett" . American Railroad Journal (Review). Second Quarto Series. XII (12): 184. Whole No. 1040, Vol. XXIX.
Zucker, Ruth (1964). "4.3.147: Elementary Transcendental Functions - Circular functions" . In Abramowitz, Milton ; Stegun, Irene A. (eds.). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. p. 78. LCCN 64-60036 .
van Haecht, Joannes (1784). "Articulus III: De secantibus circuli: Corollarium III: [109]". Geometria elementaria et practica: quam in usum auditorum (in Latin). Lovanii, e typographia academica. p. 24, foldout.
Finocchiaro, Maurice A. (2003). "Physical-Mathematical Reasoning: Galileo on the Extruding Power of Terrestrial Rotation". Synthese . 134 ( 1– 2, Logic and Mathematical Reasoning): 217– 244. doi : 10.1023/A:1022143816001 . JSTOR 20117331 .
Searles, William Henry; Ives, Howard Chapin (1915) [1880]. Field Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction (17th ed.). New York: John Wiley & Sons .
Meyer, Carl F. (1969) [1949]. Route Surveying and Design (4th ed.). Scranton, PA: International Textbook Co.
"MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. exsec function, arith.scm lines 61–63 . Retrieved 2024-04-01 .
Review: " Field Manual for Railroad Engineers . By J. C. Nagle" . The Engineer (Review). 84 : 540. 1897-12-03.
"MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. aexsec function, arith.scm lines 65–71 . Retrieved 2024-04-01 . | https://en.wikipedia.org/wiki/Inverse_coexsecant |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_cohavercosine |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_cohaversine |
In mathematics , the inverse trigonometric functions (occasionally also called antitrigonometric , [ 1 ] cyclometric , [ 2 ] or arcus functions [ 3 ] ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, [ 4 ] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. [ 1 ] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [ 5 ] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . [ 6 ]
The notations sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., as introduced by John Herschel in 1813, [ 7 ] [ 8 ] are often used as well in English-language sources, [ 1 ] much more than the also established sin [−1] ( x ) , cos [−1] ( x ) , tan [−1] ( x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin 2 ( x ) (although only sin 2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . [ 9 ]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) −1 = sec( x ) . Nevertheless, certain authors advise against using it, since it is ambiguous. [ 1 ] [ 10 ] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. [ 11 ] Although it is intended to avoid confusion with the reciprocal , which should be represented by sin −1 ( x ) , cos −1 ( x ) , etc., or, better, by sin −1 x , cos −1 x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for the standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), [ 12 ] because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
If x is allowed to be a complex number , then the range of y applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians .
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained.
Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } :
The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } :
The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :}
This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined .
Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} "
The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote [ note 1 ] for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ).
With the help of integer parity Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
And similarly for the secant function,
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same.
We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which is a shorthand way of saying that (at least) one of the following statement is true:
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ).
However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has.
An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
The equations above can be transformed by using the reflection and shift identities: [ 13 ]
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}
where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives the analogous equations for csc , sec , and cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively.
So for example, by using the equality sin ( π 2 − θ ) = cos θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,} the equation cos θ = x {\displaystyle \cos \theta =x} can be transformed into sin ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos θ = x {\displaystyle \;\cos \theta =x\;} is: θ = ( − 1 ) h + 1 arcsin ( x ) + π h + π 2 for some h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin x = π 2 − arccos x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express the right hand side of the above formula in terms of arccos x {\displaystyle \;\arccos x\;} instead of arcsin x . {\displaystyle \;\arcsin x.\;}
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .}
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
A useful form that follows directly from the table above is
It is obtained by recognizing that cos ( arctan ( x ) ) = 1 1 + x 2 = cos ( arccos ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} .
From the half-angle formula , tan ( θ 2 ) = sin ( θ ) 1 + cos ( θ ) {\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}} , we get:
This is derived from the tangent addition formula
by letting
The derivatives for complex values of z are as follows:
Only for real values of x :
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},} so
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals , but still well-defined.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series , as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition above (see Leibniz series ).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos ( x ) = π / 2 − arcsin ( x ) {\displaystyle \arccos(x)=\pi /2-\arcsin(x)} , arccsc ( x ) = arcsin ( 1 / x ) {\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)} , and so on. Another series is given by: [ 14 ]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series :
(The term in the sum for n = 0 is the empty product , so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit . [ 16 ]
Two alternatives to the power series for arctangent are these generalized continued fractions :
The second of these is valid in the cut complex plane. There are two cuts, from − i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2 , with each perfect square appearing once. The first was developed by Leonhard Euler ; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series .
For real and complex values of z :
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions :
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Using ∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du} (i.e. integration by parts ), set
Then
which by the simple substitution w = 1 − x 2 , d w = − 2 x d x {\displaystyle w=1-x^{2},\ dw=-2x\,dx} yields the final result:
Since the inverse trigonometric functions are analytic functions , they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points . One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
These functions may also be expressed using complex logarithms . This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } .
or
Simply taking the imaginary part works for any real-valued a {\displaystyle a} and b {\displaystyle b} , but if a {\displaystyle a} or b {\displaystyle b} is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of ln ( a + b i ) {\displaystyle \ln(a+bi)} also removes c {\displaystyle c} from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z {\displaystyle z} , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ {\displaystyle \theta } that result from plugging the values into the equations θ = − i ln ( a + i b c ) {\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)} above and simplifying.
The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the Im ( ln z ) ∈ ( − π , π ] {\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]} and Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0} principal branch for every function except arccotangent in the θ {\displaystyle \theta } column. Arccotangent in the θ {\displaystyle \theta } column will output on its usual principal branch by using the Im ( ln z ) ∈ [ 0 , 2 π ) {\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )} and Im ( z ) ≥ 0 {\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0} convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions . It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
Using the exponential definition of sine , and letting ξ = e i ϕ , {\displaystyle \xi =e^{i\phi },}
(the positive branch is chosen)
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem : a 2 + b 2 = h 2 {\displaystyle a^{2}+b^{2}=h^{2}} where h {\displaystyle h} is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
The two-argument atan2 function computes the arctangent of y / x given y and x , but with a range of (−π, π] . In other words, atan2( y , x ) is the angle between the positive x -axis of a plane and the point ( x , y ) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0 ), and negative sign for clockwise angles (lower half-plane, y < 0 ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (−π/2, π/2) , it can be expressed as follows:
atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y ≥ 0 , x < 0 arctan ( y x ) − π y < 0 , x < 0 π 2 y > 0 , x = 0 − π 2 y < 0 , x = 0 undefined y = 0 , x = 0 {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}
It also equals the principal value of the argument of the complex number x + iy .
This limited version of the function above may also be defined using the tangent half-angle formulae as follows: atan2 ( y , x ) = 2 arctan ( y x 2 + y 2 + x ) {\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)} provided that either x > 0 or y ≠ 0 . However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order ( y , x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language , but a few authors may use the opposite convention ( x , y ) so some caution is warranted. (See variations at atan2 § Realizations of the function in common computer languages .)
In many applications [ 17 ] the solution y {\displaystyle y} of the equation x = tan ( y ) {\displaystyle x=\tan(y)} is to come as close as possible to a given value − ∞ < η < ∞ {\displaystyle -\infty <\eta <\infty } . The adequate solution is produced by the parameter modified arctangent function
The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer.
For angles near 0 and π , arccosine is ill-conditioned , and similarly with arcsine for angles near − π /2 and π /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. [ 18 ] | https://en.wikipedia.org/wiki/Inverse_cosecant |
In mathematics , the inverse trigonometric functions (occasionally also called antitrigonometric , [ 1 ] cyclometric , [ 2 ] or arcus functions [ 3 ] ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, [ 4 ] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. [ 1 ] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [ 5 ] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . [ 6 ]
The notations sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., as introduced by John Herschel in 1813, [ 7 ] [ 8 ] are often used as well in English-language sources, [ 1 ] much more than the also established sin [−1] ( x ) , cos [−1] ( x ) , tan [−1] ( x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin 2 ( x ) (although only sin 2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . [ 9 ]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) −1 = sec( x ) . Nevertheless, certain authors advise against using it, since it is ambiguous. [ 1 ] [ 10 ] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. [ 11 ] Although it is intended to avoid confusion with the reciprocal , which should be represented by sin −1 ( x ) , cos −1 ( x ) , etc., or, better, by sin −1 x , cos −1 x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for the standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), [ 12 ] because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
If x is allowed to be a complex number , then the range of y applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians .
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained.
Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } :
The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } :
The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :}
This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined .
Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} "
The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote [ note 1 ] for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ).
With the help of integer parity Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
And similarly for the secant function,
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same.
We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which is a shorthand way of saying that (at least) one of the following statement is true:
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ).
However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has.
An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
The equations above can be transformed by using the reflection and shift identities: [ 13 ]
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}
where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives the analogous equations for csc , sec , and cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively.
So for example, by using the equality sin ( π 2 − θ ) = cos θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,} the equation cos θ = x {\displaystyle \cos \theta =x} can be transformed into sin ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos θ = x {\displaystyle \;\cos \theta =x\;} is: θ = ( − 1 ) h + 1 arcsin ( x ) + π h + π 2 for some h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin x = π 2 − arccos x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express the right hand side of the above formula in terms of arccos x {\displaystyle \;\arccos x\;} instead of arcsin x . {\displaystyle \;\arcsin x.\;}
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .}
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
A useful form that follows directly from the table above is
It is obtained by recognizing that cos ( arctan ( x ) ) = 1 1 + x 2 = cos ( arccos ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} .
From the half-angle formula , tan ( θ 2 ) = sin ( θ ) 1 + cos ( θ ) {\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}} , we get:
This is derived from the tangent addition formula
by letting
The derivatives for complex values of z are as follows:
Only for real values of x :
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},} so
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals , but still well-defined.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series , as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition above (see Leibniz series ).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos ( x ) = π / 2 − arcsin ( x ) {\displaystyle \arccos(x)=\pi /2-\arcsin(x)} , arccsc ( x ) = arcsin ( 1 / x ) {\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)} , and so on. Another series is given by: [ 14 ]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series :
(The term in the sum for n = 0 is the empty product , so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit . [ 16 ]
Two alternatives to the power series for arctangent are these generalized continued fractions :
The second of these is valid in the cut complex plane. There are two cuts, from − i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2 , with each perfect square appearing once. The first was developed by Leonhard Euler ; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series .
For real and complex values of z :
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions :
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Using ∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du} (i.e. integration by parts ), set
Then
which by the simple substitution w = 1 − x 2 , d w = − 2 x d x {\displaystyle w=1-x^{2},\ dw=-2x\,dx} yields the final result:
Since the inverse trigonometric functions are analytic functions , they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points . One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
These functions may also be expressed using complex logarithms . This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } .
or
Simply taking the imaginary part works for any real-valued a {\displaystyle a} and b {\displaystyle b} , but if a {\displaystyle a} or b {\displaystyle b} is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of ln ( a + b i ) {\displaystyle \ln(a+bi)} also removes c {\displaystyle c} from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z {\displaystyle z} , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ {\displaystyle \theta } that result from plugging the values into the equations θ = − i ln ( a + i b c ) {\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)} above and simplifying.
The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the Im ( ln z ) ∈ ( − π , π ] {\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]} and Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0} principal branch for every function except arccotangent in the θ {\displaystyle \theta } column. Arccotangent in the θ {\displaystyle \theta } column will output on its usual principal branch by using the Im ( ln z ) ∈ [ 0 , 2 π ) {\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )} and Im ( z ) ≥ 0 {\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0} convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions . It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
Using the exponential definition of sine , and letting ξ = e i ϕ , {\displaystyle \xi =e^{i\phi },}
(the positive branch is chosen)
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem : a 2 + b 2 = h 2 {\displaystyle a^{2}+b^{2}=h^{2}} where h {\displaystyle h} is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
The two-argument atan2 function computes the arctangent of y / x given y and x , but with a range of (−π, π] . In other words, atan2( y , x ) is the angle between the positive x -axis of a plane and the point ( x , y ) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0 ), and negative sign for clockwise angles (lower half-plane, y < 0 ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (−π/2, π/2) , it can be expressed as follows:
atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y ≥ 0 , x < 0 arctan ( y x ) − π y < 0 , x < 0 π 2 y > 0 , x = 0 − π 2 y < 0 , x = 0 undefined y = 0 , x = 0 {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}
It also equals the principal value of the argument of the complex number x + iy .
This limited version of the function above may also be defined using the tangent half-angle formulae as follows: atan2 ( y , x ) = 2 arctan ( y x 2 + y 2 + x ) {\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)} provided that either x > 0 or y ≠ 0 . However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order ( y , x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language , but a few authors may use the opposite convention ( x , y ) so some caution is warranted. (See variations at atan2 § Realizations of the function in common computer languages .)
In many applications [ 17 ] the solution y {\displaystyle y} of the equation x = tan ( y ) {\displaystyle x=\tan(y)} is to come as close as possible to a given value − ∞ < η < ∞ {\displaystyle -\infty <\eta <\infty } . The adequate solution is produced by the parameter modified arctangent function
The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer.
For angles near 0 and π , arccosine is ill-conditioned , and similarly with arcsine for angles near − π /2 and π /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. [ 18 ] | https://en.wikipedia.org/wiki/Inverse_cosine |
In mathematics , the inverse trigonometric functions (occasionally also called antitrigonometric , [ 1 ] cyclometric , [ 2 ] or arcus functions [ 3 ] ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, [ 4 ] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. [ 1 ] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [ 5 ] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . [ 6 ]
The notations sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., as introduced by John Herschel in 1813, [ 7 ] [ 8 ] are often used as well in English-language sources, [ 1 ] much more than the also established sin [−1] ( x ) , cos [−1] ( x ) , tan [−1] ( x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin 2 ( x ) (although only sin 2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . [ 9 ]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) −1 = sec( x ) . Nevertheless, certain authors advise against using it, since it is ambiguous. [ 1 ] [ 10 ] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. [ 11 ] Although it is intended to avoid confusion with the reciprocal , which should be represented by sin −1 ( x ) , cos −1 ( x ) , etc., or, better, by sin −1 x , cos −1 x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for the standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), [ 12 ] because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
If x is allowed to be a complex number , then the range of y applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians .
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained.
Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } :
The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } :
The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :}
This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined .
Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} "
The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote [ note 1 ] for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ).
With the help of integer parity Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
And similarly for the secant function,
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same.
We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which is a shorthand way of saying that (at least) one of the following statement is true:
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ).
However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has.
An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
The equations above can be transformed by using the reflection and shift identities: [ 13 ]
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}
where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives the analogous equations for csc , sec , and cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively.
So for example, by using the equality sin ( π 2 − θ ) = cos θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,} the equation cos θ = x {\displaystyle \cos \theta =x} can be transformed into sin ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos θ = x {\displaystyle \;\cos \theta =x\;} is: θ = ( − 1 ) h + 1 arcsin ( x ) + π h + π 2 for some h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin x = π 2 − arccos x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express the right hand side of the above formula in terms of arccos x {\displaystyle \;\arccos x\;} instead of arcsin x . {\displaystyle \;\arcsin x.\;}
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .}
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
A useful form that follows directly from the table above is
It is obtained by recognizing that cos ( arctan ( x ) ) = 1 1 + x 2 = cos ( arccos ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} .
From the half-angle formula , tan ( θ 2 ) = sin ( θ ) 1 + cos ( θ ) {\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}} , we get:
This is derived from the tangent addition formula
by letting
The derivatives for complex values of z are as follows:
Only for real values of x :
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},} so
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals , but still well-defined.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series , as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition above (see Leibniz series ).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos ( x ) = π / 2 − arcsin ( x ) {\displaystyle \arccos(x)=\pi /2-\arcsin(x)} , arccsc ( x ) = arcsin ( 1 / x ) {\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)} , and so on. Another series is given by: [ 14 ]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series :
(The term in the sum for n = 0 is the empty product , so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit . [ 16 ]
Two alternatives to the power series for arctangent are these generalized continued fractions :
The second of these is valid in the cut complex plane. There are two cuts, from − i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2 , with each perfect square appearing once. The first was developed by Leonhard Euler ; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series .
For real and complex values of z :
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions :
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Using ∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du} (i.e. integration by parts ), set
Then
which by the simple substitution w = 1 − x 2 , d w = − 2 x d x {\displaystyle w=1-x^{2},\ dw=-2x\,dx} yields the final result:
Since the inverse trigonometric functions are analytic functions , they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points . One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
These functions may also be expressed using complex logarithms . This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } .
or
Simply taking the imaginary part works for any real-valued a {\displaystyle a} and b {\displaystyle b} , but if a {\displaystyle a} or b {\displaystyle b} is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of ln ( a + b i ) {\displaystyle \ln(a+bi)} also removes c {\displaystyle c} from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z {\displaystyle z} , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ {\displaystyle \theta } that result from plugging the values into the equations θ = − i ln ( a + i b c ) {\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)} above and simplifying.
The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the Im ( ln z ) ∈ ( − π , π ] {\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]} and Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0} principal branch for every function except arccotangent in the θ {\displaystyle \theta } column. Arccotangent in the θ {\displaystyle \theta } column will output on its usual principal branch by using the Im ( ln z ) ∈ [ 0 , 2 π ) {\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )} and Im ( z ) ≥ 0 {\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0} convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions . It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
Using the exponential definition of sine , and letting ξ = e i ϕ , {\displaystyle \xi =e^{i\phi },}
(the positive branch is chosen)
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem : a 2 + b 2 = h 2 {\displaystyle a^{2}+b^{2}=h^{2}} where h {\displaystyle h} is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
The two-argument atan2 function computes the arctangent of y / x given y and x , but with a range of (−π, π] . In other words, atan2( y , x ) is the angle between the positive x -axis of a plane and the point ( x , y ) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0 ), and negative sign for clockwise angles (lower half-plane, y < 0 ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (−π/2, π/2) , it can be expressed as follows:
atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y ≥ 0 , x < 0 arctan ( y x ) − π y < 0 , x < 0 π 2 y > 0 , x = 0 − π 2 y < 0 , x = 0 undefined y = 0 , x = 0 {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}
It also equals the principal value of the argument of the complex number x + iy .
This limited version of the function above may also be defined using the tangent half-angle formulae as follows: atan2 ( y , x ) = 2 arctan ( y x 2 + y 2 + x ) {\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)} provided that either x > 0 or y ≠ 0 . However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order ( y , x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language , but a few authors may use the opposite convention ( x , y ) so some caution is warranted. (See variations at atan2 § Realizations of the function in common computer languages .)
In many applications [ 17 ] the solution y {\displaystyle y} of the equation x = tan ( y ) {\displaystyle x=\tan(y)} is to come as close as possible to a given value − ∞ < η < ∞ {\displaystyle -\infty <\eta <\infty } . The adequate solution is produced by the parameter modified arctangent function
The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer.
For angles near 0 and π , arccosine is ill-conditioned , and similarly with arcsine for angles near − π /2 and π /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. [ 18 ] | https://en.wikipedia.org/wiki/Inverse_cotangent |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_covercosine |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_coversine |
The inverse electron demand Diels–Alder reaction , or DA INV or IEDDA [ 1 ] is an organic chemical reaction, in which two new chemical bonds and a six-membered ring are formed. It is related to the Diels–Alder reaction , but unlike the Diels–Alder (or DA) reaction, the DA INV is a cycloaddition between an electron-rich dienophile and an electron-poor diene . [ 2 ] During a DA INV reaction, three pi-bonds are broken, and two sigma bonds and one new pi-bond are formed. A prototypical DA INV reaction is shown on the right.
DA INV reactions often involve heteroatoms , and can be used to form heterocyclic compounds . This makes the DA INV reaction particularly useful in natural product syntheses, where the target compounds often contain heterocycles. Recently, the DA INV reaction has been used to synthesize a drug transport system which targets prostate cancer . [ 3 ]
The Diels–Alder reaction was first reported in 1928 by Otto Diels and Kurt Alder ; they were awarded the Nobel Prize in chemistry for their work in 1950. Since that time, use of the Diels–Alder reaction has become widespread. Conversely, DA INV does not have a clear date of inception, and lacks the comparative prominence of the standard Diels-Alder reaction. DA INV does not have a clear date of discovery, because of the difficulty that chemists had in differentiating normal from inverse electron-demand Diels-Alder reactions before the advent of modern computational methods. [ 4 ] Much of the work in this area is attributed to Dale Boger , though other authors have published numerous papers on the subject. [ 2 ] [ 5 ]
The mechanism of the DA INV reaction is controversial. While it is accepted as a formal [4+2] cycloaddition , it is not well understood whether or not the reaction is truly concerted . The accepted view is that most DA INV reactions occur via an asynchronous mechanism. The reaction proceeds via a single transition state, but not all bonds are formed or broken at the same time, as would be the case in a concerted mechanism . [ 2 ]
The formal DA INV mechanism for the reaction of acrolein and methyl vinyl ether is shown in the figure to the right. Though not entirely accurate, it provides a useful model for the reaction. During the course of the reaction, three pi-bonds (labeled with red) are broken, and three new bonds are formed (labeled in blue): two sigma bonds and one new pi-bond . [ 6 ]
Like the standard DA, DA INV reactions proceed via a single boat transition state , despite not being concerted. The single boat transition state is a simplification, but DFT calculations suggest
that the time difference in bond scission and formation is minimal, and that despite potential asynchronicity, the reaction is concerted, with relevant bonds being either partially broken or partially formed at some point during the reaction. [ 7 ] The near synchronicity of the DA INV means it can be treated similarly to the standard Diels-Alder reaction. [ 2 ]
The reaction can be modeled using a closed, boat-like transition state, with all bonds being in the process of forming or breaking at some given point, and therefore must obey the Woodward–Hoffman general selection rules. This means that for a three component, six electron system, all components must interact in a suprafacial manner (or one suprafacial and two antarfacial ). With all components being suprafacial, the allowed transition state is boat-like; a chair-like transition state would result in three two-electron antarafacial components. The chair-like case is thermally disallowed by the Woodward-Hoffman rules. [ 6 ]
In the standard Diels-Alder reaction, there are two components: the diene , which is electron rich, and the dienophile , which is electron poor. The relative electron-richness and electron deficiency of the reactants can best be described visually, in a molecular orbital diagram . In the standard Diels–Alder, the electron rich diene has molecular orbitals that are higher in energy than the orbitals of the electron poor dienophile . This difference in relative orbital energies means that, of the frontier molecular orbitals the HOMO of the diene (HOMO diene ) and the LUMO of the dienophile (LUMO dienophile ) are more similar in energy than the HOMO dienophile and the LUMO diene . [ 2 ] [ 8 ] The strongest orbital interaction is between the most similar frontier molecular orbitals: HOMO diene and LUMO dienophile .
Dimerization reactions are neither normally or inversely accelerated, and are usually low yielding. In this case, two monomers react in a DA fashion. Because the orbital energies are identical, there is no preference for interaction of the HOMO or the LUMO of either the diene or dienophile. The low yield of dimerization reactions is explained by second-order perturbation theory . The LUMO and HOMO of each species are farther apart in energy in a dimerization than in either normally or inversely accelerated Diels–Alder. This means that the orbitals interact less, and there is a lower thermodynamic drive for dimerization. [ 2 ]
In the dimerization reactions, the diene and dienophile were equally electron rich (or equally electron poor). If the diene becomes any less electron rich, or the dienophile any more so, the possible [4+2] cycloaddition reaction will then be a DA INV reaction. In the DA INV reaction, the LUMO diene and HOMO dienophile are closer in energy than the HOMO diene and LUMO dienophile . Thus, the LUMO diene and HOMO dienophile are the frontier orbitals that interact the most strongly, and result in the most energetically favourable bond formation. [ 2 ] [ 7 ] [ 9 ]
Regiochemistry in DA INV reactions can be reliably predicted in many cases. This can be done one of two ways, either by electrostatic (charge) control, or orbital control. [ 2 ] [ 7 ] [ 9 ] To predict the regiochemistry via charge control, one must consider the resonance forms of the reactants. These resonance forms can be used to assign partial charges to each of the atoms. Partially negative atoms on the diene will bond to partially positive atoms on the dienophile, and vice versa.
Predicting the regiochemistry of the reaction via orbital control requires one to calculate the relative orbital coefficients on each atom of the reactants. [ 7 ] The HOMO of the dienophile reacts with the LUMO of the diene. The relative orbital size on each atom is represented by orbital coefficients in the Frontier molecular orbital theory (FMO). Orbitals will align to maximize the bonding interactions, and minimize the anti-bonding interactions.
The Alder–Stein principle states that the stereochemistry of the reactants is maintained in the stereochemistry of the products during a Diels–Alder reaction. This means that groups which were cis in relation to one another in the starting materials will be syn to one another in the product, and groups that were trans to one another in the starting material will be anti in the product.
The Alder–Stein principle has no bearing on the relative orientation of groups on the two starting materials. One cannot predict, via this principle, whether a substituent on the diene will be syn or anti to a substituent on the dienophile. The Alder–Stein principle is only consistent across the self-same starting materials. The relationship is only valid for the groups on the diene alone, or the groups on the dienophile, alone. The relative orientation of groups between the two reactants can be predicted by the endo selection rule .
Similarly to the standard Diels–Alder reaction, the DA INV also obeys a general endo selection rule. In the standard Diels–Alder, it is known that electron withdrawing groups on the dienophile will approach endo, with respect to the diene. The exact cause of this selectivity is still debated, but the most accepted view is that endo approach maximizes secondary orbital overlap. [ 10 ] The DA INV favors an endo orientation of electron donating substituents on the dienophile. Since all Diels–Alder reactions proceed through a boat transition state , there is an "inside" and an "outside" of the transition state (inside and outside the "boat"). The substituents on the dienophile are considered "endo" if they are 'inside' the boat, and "exo" if they are on the outside.
The exo pathway would be favored by sterics, so a different explanation is needed to justify the general predominance of endo products. Frontier molecular orbital theory can be used to explain this outcome. When the substituents of the dienophile are exo, there is no interaction between those substituents and the diene. However, when the dienophile substituents are endo, there is considerable orbital overlap with the diene. In the case of DA INV the overlap of the orbitals of the electron withdrawing substituents with the orbitals of the diene create a favorable bonding interaction , stabilizing the transition state relative to the exo transition state. [ 7 ] The reaction with the lower activation energy will proceed at a greater rate. [ 7 ]
The dienes used in Inverse electron demand Diels-Alder are relatively electron-deficient species; compared to the standard Diels-Alder, where the diene is electron rich. These electron-poor species have lower molecular orbital energies than their standard DA counterparts. This lowered energy results from the inclusion of either: A) electron withdrawing group, or B) electronegative heteroatoms. Aromatic compounds can also react in DA INV reactions, such as triazines and tetrazines . Other common classes of dienes are oxo- and aza - butadienes. [ 9 ] [ 11 ]
The key quality of a good DA INV diene is a significantly lowered HOMO and LUMO, as compared to standard DA dienes. Below is a table showing a few commonly used DA INV dienes, their HOMO and LUMO energies, and some standard DA dienes, along with their respective MO energies. [ 2 ] [ 12 ] [ 13 ] [ 14 ]
The dienophiles used in inverse electron demand Diels-Alder reactions are, unlike in the standard DA, very electron rich, containing one or more electron donating groups . This results in higher orbital energies, and thus more orbital overlap with the LUMO of the diene. Common classes of dieneophiles for DA INV reaction include vinyl ethers and vinyl acetals, imine, enamines, alkynes and highly strained olefins. [ 11 ] [ 14 ]
The most important consideration in choice of dienophile is its relative orbital energies. Both HOMO and LUMO impact the rate and selectivity of the reaction. A table of common DA INV dienophiles, standard DA dienophiles, and their respective MO energies can be seen below. [ 2 ] [ 7 ] [ 12 ]
A second table shows how electron richness in the dienophiles affects the rate of reaction with a very electron poor diene, namely hexachlorocyclopentadiene . The more electron rich the dienophile is, the higher the rate of the reaction will be. This is very clear when comparing the relative rates of reaction for styrene and the less electron rich p-nitrostyrene ; the more electron rich styrene reactions roughly 40% faster than p-nitrostyrene. [ 5 ]
hexachlorocyclopentadiene
DA INV reactions provide a pathway to a rich library of synthetic targets, [ 7 ] [ 11 ] and have been utilized to form many highly functionalized systems, including selectively protected sugars, an important contribution to the field of sugar chemistry. [ 15 ] In addition, DA INV reactions can produce an array of different products from a single starting material, such as tetrazine. [ 2 ] [ 13 ]
DA INV reactions have been utilized for the synthesis of several natural products, including (-)-CC-1065, a parent compound in the Duocarmycin series, which found use as an anticancer treatment. Several drug candidates in this series have progress into clinical trials. The DA INV reaction was used to synthesise the PDE-I and PDE-II sections of (-)-CC-1065. The first reaction in the sequence is a DA INV reaction between the tetrazine and vinyl acetal , followed by a retro-Diels–Alder reaction to afford a 1,2- diazine product. After several more steps, an intramolecular DA INV reaction occurs, followed again by a retro Diels-Alder in situ, to afford an indoline product. This indoline is a converted into either PDE-I or PDE-II in a few synthetic steps.
DA INV reaction between 2,3,4,5-tetrachlorothiophene-1,1-dioxide (diene) and 4,7-dihydroisoindole derivative (dienophile) afforded a new precursor for tetranaphthoporphyrins (TNP) bearing perchlorinated aromatic rings. This precursor can be transformed into corresponding porphyrins by Lewis acid -catalyzed condensation with aromatic aldehydes and further oxidation by DDQ . Polychlorination of the TNP system has a profound favorable effect on its solubility. Heavy aggregation and poor solubility of the parent tetranaphthoporphyrins severely degrade the usefulness of this potentially very valuable porphyrin family. Thus, the observed effect of polychlorination is very welcome. Besides the effect on the solubility, polychlorination also turned out to improve substantially the stability of these compounds towards photooxidation , which has been known to be another serious drawback of tetranaphthoporphyrins. [ 16 ] | https://en.wikipedia.org/wiki/Inverse_electron-demand_Diels–Alder_reaction |
The external secant function (abbreviated exsecant , symbolized exsec ) is a trigonometric function defined in terms of the secant function:
exsec θ = sec θ − 1 = 1 cos θ − 1. {\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}
It was introduced in 1855 by American civil engineer Charles Haslett , who used it in conjunction with the existing versine function, vers θ = 1 − cos θ , {\displaystyle \operatorname {vers} \theta =1-\cos \theta ,} for designing and measuring circular sections of railroad track. [ 3 ] It was adopted by surveyors and civil engineers in the United States for railroad and road design , and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. [ 4 ] For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc ), coexsec θ = {\displaystyle \operatorname {coexsec} \theta ={}} csc θ − 1 , {\displaystyle \csc \theta -1,} the exsecant of the complementary angle , [ 5 ] [ 6 ] though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest. [ 7 ]
As a line segment , an external secant of a circle has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of the central angle between the segment's inner endpoint and the point of tangency for a line through the outer endpoint and tangent to the circle.
The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements , as used e.g. in the intersecting secants theorem . 18th century sources in Latin called any non- tangential line segment external to a circle with one endpoint on the circumference a secans exterior . [ 8 ]
The trigonometric secant , named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under the name secant . [ 9 ]
In the 19th century, most railroad tracks were constructed out of arcs of circles , called simple curves . [ 10 ] Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups. [ 11 ]
The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, R exsec 1 2 Δ . {\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .} [ 12 ] By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to the track [ 13 ] – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, R vers 1 2 Δ . {\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .} These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. [ 3 ] The same idea was adopted by other authors, such as Searles (1880). [ 14 ] By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants". [ 15 ]
In the late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines. [ 15 ] [ 16 ]
Solving the same types of problems is required when surveying circular sections of canals [ 17 ] and roads, and the exsecant was still used in mid-20th century books about road surveying. [ 18 ]
The exsecant has sometimes been used for other applications, such as beam theory [ 19 ] and depth sounding with a wire. [ 20 ]
In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. [ 21 ] Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries ), [ 22 ] and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.
Naïvely evaluating the expressions 1 − cos θ {\displaystyle 1-\cos \theta } (versine) and sec θ − 1 {\displaystyle \sec \theta -1} (exsecant) is problematic for small angles where sec θ ≈ cos θ ≈ 1. {\displaystyle \sec \theta \approx \cos \theta \approx 1.} Computing the difference between two approximately equal quantities results in catastrophic cancellation : because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.
For example, the secant of 1° is approximately 1.000 152 , with the leading several digits wasted on zeros, while the common logarithm of the exsecant of 1° is approximately −3.817 220 , [ 23 ] all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place trigonometric table and then subtracting 1 , the difference sec 1° − 1 ≈ 0.000 152 has only 3 significant digits , and after computing the logarithm only three digits are correct, log(sec 1° − 1) ≈ −3.81 8 156 . [ 24 ] For even smaller angles loss of precision is worse.
If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as exsec θ = tan θ tan 1 2 θ | , {\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},} or using versine, exsec θ = vers θ sec θ , {\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,} which can itself be computed as vers θ = 2 ( sin 1 2 θ ) ) 2 | = {\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}} sin θ tan 1 2 θ | {\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}} ; Haslett used these identities to compute his 1855 exsecant and versine tables. [ 25 ] [ 26 ]
For a sufficiently small angle, a circular arc is approximately shaped like a parabola , and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength. [ 27 ]
The inverse of the exsecant function, which might be symbolized arcexsec , [ 6 ] is well defined if its argument y ≥ 0 {\displaystyle y\geq 0} or y ≤ − 2 {\displaystyle y\leq -2} and can be expressed in terms of other inverse trigonometric functions (using radians for the angle):
arcexsec y = arcsec ( y + 1 ) = { arctan ( y 2 + 2 y ) if y ≥ 0 , undefined if − 2 < y < 0 , π − arctan ( y 2 + 2 y ) if y ≤ − 2 ; . {\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\[6mu]{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\[4mu]\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}
the arctangent expression is well behaved for small angles. [ 28 ]
While historical uses of the exsecant did not explicitly involve calculus , its derivative and antiderivative (for x in radians) are: [ 29 ]
d d x exsec x = tan x sec x , ∫ exsec x d x = ln | sec x + tan x | − x + C , ∫ | {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\[10mu]\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}}
where ln is the natural logarithm . See also Integral of the secant function .
The exsecant of twice an angle is: [ 6 ]
exsec 2 θ = 2 sin 2 θ 1 − 2 sin 2 θ . {\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.}
Van Sickle, Jenna (2011). "The history of one definition: Teaching trigonometry in the US before 1900" . International Journal for the History of Mathematics Education . 6 (2): 55– 70.
Review: Poor, Henry Varnum , ed. (1856-03-22). " Practical Book of Reference, and Engineer's Field Book . By Charles Haslett" . American Railroad Journal (Review). Second Quarto Series. XII (12): 184. Whole No. 1040, Vol. XXIX.
Zucker, Ruth (1964). "4.3.147: Elementary Transcendental Functions - Circular functions" . In Abramowitz, Milton ; Stegun, Irene A. (eds.). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. p. 78. LCCN 64-60036 .
van Haecht, Joannes (1784). "Articulus III: De secantibus circuli: Corollarium III: [109]". Geometria elementaria et practica: quam in usum auditorum (in Latin). Lovanii, e typographia academica. p. 24, foldout.
Finocchiaro, Maurice A. (2003). "Physical-Mathematical Reasoning: Galileo on the Extruding Power of Terrestrial Rotation". Synthese . 134 ( 1– 2, Logic and Mathematical Reasoning): 217– 244. doi : 10.1023/A:1022143816001 . JSTOR 20117331 .
Searles, William Henry; Ives, Howard Chapin (1915) [1880]. Field Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction (17th ed.). New York: John Wiley & Sons .
Meyer, Carl F. (1969) [1949]. Route Surveying and Design (4th ed.). Scranton, PA: International Textbook Co.
"MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. exsec function, arith.scm lines 61–63 . Retrieved 2024-04-01 .
Review: " Field Manual for Railroad Engineers . By J. C. Nagle" . The Engineer (Review). 84 : 540. 1897-12-03.
"MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. aexsec function, arith.scm lines 65–71 . Retrieved 2024-04-01 . | https://en.wikipedia.org/wiki/Inverse_excosecant |
The external secant function (abbreviated exsecant , symbolized exsec ) is a trigonometric function defined in terms of the secant function:
exsec θ = sec θ − 1 = 1 cos θ − 1. {\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}
It was introduced in 1855 by American civil engineer Charles Haslett , who used it in conjunction with the existing versine function, vers θ = 1 − cos θ , {\displaystyle \operatorname {vers} \theta =1-\cos \theta ,} for designing and measuring circular sections of railroad track. [ 3 ] It was adopted by surveyors and civil engineers in the United States for railroad and road design , and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. [ 4 ] For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc ), coexsec θ = {\displaystyle \operatorname {coexsec} \theta ={}} csc θ − 1 , {\displaystyle \csc \theta -1,} the exsecant of the complementary angle , [ 5 ] [ 6 ] though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest. [ 7 ]
As a line segment , an external secant of a circle has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of the central angle between the segment's inner endpoint and the point of tangency for a line through the outer endpoint and tangent to the circle.
The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements , as used e.g. in the intersecting secants theorem . 18th century sources in Latin called any non- tangential line segment external to a circle with one endpoint on the circumference a secans exterior . [ 8 ]
The trigonometric secant , named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under the name secant . [ 9 ]
In the 19th century, most railroad tracks were constructed out of arcs of circles , called simple curves . [ 10 ] Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups. [ 11 ]
The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, R exsec 1 2 Δ . {\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .} [ 12 ] By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to the track [ 13 ] – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, R vers 1 2 Δ . {\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .} These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. [ 3 ] The same idea was adopted by other authors, such as Searles (1880). [ 14 ] By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants". [ 15 ]
In the late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines. [ 15 ] [ 16 ]
Solving the same types of problems is required when surveying circular sections of canals [ 17 ] and roads, and the exsecant was still used in mid-20th century books about road surveying. [ 18 ]
The exsecant has sometimes been used for other applications, such as beam theory [ 19 ] and depth sounding with a wire. [ 20 ]
In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. [ 21 ] Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries ), [ 22 ] and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.
Naïvely evaluating the expressions 1 − cos θ {\displaystyle 1-\cos \theta } (versine) and sec θ − 1 {\displaystyle \sec \theta -1} (exsecant) is problematic for small angles where sec θ ≈ cos θ ≈ 1. {\displaystyle \sec \theta \approx \cos \theta \approx 1.} Computing the difference between two approximately equal quantities results in catastrophic cancellation : because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.
For example, the secant of 1° is approximately 1.000 152 , with the leading several digits wasted on zeros, while the common logarithm of the exsecant of 1° is approximately −3.817 220 , [ 23 ] all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place trigonometric table and then subtracting 1 , the difference sec 1° − 1 ≈ 0.000 152 has only 3 significant digits , and after computing the logarithm only three digits are correct, log(sec 1° − 1) ≈ −3.81 8 156 . [ 24 ] For even smaller angles loss of precision is worse.
If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as exsec θ = tan θ tan 1 2 θ | , {\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},} or using versine, exsec θ = vers θ sec θ , {\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,} which can itself be computed as vers θ = 2 ( sin 1 2 θ ) ) 2 | = {\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}} sin θ tan 1 2 θ | {\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}} ; Haslett used these identities to compute his 1855 exsecant and versine tables. [ 25 ] [ 26 ]
For a sufficiently small angle, a circular arc is approximately shaped like a parabola , and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength. [ 27 ]
The inverse of the exsecant function, which might be symbolized arcexsec , [ 6 ] is well defined if its argument y ≥ 0 {\displaystyle y\geq 0} or y ≤ − 2 {\displaystyle y\leq -2} and can be expressed in terms of other inverse trigonometric functions (using radians for the angle):
arcexsec y = arcsec ( y + 1 ) = { arctan ( y 2 + 2 y ) if y ≥ 0 , undefined if − 2 < y < 0 , π − arctan ( y 2 + 2 y ) if y ≤ − 2 ; . {\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\[6mu]{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\[4mu]\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}
the arctangent expression is well behaved for small angles. [ 28 ]
While historical uses of the exsecant did not explicitly involve calculus , its derivative and antiderivative (for x in radians) are: [ 29 ]
d d x exsec x = tan x sec x , ∫ exsec x d x = ln | sec x + tan x | − x + C , ∫ | {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\[10mu]\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}}
where ln is the natural logarithm . See also Integral of the secant function .
The exsecant of twice an angle is: [ 6 ]
exsec 2 θ = 2 sin 2 θ 1 − 2 sin 2 θ . {\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.}
Van Sickle, Jenna (2011). "The history of one definition: Teaching trigonometry in the US before 1900" . International Journal for the History of Mathematics Education . 6 (2): 55– 70.
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"MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. aexsec function, arith.scm lines 65–71 . Retrieved 2024-04-01 . | https://en.wikipedia.org/wiki/Inverse_exsecant |
In calculus , the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f . More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ( x ) = y {\displaystyle f(x)=y} , then the inverse function rule is, in Lagrange's notation ,
This formula holds in general whenever f {\displaystyle f} is continuous and injective on an interval I , with f {\displaystyle f} being differentiable at f − 1 ( y ) {\displaystyle f^{-1}(y)} ( ∈ I {\displaystyle \in I} ) and where f ′ ( f − 1 ( y ) ) ≠ 0 {\displaystyle f'(f^{-1}(y))\neq 0} . The same formula is also equivalent to the expression
where D {\displaystyle {\mathcal {D}}} denotes the unary derivative operator (on the space of functions) and ∘ {\displaystyle \circ } denotes function composition .
Geometrically, a function and inverse function have graphs that are reflections , in the line y = x {\displaystyle y=x} . This reflection operation turns the gradient of any line into its reciprocal . [ 1 ]
Assuming that f {\displaystyle f} has an inverse in a neighbourhood of x {\displaystyle x} and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at x {\displaystyle x} and have a derivative given by the above formula.
The inverse function rule may also be expressed in Leibniz's notation . As that notation suggests,
This relation is obtained by differentiating the equation f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} in terms of x and applying the chain rule , yielding that:
considering that the derivative of x with respect to x is 1.
Let f {\displaystyle f} be an invertible (bijective) function, let x {\displaystyle x} be in the domain of f {\displaystyle f} , and let y = f ( x ) . {\displaystyle y=f(x).} Let g = f − 1 . {\displaystyle g=f^{-1}.} So, f ( g ( y ) ) = y . {\displaystyle f(g(y))=y.} Derivating this equation with respect to y {\displaystyle y} , and using the chain rule , one gets
That is,
or
At x = 0 {\displaystyle x=0} , however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.
Let z = f ′ ( x ) {\displaystyle z=f'(x)} then we have, assuming f ″ ( x ) ≠ 0 {\displaystyle f''(x)\neq 0} : d ( f ′ ) − 1 ( z ) d z = 1 f ″ ( x ) {\displaystyle {\frac {d(f')^{-1}(z)}{dz}}={\frac {1}{f''(x)}}} This can be shown using the previous notation y = f ( x ) {\displaystyle y=f(x)} . Then we have:
By induction, we can generalize this result for any integer n ≥ 1 {\displaystyle n\geq 1} , with z = f ( n ) ( x ) {\displaystyle z=f^{(n)}(x)} , the nth derivative of f(x), and y = f ( n − 1 ) ( x ) {\displaystyle y=f^{(n-1)}(x)} , assuming f ( i ) ( x ) ≠ 0 for 0 < i ≤ n + 1 {\displaystyle f^{(i)}(x)\neq 0{\text{ for }}0<i\leq n+1} :
The chain rule given above is obtained by differentiating the identity f − 1 ( f ( x ) ) = x {\displaystyle f^{-1}(f(x))=x} with respect to x . One can continue the same process for higher derivatives. Differentiating the identity twice with respect to x , one obtains
that is simplified further by the chain rule as
Replacing the first derivative, using the identity obtained earlier, we get
Similarly for the third derivative:
or using the formula for the second derivative,
These formulas are generalized by the Faà di Bruno's formula .
These formulas can also be written using Lagrange's notation. If f and g are inverses, then
so that
which agrees with the direct calculation. | https://en.wikipedia.org/wiki/Inverse_function_rule |
In mathematics , the inverse function theorem is a theorem that asserts that, if a real function f has a continuous derivative near a point where its derivative is nonzero, then, near this point, f has an inverse function . The inverse function is also differentiable , and the inverse function rule expresses its derivative as the multiplicative inverse of the derivative of f .
The theorem applies verbatim to complex-valued functions of a complex variable . It generalizes to functions from n - tuples (of real or complex numbers) to n -tuples, and to functions between vector spaces of the same finite dimension, by replacing "derivative" with " Jacobian matrix " and "nonzero derivative" with "nonzero Jacobian determinant ".
If the function of the theorem belongs to a higher differentiability class , the same is true for the inverse function. There are also versions of the inverse function theorem for holomorphic functions , for differentiable maps between manifolds , for differentiable functions between Banach spaces , and so forth.
The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem .
For functions of a single variable , the theorem states that if f {\displaystyle f} is a continuously differentiable function with nonzero derivative at the point a {\displaystyle a} ; then f {\displaystyle f} is injective (or bijective onto the image) in a neighborhood of a {\displaystyle a} , the inverse is continuously differentiable near b = f ( a ) {\displaystyle b=f(a)} , and the derivative of the inverse function at b {\displaystyle b} is the reciprocal of the derivative of f {\displaystyle f} at a {\displaystyle a} : ( f − 1 ) ′ ( b ) = 1 f ′ ( a ) = 1 f ′ ( f − 1 ( b ) ) . {\displaystyle {\bigl (}f^{-1}{\bigr )}'(b)={\frac {1}{f'(a)}}={\frac {1}{f'(f^{-1}(b))}}.}
It can happen that a function f {\displaystyle f} may be injective near a point a {\displaystyle a} while f ′ ( a ) = 0 {\displaystyle f'(a)=0} . An example is f ( x ) = ( x − a ) 3 {\displaystyle f(x)=(x-a)^{3}} . In fact, for such a function, the inverse cannot be differentiable at b = f ( a ) {\displaystyle b=f(a)} , since if f − 1 {\displaystyle f^{-1}} were differentiable at b {\displaystyle b} , then, by the chain rule, 1 = ( f − 1 ∘ f ) ′ ( a ) = ( f − 1 ) ′ ( b ) f ′ ( a ) {\displaystyle 1=(f^{-1}\circ f)'(a)=(f^{-1})'(b)f'(a)} , which implies f ′ ( a ) ≠ 0 {\displaystyle f'(a)\neq 0} . (The situation is different for holomorphic functions; see #Holomorphic inverse function theorem below.)
For functions of more than one variable, the theorem states that if f {\displaystyle f} is a continuously differentiable function from an open subset A {\displaystyle A} of R n {\displaystyle \mathbb {R} ^{n}} into R n {\displaystyle \mathbb {R} ^{n}} , and the derivative f ′ ( a ) {\displaystyle f'(a)} is invertible at a point a (that is, the determinant of the Jacobian matrix of f at a is non-zero), then there exist neighborhoods U {\displaystyle U} of a {\displaystyle a} in A {\displaystyle A} and V {\displaystyle V} of b = f ( a ) {\displaystyle b=f(a)} such that f ( U ) ⊂ V {\displaystyle f(U)\subset V} and f : U → V {\displaystyle f:U\to V} is bijective. [ 1 ] Writing f = ( f 1 , … , f n ) {\displaystyle f=(f_{1},\ldots ,f_{n})} , this means that the system of n equations y i = f i ( x 1 , … , x n ) {\displaystyle y_{i}=f_{i}(x_{1},\dots ,x_{n})} has a unique solution for x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} in terms of y 1 , … , y n {\displaystyle y_{1},\dots ,y_{n}} when x ∈ U , y ∈ V {\displaystyle x\in U,y\in V} . Note that the theorem does not say f {\displaystyle f} is bijective onto the image where f ′ {\displaystyle f'} is invertible but that it is locally bijective where f ′ {\displaystyle f'} is invertible.
Moreover, the theorem says that the inverse function f − 1 : V → U {\displaystyle f^{-1}:V\to U} is continuously differentiable, and its derivative at b = f ( a ) {\displaystyle b=f(a)} is the inverse map of f ′ ( a ) {\displaystyle f'(a)} ; i.e.,
In other words, if J f − 1 ( b ) , J f ( a ) {\displaystyle Jf^{-1}(b),Jf(a)} are the Jacobian matrices representing ( f − 1 ) ′ ( b ) , f ′ ( a ) {\displaystyle (f^{-1})'(b),f'(a)} , this means:
The hard part of the theorem is the existence and differentiability of f − 1 {\displaystyle f^{-1}} . Assuming this, the inverse derivative formula follows from the chain rule applied to f − 1 ∘ f = I {\displaystyle f^{-1}\circ f=I} . (Indeed, 1 = I ′ ( a ) = ( f − 1 ∘ f ) ′ ( a ) = ( f − 1 ) ′ ( b ) ∘ f ′ ( a ) . {\displaystyle 1=I'(a)=(f^{-1}\circ f)'(a)=(f^{-1})'(b)\circ f'(a).} ) Since taking the inverse is infinitely differentiable, the formula for the derivative of the inverse shows that if f {\displaystyle f} is continuously k {\displaystyle k} times differentiable, with invertible derivative at the point a , then the inverse is also continuously k {\displaystyle k} times differentiable. Here k {\displaystyle k} is a positive integer or ∞ {\displaystyle \infty } .
There are two variants of the inverse function theorem. [ 1 ] Given a continuously differentiable map f : U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} , the first is
and the second is
In the first case (when f ′ ( a ) {\displaystyle f'(a)} is surjective), the point b = f ( a ) {\displaystyle b=f(a)} is called a regular value . Since m = dim ker ( f ′ ( a ) ) + dim im ( f ′ ( a ) ) {\displaystyle m=\dim \ker(f'(a))+\dim \operatorname {im} (f'(a))} , the first case is equivalent to saying b = f ( a ) {\displaystyle b=f(a)} is not in the image of critical points a {\displaystyle a} (a critical point is a point a {\displaystyle a} such that the kernel of f ′ ( a ) {\displaystyle f'(a)} is nonzero). The statement in the first case is a special case of the submersion theorem .
These variants are restatements of the inverse functions theorem. Indeed, in the first case when f ′ ( a ) {\displaystyle f'(a)} is surjective, we can find an (injective) linear map T {\displaystyle T} such that f ′ ( a ) ∘ T = I {\displaystyle f'(a)\circ T=I} . Define h ( x ) = a + T x {\displaystyle h(x)=a+Tx} so that we have:
Thus, by the inverse function theorem, f ∘ h {\displaystyle f\circ h} has inverse near 0 {\displaystyle 0} ; i.e., f ∘ h ∘ ( f ∘ h ) − 1 = I {\displaystyle f\circ h\circ (f\circ h)^{-1}=I} near b {\displaystyle b} . The second case ( f ′ ( a ) {\displaystyle f'(a)} is injective) is seen in the similar way.
Consider the vector-valued function F : R 2 → R 2 {\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} ^{2}\!} defined by:
The Jacobian matrix of it at ( x , y ) {\displaystyle (x,y)} is:
with the determinant:
The determinant e 2 x {\displaystyle e^{2x}\!} is nonzero everywhere. Thus the theorem guarantees that, for every point p in R 2 {\displaystyle \mathbb {R} ^{2}\!} , there exists a neighborhood about p over which F is invertible. This does not mean F is invertible over its entire domain: in this case F is not even injective since it is periodic: F ( x , y ) = F ( x , y + 2 π ) {\displaystyle F(x,y)=F(x,y+2\pi )\!} .
If one drops the assumption that the derivative is continuous, the function no longer need be invertible. For example f ( x ) = x + 2 x 2 sin ( 1 x ) {\displaystyle f(x)=x+2x^{2}\sin({\tfrac {1}{x}})} and f ( 0 ) = 0 {\displaystyle f(0)=0} has discontinuous derivative f ′ ( x ) = 1 − 2 cos ( 1 x ) + 4 x sin ( 1 x ) {\displaystyle f'\!(x)=1-2\cos({\tfrac {1}{x}})+4x\sin({\tfrac {1}{x}})} and f ′ ( 0 ) = 1 {\displaystyle f'\!(0)=1} , which vanishes arbitrarily close to x = 0 {\displaystyle x=0} . These critical points are local max/min points of f {\displaystyle f} , so f {\displaystyle f} is not one-to-one (and not invertible) on any interval containing x = 0 {\displaystyle x=0} . Intuitively, the slope f ′ ( 0 ) = 1 {\displaystyle f'\!(0)=1} does not propagate to nearby points, where the slopes are governed by a weak but rapid oscillation.
As an important result, the inverse function theorem has been given numerous proofs. The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed-point theorem (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations ). [ 2 ] [ 3 ]
Since the fixed point theorem applies in infinite-dimensional (Banach space) settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem [ 4 ] (see Generalizations below).
An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set . [ 5 ] This approach has an advantage that the proof generalizes to a situation where there is no Cauchy completeness (see § Over a real closed field ).
Yet another proof uses Newton's method , which has the advantage of providing an effective version of the theorem: bounds on the derivative of the function imply an estimate of the size of the neighborhood on which the function is invertible. [ 6 ]
We want to prove the following: Let D ⊆ R {\displaystyle D\subseteq \mathbb {R} } be an open set with x 0 ∈ D , f : D → R {\displaystyle x_{0}\in D,f:D\to \mathbb {R} } a continuously differentiable function defined on D {\displaystyle D} , and suppose that f ′ ( x 0 ) ≠ 0 {\displaystyle f'(x_{0})\neq 0} . Then there exists an open interval I {\displaystyle I} with x 0 ∈ I {\displaystyle x_{0}\in I} such that f {\displaystyle f} maps I {\displaystyle I} bijectively onto the open interval J = f ( I ) {\displaystyle J=f(I)} , and such that the inverse function f − 1 : J → I {\displaystyle f^{-1}:J\to I} is continuously differentiable, and for any y ∈ J {\displaystyle y\in J} , if x ∈ I {\displaystyle x\in I} is such that f ( x ) = y {\displaystyle f(x)=y} , then ( f − 1 ) ′ ( y ) = 1 f ′ ( x ) {\displaystyle (f^{-1})'(y)={\dfrac {1}{f'(x)}}} .
We may without loss of generality assume that f ′ ( x 0 ) > 0 {\displaystyle f'(x_{0})>0} . Given that D {\displaystyle D} is an open set and f ′ {\displaystyle f'} is continuous at x 0 {\displaystyle x_{0}} , there exists r > 0 {\displaystyle r>0} such that ( x 0 − r , x 0 + r ) ⊆ D {\displaystyle (x_{0}-r,x_{0}+r)\subseteq D} and | f ′ ( x ) − f ′ ( x 0 ) | < f ′ ( x 0 ) 2 for all | x − x 0 | < r . {\displaystyle |f'(x)-f'(x_{0})|<{\dfrac {f'(x_{0})}{2}}\qquad {\text{for all }}|x-x_{0}|<r.}
In particular, f ′ ( x ) > f ′ ( x 0 ) 2 > 0 for all | x − x 0 | < r . {\displaystyle f'(x)>{\dfrac {f'(x_{0})}{2}}>0\qquad {\text{for all }}|x-x_{0}|<r.}
This shows that f {\displaystyle f} is strictly increasing for all | x − x 0 | < r {\displaystyle |x-x_{0}|<r} . Let δ > 0 {\displaystyle \delta >0} be such that δ < r {\displaystyle \delta <r} . Then [ x − δ , x + δ ] ⊆ ( x 0 − r , x 0 + r ) {\displaystyle [x-\delta ,x+\delta ]\subseteq (x_{0}-r,x_{0}+r)} . By the intermediate value theorem, we find that f {\displaystyle f} maps the interval [ x − δ , x + δ ] {\displaystyle [x-\delta ,x+\delta ]} bijectively onto [ f ( x − δ ) , f ( x + δ ) ] {\displaystyle [f(x-\delta ),f(x+\delta )]} . Denote by I = ( x − δ , x + δ ) {\displaystyle I=(x-\delta ,x+\delta )} and J = ( f ( x − δ ) , f ( x + δ ) ) {\displaystyle J=(f(x-\delta ),f(x+\delta ))} . Then f : I → J {\displaystyle f:I\to J} is a bijection and the inverse f − 1 : J → I {\displaystyle f^{-1}:J\to I} exists. The fact that f − 1 : J → I {\displaystyle f^{-1}:J\to I} is differentiable follows from the differentiability of f {\displaystyle f} . In particular, the result follows from the fact that if f : I → R {\displaystyle f:I\to \mathbb {R} } is a strictly monotonic and continuous function that is differentiable at x 0 ∈ I {\displaystyle x_{0}\in I} with f ′ ( x 0 ) ≠ 0 {\displaystyle f'(x_{0})\neq 0} , then f − 1 : f ( I ) → R {\displaystyle f^{-1}:f(I)\to \mathbb {R} } is differentiable with ( f − 1 ) ′ ( y 0 ) = 1 f ′ ( y 0 ) {\displaystyle (f^{-1})'(y_{0})={\dfrac {1}{f'(y_{0})}}} , where y 0 = f ( x 0 ) {\displaystyle y_{0}=f(x_{0})} (a standard result in analysis). This completes the proof.
To prove existence, it can be assumed after an affine transformation that f ( 0 ) = 0 {\displaystyle f(0)=0} and f ′ ( 0 ) = I {\displaystyle f^{\prime }(0)=I} , so that a = b = 0 {\displaystyle a=b=0} .
By the mean value theorem for vector-valued functions , for a differentiable function u : [ 0 , 1 ] → R m {\displaystyle u:[0,1]\to \mathbb {R} ^{m}} , ‖ u ( 1 ) − u ( 0 ) ‖ ≤ sup 0 ≤ t ≤ 1 ‖ u ′ ( t ) ‖ {\textstyle \|u(1)-u(0)\|\leq \sup _{0\leq t\leq 1}\|u^{\prime }(t)\|} . Setting u ( t ) = f ( x + t ( x ′ − x ) ) − x − t ( x ′ − x ) {\displaystyle u(t)=f(x+t(x^{\prime }-x))-x-t(x^{\prime }-x)} , it follows that
Now choose δ > 0 {\displaystyle \delta >0} so that ‖ f ′ ( x ) − I ‖ < 1 2 {\textstyle \|f'(x)-I\|<{1 \over 2}} for ‖ x ‖ < δ {\displaystyle \|x\|<\delta } . Suppose that ‖ y ‖ < δ / 2 {\displaystyle \|y\|<\delta /2} and define x n {\displaystyle x_{n}} inductively by x 0 = 0 {\displaystyle x_{0}=0} and x n + 1 = x n + y − f ( x n ) {\displaystyle x_{n+1}=x_{n}+y-f(x_{n})} . The assumptions show that if ‖ x ‖ , ‖ x ′ ‖ < δ {\displaystyle \|x\|,\,\,\|x^{\prime }\|<\delta } then
In particular f ( x ) = f ( x ′ ) {\displaystyle f(x)=f(x^{\prime })} implies x = x ′ {\displaystyle x=x^{\prime }} . In the inductive scheme ‖ x n ‖ < δ {\displaystyle \|x_{n}\|<\delta } and ‖ x n + 1 − x n ‖ < δ / 2 n {\displaystyle \|x_{n+1}-x_{n}\|<\delta /2^{n}} . Thus ( x n ) {\displaystyle (x_{n})} is a Cauchy sequence tending to x {\displaystyle x} . By construction f ( x ) = y {\displaystyle f(x)=y} as required.
To check that g = f − 1 {\displaystyle g=f^{-1}} is C 1 , write g ( y + k ) = x + h {\displaystyle g(y+k)=x+h} so that f ( x + h ) = f ( x ) + k {\displaystyle f(x+h)=f(x)+k} . By the inequalities above, ‖ h − k ‖ < ‖ h ‖ / 2 {\displaystyle \|h-k\|<\|h\|/2} so that ‖ h ‖ / 2 < ‖ k ‖ < 2 ‖ h ‖ {\displaystyle \|h\|/2<\|k\|<2\|h\|} .
On the other hand if A = f ′ ( x ) {\displaystyle A=f^{\prime }(x)} , then ‖ A − I ‖ < 1 / 2 {\displaystyle \|A-I\|<1/2} . Using the geometric series for B = I − A {\displaystyle B=I-A} , it follows that ‖ A − 1 ‖ < 2 {\displaystyle \|A^{-1}\|<2} . But then
tends to 0 as k {\displaystyle k} and h {\displaystyle h} tend to 0, proving that g {\displaystyle g} is C 1 with g ′ ( y ) = f ′ ( g ( y ) ) − 1 {\displaystyle g^{\prime }(y)=f^{\prime }(g(y))^{-1}} .
The proof above is presented for a finite-dimensional space, but applies equally well for Banach spaces . If an invertible function f {\displaystyle f} is C k with k > 1 {\displaystyle k>1} , then so too is its inverse. This follows by induction using the fact that the map F ( A ) = A − 1 {\displaystyle F(A)=A^{-1}} on operators is C k for any k {\displaystyle k} (in the finite-dimensional case this is an elementary fact because the inverse of a matrix is given as the adjugate matrix divided by its determinant ). [ 1 ] [ 7 ] The method of proof here can be found in the books of Henri Cartan , Jean Dieudonné , Serge Lang , Roger Godement and Lars Hörmander .
Here is a proof based on the contraction mapping theorem . Specifically, following T. Tao, [ 8 ] it uses the following consequence of the contraction mapping theorem.
Lemma — Let B ( 0 , r ) {\displaystyle B(0,r)} denote an open ball of radius r in R n {\displaystyle \mathbb {R} ^{n}} with center 0 and g : B ( 0 , r ) → R n {\displaystyle g:B(0,r)\to \mathbb {R} ^{n}} a map with a constant 0 < c < 1 {\displaystyle 0<c<1} such that
for all x , y {\displaystyle x,y} in B ( 0 , r ) {\displaystyle B(0,r)} . Then for f = I + g {\displaystyle f=I+g} on B ( 0 , r ) {\displaystyle B(0,r)} , we have
in particular, f is injective. If, moreover, g ( 0 ) = 0 {\displaystyle g(0)=0} , then
More generally, the statement remains true if R n {\displaystyle \mathbb {R} ^{n}} is replaced by a Banach space. Also, the first part of the lemma is true for any normed space.
Basically, the lemma says that a small perturbation of the identity map by a contraction map is injective and preserves a ball in some sense. Assuming the lemma for a moment, we prove the theorem first. As in the above proof, it is enough to prove the special case when a = 0 , b = f ( a ) = 0 {\displaystyle a=0,b=f(a)=0} and f ′ ( 0 ) = I {\displaystyle f'(0)=I} . Let g = f − I {\displaystyle g=f-I} . The mean value inequality applied to t ↦ g ( x + t ( y − x ) ) {\displaystyle t\mapsto g(x+t(y-x))} says:
Since g ′ ( 0 ) = I − I = 0 {\displaystyle g'(0)=I-I=0} and g ′ {\displaystyle g'} is continuous, we can find an r > 0 {\displaystyle r>0} such that
for all x , y {\displaystyle x,y} in B ( 0 , r ) {\displaystyle B(0,r)} . Then the early lemma says that f = g + I {\displaystyle f=g+I} is injective on B ( 0 , r ) {\displaystyle B(0,r)} and B ( 0 , r / 2 ) ⊂ f ( B ( 0 , r ) ) {\displaystyle B(0,r/2)\subset f(B(0,r))} . Then
is bijective and thus has an inverse. Next, we show the inverse f − 1 {\displaystyle f^{-1}} is continuously differentiable (this part of the argument is the same as that in the previous proof). This time, let g = f − 1 {\displaystyle g=f^{-1}} denote the inverse of f {\displaystyle f} and A = f ′ ( x ) {\displaystyle A=f'(x)} . For x = g ( y ) {\displaystyle x=g(y)} , we write g ( y + k ) = x + h {\displaystyle g(y+k)=x+h} or y + k = f ( x + h ) {\displaystyle y+k=f(x+h)} . Now, by the early estimate, we have
and so | h | / 2 ≤ | k | {\displaystyle |h|/2\leq |k|} . Writing ‖ ⋅ ‖ {\displaystyle \|\cdot \|} for the operator norm,
As k → 0 {\displaystyle k\to 0} , we have h → 0 {\displaystyle h\to 0} and | h | / | k | {\displaystyle |h|/|k|} is bounded. Hence, g {\displaystyle g} is differentiable at y {\displaystyle y} with the derivative g ′ ( y ) = f ′ ( g ( y ) ) − 1 {\displaystyle g'(y)=f'(g(y))^{-1}} . Also, g ′ {\displaystyle g'} is the same as the composition ι ∘ f ′ ∘ g {\displaystyle \iota \circ f'\circ g} where ι : T ↦ T − 1 {\displaystyle \iota :T\mapsto T^{-1}} ; so g ′ {\displaystyle g'} is continuous.
It remains to show the lemma. First, we have:
which is to say
This proves the first part. Next, we show f ( B ( 0 , r ) ) ⊃ B ( 0 , ( 1 − c ) r ) {\displaystyle f(B(0,r))\supset B(0,(1-c)r)} . The idea is to note that this is equivalent to, given a point y {\displaystyle y} in B ( 0 , ( 1 − c ) r ) {\displaystyle B(0,(1-c)r)} , find a fixed point of the map
where 0 < r ′ < r {\displaystyle 0<r'<r} such that | y | ≤ ( 1 − c ) r ′ {\displaystyle |y|\leq (1-c)r'} and the bar means a closed ball. To find a fixed point, we use the contraction mapping theorem and checking that F {\displaystyle F} is a well-defined strict-contraction mapping is straightforward. Finally, we have: f ( B ( 0 , r ) ) ⊂ B ( 0 , ( 1 + c ) r ) {\displaystyle f(B(0,r))\subset B(0,(1+c)r)} since
As might be clear, this proof is not substantially different from the previous one, as the proof of the contraction mapping theorem is by successive approximation.
The inverse function theorem can be used to solve a system of equations
i.e., expressing y 1 , … , y n {\displaystyle y_{1},\dots ,y_{n}} as functions of x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\dots ,x_{n})} , provided the Jacobian matrix is invertible. The implicit function theorem allows to solve a more general system of equations:
for y {\displaystyle y} in terms of x {\displaystyle x} . Though more general, the theorem is actually a consequence of the inverse function theorem. First, the precise statement of the implicit function theorem is as follows: [ 9 ]
To see this, consider the map F ( x , y ) = ( x , f ( x , y ) ) {\displaystyle F(x,y)=(x,f(x,y))} . By the inverse function theorem, F : U × V → W {\displaystyle F:U\times V\to W} has the inverse G {\displaystyle G} for some neighborhoods U , V , W {\displaystyle U,V,W} . We then have:
implying x = G 1 ( x , y ) {\displaystyle x=G_{1}(x,y)} and y = f ( x , G 2 ( x , y ) ) . {\displaystyle y=f(x,G_{2}(x,y)).} Thus g ( x ) = G 2 ( x , 0 ) {\displaystyle g(x)=G_{2}(x,0)} has the required property. ◻ {\displaystyle \square }
In differential geometry, the inverse function theorem is used to show that the pre-image of a regular value under a smooth map is a manifold. [ 10 ] Indeed, let f : U → R r {\displaystyle f:U\to \mathbb {R} ^{r}} be such a smooth map from an open subset of R n {\displaystyle \mathbb {R} ^{n}} (since the result is local, there is no loss of generality with considering such a map). Fix a point a {\displaystyle a} in f − 1 ( b ) {\displaystyle f^{-1}(b)} and then, by permuting the coordinates on R n {\displaystyle \mathbb {R} ^{n}} , assume the matrix [ ∂ f i ∂ x j ( a ) ] 1 ≤ i , j ≤ r {\displaystyle \left[{\frac {\partial f_{i}}{\partial x_{j}}}(a)\right]_{1\leq i,j\leq r}} has rank r {\displaystyle r} . Then the map F : U → R r × R n − r = R n , x ↦ ( f ( x ) , x r + 1 , … , x n ) {\displaystyle F:U\to \mathbb {R} ^{r}\times \mathbb {R} ^{n-r}=\mathbb {R} ^{n},\,x\mapsto (f(x),x_{r+1},\dots ,x_{n})} is such that F ′ ( a ) {\displaystyle F'(a)} has rank n {\displaystyle n} . Hence, by the inverse function theorem, we find the smooth inverse G {\displaystyle G} of F {\displaystyle F} defined in a neighborhood V × W {\displaystyle V\times W} of ( b , a r + 1 , … , a n ) {\displaystyle (b,a_{r+1},\dots ,a_{n})} . We then have
which implies
That is, after the change of coordinates by G {\displaystyle G} , f {\displaystyle f} is a coordinate projection (this fact is known as the submersion theorem ). Moreover, since G : V × W → U ′ = G ( V × W ) {\displaystyle G:V\times W\to U'=G(V\times W)} is bijective, the map
is bijective with the smooth inverse. That is to say, g {\displaystyle g} gives a local parametrization of f − 1 ( b ) {\displaystyle f^{-1}(b)} around a {\displaystyle a} . Hence, f − 1 ( b ) {\displaystyle f^{-1}(b)} is a manifold. ◻ {\displaystyle \square } (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.)
More generally, the theorem shows that if a smooth map f : P → E {\displaystyle f:P\to E} is transversal to a submanifold M ⊂ E {\displaystyle M\subset E} , then the pre-image f − 1 ( M ) ↪ P {\displaystyle f^{-1}(M)\hookrightarrow P} is a submanifold. [ 11 ]
The inverse function theorem is a local result; it applies to each point. A priori , the theorem thus only shows the function f {\displaystyle f} is locally bijective (or locally diffeomorphic of some class). The next topological lemma can be used to upgrade local injectivity to injectivity that is global to some extent.
Lemma — [ 12 ] [ full citation needed ] [ 13 ] If A {\displaystyle A} is a closed subset of a (second-countable) topological manifold X {\displaystyle X} (or, more generally, a topological space admitting an exhaustion by compact subsets ) and f : X → Z {\displaystyle f:X\to Z} , Z {\displaystyle Z} some topological space, is a local homeomorphism that is injective on A {\displaystyle A} , then f {\displaystyle f} is injective on some neighborhood of A {\displaystyle A} .
Proof: [ 14 ] First assume X {\displaystyle X} is compact . If the conclusion of the theorem is false, we can find two sequences x i ≠ y i {\displaystyle x_{i}\neq y_{i}} such that f ( x i ) = f ( y i ) {\displaystyle f(x_{i})=f(y_{i})} and x i , y i {\displaystyle x_{i},y_{i}} each converge to some points x , y {\displaystyle x,y} in A {\displaystyle A} . Since f {\displaystyle f} is injective on A {\displaystyle A} , x = y {\displaystyle x=y} . Now, if i {\displaystyle i} is large enough, x i , y i {\displaystyle x_{i},y_{i}} are in a neighborhood of x = y {\displaystyle x=y} where f {\displaystyle f} is injective; thus, x i = y i {\displaystyle x_{i}=y_{i}} , a contradiction.
In general, consider the set E = { ( x , y ) ∈ X 2 ∣ x ≠ y , f ( x ) = f ( y ) } {\displaystyle E=\{(x,y)\in X^{2}\mid x\neq y,f(x)=f(y)\}} . It is disjoint from S × S {\displaystyle S\times S} for any subset S ⊂ X {\displaystyle S\subset X} where f {\displaystyle f} is injective. Let X 1 ⊂ X 2 ⊂ ⋯ {\displaystyle X_{1}\subset X_{2}\subset \cdots } be an increasing sequence of compact subsets with union X {\displaystyle X} and with X i {\displaystyle X_{i}} contained in the interior of X i + 1 {\displaystyle X_{i+1}} . Then, by the first part of the proof, for each i {\displaystyle i} , we can find a neighborhood U i {\displaystyle U_{i}} of A ∩ X i {\displaystyle A\cap X_{i}} such that U i 2 ⊂ X 2 − E {\displaystyle U_{i}^{2}\subset X^{2}-E} . Then U = ⋃ i U i {\displaystyle U=\bigcup _{i}U_{i}} has the required property. ◻ {\displaystyle \square } (See also [ 15 ] for an alternative approach.)
The lemma implies the following (a sort of) global version of the inverse function theorem:
Inverse function theorem — [ 16 ] Let f : U → V {\displaystyle f:U\to V} be a map between open subsets of R n {\displaystyle \mathbb {R} ^{n}} or more generally of manifolds. Assume f {\displaystyle f} is continuously differentiable (or is C k {\displaystyle C^{k}} ). If f {\displaystyle f} is injective on a closed subset A ⊂ U {\displaystyle A\subset U} and if the Jacobian matrix of f {\displaystyle f} is invertible at each point of A {\displaystyle A} , then f {\displaystyle f} is injective on a neighborhood A ′ {\displaystyle A'} of A {\displaystyle A} and f − 1 : f ( A ′ ) → A ′ {\displaystyle f^{-1}:f(A')\to A'} is continuously differentiable (or is C k {\displaystyle C^{k}} ).
Note that if A {\displaystyle A} is a point, then the above is the usual inverse function theorem.
There is a version of the inverse function theorem for holomorphic maps .
Theorem — [ 17 ] [ 18 ] Let U , V ⊂ C n {\displaystyle U,V\subset \mathbb {C} ^{n}} be open subsets such that 0 ∈ U {\displaystyle 0\in U} and f : U → V {\displaystyle f:U\to V} a holomorphic map whose Jacobian matrix in variables z i , z ¯ i {\displaystyle z_{i},{\overline {z}}_{i}} is invertible (the determinant is nonzero) at 0 {\displaystyle 0} . Then f {\displaystyle f} is injective in some neighborhood W {\displaystyle W} of 0 {\displaystyle 0} and the inverse f − 1 : f ( W ) → W {\displaystyle f^{-1}:f(W)\to W} is holomorphic.
The theorem follows from the usual inverse function theorem. Indeed, let J R ( f ) {\displaystyle J_{\mathbb {R} }(f)} denote the Jacobian matrix of f {\displaystyle f} in variables x i , y i {\displaystyle x_{i},y_{i}} and J ( f ) {\displaystyle J(f)} for that in z j , z ¯ j {\displaystyle z_{j},{\overline {z}}_{j}} . Then we have det J R ( f ) = | det J ( f ) | 2 {\displaystyle \det J_{\mathbb {R} }(f)=|\det J(f)|^{2}} , which is nonzero by assumption. Hence, by the usual inverse function theorem, f {\displaystyle f} is injective near 0 {\displaystyle 0} with continuously differentiable inverse. By chain rule, with w = f ( z ) {\displaystyle w=f(z)} ,
where the left-hand side and the first term on the right vanish since f j − 1 ∘ f {\displaystyle f_{j}^{-1}\circ f} and f k {\displaystyle f_{k}} are holomorphic. Thus, ∂ f j − 1 ∂ w ¯ k ( w ) = 0 {\displaystyle {\frac {\partial f_{j}^{-1}}{\partial {\overline {w}}_{k}}}(w)=0} for each k {\displaystyle k} . ◻ {\displaystyle \square }
Similarly, there is the implicit function theorem for holomorphic functions. [ 19 ]
As already noted earlier, it can happen that an injective smooth function has the inverse that is not smooth (e.g., f ( x ) = x 3 {\displaystyle f(x)=x^{3}} in a real variable). This is not the case for holomorphic functions because of:
Proposition — [ 19 ] If f : U → V {\displaystyle f:U\to V} is an injective holomorphic map between open subsets of C n {\displaystyle \mathbb {C} ^{n}} , then f − 1 : f ( U ) → U {\displaystyle f^{-1}:f(U)\to U} is holomorphic.
The inverse function theorem can be rephrased in terms of differentiable maps between differentiable manifolds . In this context the theorem states that for a differentiable map F : M → N {\displaystyle F:M\to N} (of class C 1 {\displaystyle C^{1}} ), if the differential of F {\displaystyle F} ,
is a linear isomorphism at a point p {\displaystyle p} in M {\displaystyle M} then there exists an open neighborhood U {\displaystyle U} of p {\displaystyle p} such that
is a diffeomorphism . Note that this implies that the connected components of M and N containing p and F ( p ) have the same dimension, as is already directly implied from the assumption that dF p is an isomorphism.
If the derivative of F is an isomorphism at all points p in M then the map F is a local diffeomorphism .
The inverse function theorem can also be generalized to differentiable maps between Banach spaces X and Y . [ 20 ] Let U be an open neighbourhood of the origin in X and F : U → Y {\displaystyle F:U\to Y\!} a continuously differentiable function, and assume that the Fréchet derivative d F 0 : X → Y {\displaystyle dF_{0}:X\to Y\!} of F at 0 is a bounded linear isomorphism of X onto Y . Then there exists an open neighbourhood V of F ( 0 ) {\displaystyle F(0)\!} in Y and a continuously differentiable map G : V → X {\displaystyle G:V\to X\!} such that F ( G ( y ) ) = y {\displaystyle F(G(y))=y} for all y in V . Moreover, G ( y ) {\displaystyle G(y)\!} is the only sufficiently small solution x of the equation F ( x ) = y {\displaystyle F(x)=y\!} .
There is also the inverse function theorem for Banach manifolds . [ 21 ]
The inverse function theorem (and the implicit function theorem ) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant rank near a point can be put in a particular normal form near that point. [ 22 ] Specifically, if F : M → N {\displaystyle F:M\to N} has constant rank near a point p ∈ M {\displaystyle p\in M\!} , then there are open neighborhoods U of p and V of F ( p ) {\displaystyle F(p)\!} and there are diffeomorphisms u : T p M → U {\displaystyle u:T_{p}M\to U\!} and v : T F ( p ) N → V {\displaystyle v:T_{F(p)}N\to V\!} such that F ( U ) ⊆ V {\displaystyle F(U)\subseteq V\!} and such that the derivative d F p : T p M → T F ( p ) N {\displaystyle dF_{p}:T_{p}M\to T_{F(p)}N\!} is equal to v − 1 ∘ F ∘ u {\displaystyle v^{-1}\circ F\circ u\!} . That is, F "looks like" its derivative near p . The set of points p ∈ M {\displaystyle p\in M} such that the rank is constant in a neighborhood of p {\displaystyle p} is an open dense subset of M ; this is a consequence of semicontinuity of the rank function. Thus the constant rank theorem applies to a generic point of the domain.
When the derivative of F is injective (resp. surjective) at a point p , it is also injective (resp. surjective) in a neighborhood of p , and hence the rank of F is constant on that neighborhood, and the constant rank theorem applies.
If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true or false, even in the case of two variables. This is a major open problem in the theory of polynomials.
When f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} with m ≤ n {\displaystyle m\leq n} , f {\displaystyle f} is k {\displaystyle k} times continuously differentiable , and the Jacobian A = ∇ f ( x ¯ ) {\displaystyle A=\nabla f({\overline {x}})} at a point x ¯ {\displaystyle {\overline {x}}} is of rank m {\displaystyle m} , the inverse of f {\displaystyle f} may not be unique. However, there exists a local selection function s {\displaystyle s} such that f ( s ( y ) ) = y {\displaystyle f(s(y))=y} for all y {\displaystyle y} in a neighborhood of y ¯ = f ( x ¯ ) {\displaystyle {\overline {y}}=f({\overline {x}})} , s ( y ¯ ) = x ¯ {\displaystyle s({\overline {y}})={\overline {x}}} , s {\displaystyle s} is k {\displaystyle k} times continuously differentiable in this neighborhood, and ∇ s ( y ¯ ) = A T ( A A T ) − 1 {\displaystyle \nabla s({\overline {y}})=A^{T}(AA^{T})^{-1}} ( ∇ s ( y ¯ ) {\displaystyle \nabla s({\overline {y}})} is the Moore–Penrose pseudoinverse of A {\displaystyle A} ). [ 23 ]
The inverse function theorem also holds over a real closed field k (or an O-minimal structure ). [ 24 ] Precisely, the theorem holds for a semialgebraic (or definable) map between open subsets of k n {\displaystyle k^{n}} that is continuously differentiable.
The usual proof of the IFT uses Banach's fixed point theorem, which relies on the Cauchy completeness. That part of the argument is replaced by the use of the extreme value theorem , which does not need completeness. Explicitly, in § A proof using the contraction mapping principle , the Cauchy completeness is used only to establish the inclusion B ( 0 , r / 2 ) ⊂ f ( B ( 0 , r ) ) {\displaystyle B(0,r/2)\subset f(B(0,r))} . Here, we shall directly show B ( 0 , r / 4 ) ⊂ f ( B ( 0 , r ) ) {\displaystyle B(0,r/4)\subset f(B(0,r))} instead (which is enough). Given a point y {\displaystyle y} in B ( 0 , r / 4 ) {\displaystyle B(0,r/4)} , consider the function P ( x ) = | f ( x ) − y | 2 {\displaystyle P(x)=|f(x)-y|^{2}} defined on a neighborhood of B ¯ ( 0 , r ) {\displaystyle {\overline {B}}(0,r)} . If P ′ ( x ) = 0 {\displaystyle P'(x)=0} , then 0 = P ′ ( x ) = 2 [ f 1 ( x ) − y 1 ⋯ f n ( x ) − y n ] f ′ ( x ) {\displaystyle 0=P'(x)=2[f_{1}(x)-y_{1}\cdots f_{n}(x)-y_{n}]f'(x)} and so f ( x ) = y {\displaystyle f(x)=y} , since f ′ ( x ) {\displaystyle f'(x)} is invertible. Now, by the extreme value theorem, P {\displaystyle P} admits a minimal at some point x 0 {\displaystyle x_{0}} on the closed ball B ¯ ( 0 , r ) {\displaystyle {\overline {B}}(0,r)} , which can be shown to lie in B ( 0 , r ) {\displaystyle B(0,r)} using 2 − 1 | x | ≤ | f ( x ) | {\displaystyle 2^{-1}|x|\leq |f(x)|} . Since P ′ ( x 0 ) = 0 {\displaystyle P'(x_{0})=0} , f ( x 0 ) = y {\displaystyle f(x_{0})=y} , which proves the claimed inclusion. ◻ {\displaystyle \square }
Alternatively, one can deduce the theorem from the one over real numbers by Tarski's principle . [ citation needed ] | https://en.wikipedia.org/wiki/Inverse_function_theorem |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_hacovercosine |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_hacoversine |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_havercosine |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_haversine |
In computer animation and robotics , inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain , such as a robot manipulator or animation character's skeleton , in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas , a process known as forward kinematics . However, the reverse operation is, in general, much more challenging. [ 1 ] [ 2 ] [ 3 ]
Inverse kinematics is also used to recover the movements of an object in the world from some other data, such as a film of those movements, or a film of the world as seen by a camera which is itself making those movements. This occurs, for example, where a human actor's filmed movements are to be duplicated by an animated character .
In robotics, inverse kinematics makes use of the kinematics equations to determine the joint parameters that provide a desired configuration (position and rotation) for each of the robot's end-effectors . [ 4 ] This is important because robot tasks are performed with the end effectors, while control effort applies to the joints. Determining the movement of a robot so that its end-effectors move from an initial configuration to a desired configuration is known as motion planning . Inverse kinematics transforms the motion plan into joint actuator trajectories for the robot. [ 2 ] Similar formulas determine the positions of the skeleton of an animated character that is to move in a particular way in a film, or of a vehicle such as a car or boat containing the camera which is shooting a scene of a film. Once a vehicle's motions are known, they can be used to determine the constantly-changing viewpoint for computer-generated imagery of objects in the landscape such as buildings, so that these objects change in perspective while themselves not appearing to move as the vehicle-borne camera goes past them.
The movement of a kinematic chain , whether it is a robot or an animated character, is modeled by the kinematics equations of the chain. These equations define the configuration of the chain in terms of its joint parameters. Forward kinematics uses the joint parameters to compute the configuration of the chain, and inverse kinematics reverses this calculation to determine the joint parameters that achieve a desired configuration. [ 5 ] [ 6 ] [ 7 ]
Kinematic analysis is one of the first steps in the design of most industrial robots. Kinematic analysis allows the designer to obtain information on the position of each component within the mechanical system. This information is necessary for subsequent dynamic analysis along with control paths.
Inverse kinematics is an example of the kinematic analysis of a constrained system of rigid bodies, or kinematic chain . The kinematic equations of a robot can be used to define the loop equations of a complex articulated system. These loop equations are non-linear constraints on the configuration parameters of the system. The independent parameters in these equations are known as the degrees of freedom of the system.
While analytical solutions to the inverse kinematics problem exist for a wide range of kinematic chains, computer modeling and animation tools often use Newton's method to solve the non-linear kinematics equations. [ 2 ] When trying to find an analytical solution it is often convenient to exploit the geometry of the system and decompose it using subproblems with known solutions . [ 8 ] [ 9 ]
Other applications of inverse kinematic algorithms include interactive manipulation , animation control and collision avoidance .
Inverse kinematics is important to game programming and 3D animation , where it is used to connect game characters physically to the world, such as feet landing firmly on top of terrain (see [ 10 ] for a comprehensive survey on Inverse Kinematics Techniques in Computer Graphics ).
An animated figure is modeled with a skeleton of rigid segments connected with joints, called a kinematic chain . The kinematics equations of the figure define the relationship between the joint angles of the figure and its pose or configuration. The forward kinematic animation problem uses the kinematics equations to determine the pose given the joint angles. The inverse kinematics problem computes the joint angles for a desired pose of the figure.
It is often easier for computer-based designers, artists, and animators to define the spatial configuration of an assembly or figure by moving parts, or arms and legs, rather than directly manipulating joint angles. Therefore, inverse kinematics is used in computer-aided design systems to animate assemblies and by computer-based artists and animators to position figures and characters.
The assembly is modeled as rigid links connected by joints that are defined as mates, or geometric constraints. Movement of one element requires the computation of the joint angles for the other elements to maintain the joint constraints . For example, inverse kinematics allows an artist to move the hand of a 3D human model to a desired position and orientation and have an algorithm select the proper angles of the wrist, elbow, and shoulder joints. Successful implementation of computer animation usually also requires that the figure move within reasonable anthropomorphic limits.
A method of comparing both forward and inverse kinematics for the animation of a character can be defined by the advantages inherent to each. For instance, blocking animation where large motion arcs are used is often more advantageous in forward kinematics. However, more delicate animation and positioning of the target end-effector in relation to other models might be easier using inverted kinematics. Modern digital creation packages (DCC) offer methods to apply both forward and inverse kinematics to models.
In some, but not all cases, there exist analytical solutions to inverse kinematic problems. One such example is for a 6- Degrees of Freedom (DoF) robot (for example, 6 revolute joints) moving in 3D space (with 3 position degrees of freedom, and 3 rotational degrees of freedom). If the degrees of freedom of the robot exceeds the degrees of freedom of the end-effector, for example with a 7 DoF robot with 7 revolute joints, then there exist infinitely many solutions to the IK problem, and an analytical solution does not exist. Further extending this example, it is possible to fix one joint and analytically solve for the other joints, but perhaps a better solution is offered by numerical methods (next section), which can instead optimize a solution given additional preferences (costs in an optimization problem).
An analytic solution to an inverse kinematics problem is a closed-form expression that takes the end-effector pose as input and gives joint positions as output, q = f ( x ) {\displaystyle q=f(x)} . Analytical inverse kinematics solvers can be significantly faster than numerical solvers and provide more than one solution, but only a finite number of solutions, for a given end-effector pose.
Many different programs (Such as FOSS programs IKFast and Inverse Kinematics Library ) are able to solve these problems quickly and efficiently using different algorithms such as the FABRIK solver . One issue with these solvers, is that they are known to not necessarily give locally smooth solutions between two adjacent configurations, which can cause instability if iterative solutions to inverse kinematics are required, such as if the IK is solved inside a high-rate control loop.
Many industrial 6DOF robots feature three rotational joints with intersecting axes ("spherical wrist"). These robots, known as robots with an "Ortho-parallel Basis and a Spherical Wrist," can be defined by 7 kinematic parameters that are distances in their assumed standard geometry. [ 11 ] These robots may have up to 8 independent solutions for any given position and rotation of the robot tool head. Open-source solutions for C++ [ 12 ] and Rust [ 13 ] exist. OPW has also been integrated into ROS framework. [ 14 ]
There are many methods of modelling and solving inverse kinematics problems. The most flexible of these methods typically rely on iterative optimization to seek out an approximate solution, due to the difficulty of inverting the forward kinematics equation and the possibility of an empty solution space . The core idea behind several of these methods is to model the forward kinematics equation using a Taylor series expansion, which can be simpler to invert and solve than the original system.
The Jacobian inverse technique is a simple yet effective way of implementing inverse kinematics. Let there be m {\displaystyle m} variables that govern the forward-kinematics equation, i.e. the position function. These variables may be joint angles, lengths, or other arbitrary real values. If, for example, the IK system lives in a 3-dimensional space, the position function can be viewed as a mapping p ( x ) : R m → R 3 {\displaystyle p(x):\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{3}} . Let p 0 = p ( x 0 ) {\displaystyle p_{0}=p(x_{0})} give the initial position of the system, and
be the goal position of the system. The Jacobian inverse technique iteratively computes an estimate of Δ x {\displaystyle \Delta x} that minimizes the error given by | | p ( x 0 + Δ x estimate ) − p 1 | | {\displaystyle ||p(x_{0}+\Delta x_{\text{estimate}})-p_{1}||} .
For small Δ x {\displaystyle \Delta x} -vectors, the series expansion of the position function gives
where J p ( x 0 ) {\displaystyle J_{p}(x_{0})} is the (3 × m) Jacobian matrix of the position function at x 0 {\displaystyle x_{0}} .
The (i, k)-th entry of the Jacobian matrix can be approximated numerically
where p i ( x ) {\displaystyle p_{i}(x)} gives the i-th component of the position function, x 0 , k + h {\displaystyle x_{0,k}+h} is simply x 0 {\displaystyle x_{0}} with a small delta added to its k-th component, and h {\displaystyle h} is a reasonably small positive value.
Taking the Moore–Penrose pseudoinverse of the Jacobian (computable using a singular value decomposition ) and re-arranging terms results in
where Δ p = p ( x 0 + Δ x ) − p ( x 0 ) {\displaystyle \Delta p=p(x_{0}+\Delta x)-p(x_{0})} .
Applying the inverse Jacobian method once will result in a very rough estimate of the desired Δ x {\displaystyle \Delta x} -vector. A line search should be used to scale this Δ x {\displaystyle \Delta x} to an acceptable value. The estimate for Δ x {\displaystyle \Delta x} can be improved via the following algorithm (known as the Newton–Raphson method ):
Once some Δ x {\displaystyle \Delta x} -vector has caused the error to drop close to zero, the algorithm should terminate. Existing methods based on the Hessian matrix of the system have been reported to converge to desired Δ x {\displaystyle \Delta x} values using fewer iterations, though, in some cases more computational resources.
The inverse kinematics problem can also be approximated using heuristic methods. These methods perform simple, iterative operations to gradually lead to an approximation of the solution. The heuristic algorithms have low computational cost (return the final pose very quickly), and usually support joint constraints. The most popular heuristic algorithms are cyclic coordinate descent (CCD) [ 15 ] and forward and backward reaching inverse kinematics (FABRIK). [ 16 ] | https://en.wikipedia.org/wiki/Inverse_kinematics |
The inverse magnetostrictive effect , magnetoelastic effect or Villari effect , after its discoverer Emilio Villari , is the change of the magnetic susceptibility of a material when subjected to a mechanical stress.
The magnetostriction λ {\displaystyle \lambda } characterizes the shape change of a ferromagnetic material during magnetization, whereas the inverse magnetostrictive effect characterizes the change of sample magnetization M {\displaystyle M} (for given magnetizing field strength H {\displaystyle H} ) when mechanical stresses σ {\displaystyle \sigma } are applied to the sample. [ 1 ]
Under a given uni-axial mechanical stress σ {\displaystyle \sigma } , the flux density B {\displaystyle B} for a given magnetizing field strength H {\displaystyle H} may increase or decrease. The way in which a material responds to stresses depends on its saturation magnetostriction λ s {\displaystyle \lambda _{s}} . For this analysis, compressive stresses σ {\displaystyle \sigma } are considered as negative, whereas tensile stresses are positive. According to Le Chatelier's principle :
( d λ d H ) σ = ( d B d σ ) H {\displaystyle \left({\frac {d\lambda }{dH}}\right)_{\sigma }=\left({\frac {dB}{d\sigma }}\right)_{H}}
This means, that when the product σ λ s {\displaystyle \sigma \lambda _{s}} is positive, the flux density B {\displaystyle B} increases under stress. On the other hand, when the product σ λ s {\displaystyle \sigma \lambda _{s}} is negative, the flux density B {\displaystyle B} decreases under stress. This effect was confirmed experimentally. [ 2 ]
In the case of a single stress σ {\displaystyle \sigma } acting upon a single magnetic domain, the magnetic strain energy density E σ {\displaystyle E_{\sigma }} can be expressed as: [ 1 ]
E σ = 3 2 λ s σ sin 2 ( θ ) {\displaystyle E_{\sigma }={\frac {3}{2}}\lambda _{s}\sigma \sin ^{2}(\theta )}
where λ s {\displaystyle \lambda _{s}} is the magnetostrictive expansion at saturation, and θ {\displaystyle \theta } is the angle between the saturation magnetization and the stress's direction.
When λ s {\displaystyle \lambda _{s}} and σ {\displaystyle \sigma } are both positive (like in iron under tension), the energy is minimum for θ {\displaystyle \theta } = 0, i.e. when tension is aligned with the saturation magnetization. Consequently, the magnetization is increased by tension.
In fact, magnetostriction is more complex and depends on the direction of the crystal axes. In iron , the [100] axes are the directions of easy magnetization, while there is little magnetization along the [111] directions (unless the magnetization becomes close to the saturation magnetization, leading to the change of the domain orientation from [111] to [100]). This magnetic anisotropy pushed authors to define two independent longitudinal magnetostrictions λ 100 {\displaystyle \lambda _{100}} and λ 111 {\displaystyle \lambda _{111}} .
Method suitable for effective testing of magnetoelastic effect in magnetic materials should fulfill the following requirements: [ 3 ]
Following testing methods were developed:
Magnetoelastic effect can be used in development of force sensors . [ 8 ] [ 9 ] This effect was used for sensors:
Inverse magnetoelastic effects have to be also considered as a side effect of accidental or intentional application of mechanical stresses to the magnetic core of inductive component, e.g. fluxgates or generator/motor stators when installed with interference fits. [ 12 ] | https://en.wikipedia.org/wiki/Inverse_magnetostrictive_effect |
Inverse photoemission spectroscopy ( IPES ) is a surface science technique used to study the unoccupied electronic structure of surfaces, thin films, and adsorbates. A well-collimated beam of electrons of a well defined energy (< 20 eV) is directed at the sample. These electrons couple to high-lying unoccupied electronic states and decay to low-lying unoccupied states, with a subset of these transitions being radiative. The photons emitted in the decay process are detected and an energy spectrum, photon counts vs. incident electron energy, is generated. Due to the low energy of the incident electrons, their penetration depth is only a few atomic layers, making inverse photoemission a particularly surface sensitive technique. As inverse photoemission probes the electronic states above the Fermi level of the system, it is a complementary technique to photoemission spectroscopy .
The energy of photons ( h ν {\displaystyle h\nu } , where h {\displaystyle h} is the Planck constant ) emitted when electrons incident on a substance using an electron beam with a constant energy ( E i {\displaystyle E_{i}} ) relax to a lower energy unoccupied state ( E f {\displaystyle E_{f}} ) is given by the conservation of energy as:
By measuring E i {\displaystyle E_{i}} and h ν {\displaystyle h\nu } , the unoccupied state ( E f {\displaystyle E_{f}} ) of the surface can be found.
Two modes can be used for this measurement. One is the isochromat mode, which scans the incident electron energy and keeps the detected photon energy constant. The other is the tunable photon energy mode, or spectrograph mode, which keeps the incident electron energy constant and measures the distribution of the detected photon energy. The latter can also measure the resonant inverse photoemission spectroscopy .
In isochromat mode, the incident electron energy is ramped and the emitted photons are detected at a fixed energy that is determined by the photon detector. Typically, an I 2 gas filled Geiger-Müller tube with an entrance window of either SrF 2 or CaF 2 is used as the photon detector. The combination of window and filling gas determines the detected photon energy, and for I 2 gas and either a SrF 2 or CaF 2 window, the photons energies are ~ 9.5 eV and ~ 9.7 eV, respectively.
In spectrograph mode, the energy of the incident electron remains fixed and a grating spectrometer is used to the detect the emitted photons over a range of photon energies. A diffraction grating is used to disperse the emitted photons that are in turn detected with a two-dimensional position sensitive detector.
One advantage of spectrograph mode is the ability to acquire IPES spectra over a wide range of photon energies simultaneously. Additionally, the incident electron energy remains fixed which allows better focusing of the electron beam on the sample. Furthermore, by changing the incident electron energy the electronic structure can be studied in great detail. Although the grating spectrometer is very stable over time, the set-up can be very complex and its maintenance can be very expensive. The advantages of isochromat mode are its low cost, simple design and higher count rates. [ 1 ] | https://en.wikipedia.org/wiki/Inverse_photoemission_spectroscopy |
Inverse polymerase chain reaction ( Inverse PCR ) is a variant of the polymerase chain reaction that is used to amplify DNA with only one known sequence. One limitation of conventional PCR is that it requires primers complementary to both termini of the target DNA, but this method allows PCR to be carried out even if only one sequence is available from which primers may be designed.
Inverse PCR is especially useful for the determination of insert locations. For example, various retroviruses and transposons randomly integrate into genomic DNA . [ 1 ] To identify the sites where they have entered, the known, "internal" viral or transposon sequences can be used to design primers that will amplify a small portion of the flanking, "external" genomic DNA. The amplified product can then be sequenced and compared with DNA databases to locate the sequence which has been disrupted.
The inverse PCR method involves a series of restriction digests and ligation , resulting in a looped fragment that can be primed for PCR from a single section of known sequence. Then, like other polymerase chain reaction processes, the DNA is amplified by the thermostable DNA polymerase :
Finally the sequence of the sequenced PCR product is compared against sequence databases.
It is used in case of chromosome crawling. | https://en.wikipedia.org/wiki/Inverse_polymerase_chain_reaction |
In probability theory , inverse probability is an old term for the probability distribution of an unobserved variable.
Today, the problem of determining an unobserved variable (by whatever method) is called inferential statistics . The method of inverse probability (assigning a probability distribution to an unobserved variable) is called Bayesian probability , the distribution of data given the unobserved variable is the likelihood function (which does not by itself give a probability distribution for the parameter), and the distribution of an unobserved variable, given both data and a prior distribution , is the posterior distribution . The development of the field and terminology from "inverse probability" to "Bayesian probability" is described by Fienberg (2006) .
The term "inverse probability" appears in an 1837 paper of De Morgan , in reference to Laplace 's method of probability (developed in a 1774 paper, which independently discovered and popularized Bayesian methods, and a 1812 book), though the term "inverse probability" does not occur in these. [ 1 ] Fisher uses the term in Fisher (1922) , referring to "the fundamental paradox of inverse probability" as the source of the confusion between statistical terms that refer to the true value to be estimated, with the actual value arrived at by the estimation method, which is subject to error. Later Jeffreys uses the term in his defense of the methods of Bayes and Laplace, in Jeffreys (1939) . The term "Bayesian", which displaced "inverse probability", was introduced by Ronald Fisher in 1950. [ 2 ] Inverse probability, variously interpreted, was the dominant approach to statistics until the development of frequentism in the early 20th century by Ronald Fisher , Jerzy Neyman and Egon Pearson . [ 3 ] Following the development of frequentism, the terms frequentist and Bayesian developed to contrast these approaches, and became common in the 1950s.
In modern terms, given a probability distribution p ( x |θ) for an observable quantity x conditional on an unobserved variable θ, the "inverse probability" is the posterior distribution p (θ| x ), which depends both on the likelihood function (the inversion of the probability distribution) and a prior distribution. The distribution p ( x |θ) itself is called the direct probability .
The inverse probability problem (in the 18th and 19th centuries) was the problem of estimating a parameter from experimental data in the experimental sciences, especially astronomy and biology . A simple example would be the problem of estimating the position of a star in the sky (at a certain time on a certain date) for purposes of navigation . Given the data, one must estimate the true position (probably by averaging). This problem would now be considered one of inferential statistics .
The terms "direct probability" and "inverse probability" were in use until the middle part of the 20th century, when the terms " likelihood function " and "posterior distribution" became prevalent. | https://en.wikipedia.org/wiki/Inverse_probability |
In mathematics , the inverse trigonometric functions (occasionally also called antitrigonometric , [ 1 ] cyclometric , [ 2 ] or arcus functions [ 3 ] ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, [ 4 ] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. [ 1 ] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [ 5 ] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . [ 6 ]
The notations sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., as introduced by John Herschel in 1813, [ 7 ] [ 8 ] are often used as well in English-language sources, [ 1 ] much more than the also established sin [−1] ( x ) , cos [−1] ( x ) , tan [−1] ( x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin 2 ( x ) (although only sin 2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . [ 9 ]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) −1 = sec( x ) . Nevertheless, certain authors advise against using it, since it is ambiguous. [ 1 ] [ 10 ] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. [ 11 ] Although it is intended to avoid confusion with the reciprocal , which should be represented by sin −1 ( x ) , cos −1 ( x ) , etc., or, better, by sin −1 x , cos −1 x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for the standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), [ 12 ] because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
If x is allowed to be a complex number , then the range of y applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians .
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained.
Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } :
The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } :
The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :}
This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined .
Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} "
The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote [ note 1 ] for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ).
With the help of integer parity Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
And similarly for the secant function,
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same.
We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which is a shorthand way of saying that (at least) one of the following statement is true:
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ).
However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has.
An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
The equations above can be transformed by using the reflection and shift identities: [ 13 ]
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}
where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives the analogous equations for csc , sec , and cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively.
So for example, by using the equality sin ( π 2 − θ ) = cos θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,} the equation cos θ = x {\displaystyle \cos \theta =x} can be transformed into sin ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos θ = x {\displaystyle \;\cos \theta =x\;} is: θ = ( − 1 ) h + 1 arcsin ( x ) + π h + π 2 for some h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin x = π 2 − arccos x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express the right hand side of the above formula in terms of arccos x {\displaystyle \;\arccos x\;} instead of arcsin x . {\displaystyle \;\arcsin x.\;}
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .}
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
A useful form that follows directly from the table above is
It is obtained by recognizing that cos ( arctan ( x ) ) = 1 1 + x 2 = cos ( arccos ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} .
From the half-angle formula , tan ( θ 2 ) = sin ( θ ) 1 + cos ( θ ) {\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}} , we get:
This is derived from the tangent addition formula
by letting
The derivatives for complex values of z are as follows:
Only for real values of x :
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},} so
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals , but still well-defined.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series , as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition above (see Leibniz series ).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos ( x ) = π / 2 − arcsin ( x ) {\displaystyle \arccos(x)=\pi /2-\arcsin(x)} , arccsc ( x ) = arcsin ( 1 / x ) {\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)} , and so on. Another series is given by: [ 14 ]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series :
(The term in the sum for n = 0 is the empty product , so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit . [ 16 ]
Two alternatives to the power series for arctangent are these generalized continued fractions :
The second of these is valid in the cut complex plane. There are two cuts, from − i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2 , with each perfect square appearing once. The first was developed by Leonhard Euler ; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series .
For real and complex values of z :
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions :
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Using ∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du} (i.e. integration by parts ), set
Then
which by the simple substitution w = 1 − x 2 , d w = − 2 x d x {\displaystyle w=1-x^{2},\ dw=-2x\,dx} yields the final result:
Since the inverse trigonometric functions are analytic functions , they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points . One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
These functions may also be expressed using complex logarithms . This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } .
or
Simply taking the imaginary part works for any real-valued a {\displaystyle a} and b {\displaystyle b} , but if a {\displaystyle a} or b {\displaystyle b} is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of ln ( a + b i ) {\displaystyle \ln(a+bi)} also removes c {\displaystyle c} from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z {\displaystyle z} , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ {\displaystyle \theta } that result from plugging the values into the equations θ = − i ln ( a + i b c ) {\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)} above and simplifying.
The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the Im ( ln z ) ∈ ( − π , π ] {\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]} and Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0} principal branch for every function except arccotangent in the θ {\displaystyle \theta } column. Arccotangent in the θ {\displaystyle \theta } column will output on its usual principal branch by using the Im ( ln z ) ∈ [ 0 , 2 π ) {\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )} and Im ( z ) ≥ 0 {\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0} convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions . It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
Using the exponential definition of sine , and letting ξ = e i ϕ , {\displaystyle \xi =e^{i\phi },}
(the positive branch is chosen)
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem : a 2 + b 2 = h 2 {\displaystyle a^{2}+b^{2}=h^{2}} where h {\displaystyle h} is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
The two-argument atan2 function computes the arctangent of y / x given y and x , but with a range of (−π, π] . In other words, atan2( y , x ) is the angle between the positive x -axis of a plane and the point ( x , y ) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0 ), and negative sign for clockwise angles (lower half-plane, y < 0 ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (−π/2, π/2) , it can be expressed as follows:
atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y ≥ 0 , x < 0 arctan ( y x ) − π y < 0 , x < 0 π 2 y > 0 , x = 0 − π 2 y < 0 , x = 0 undefined y = 0 , x = 0 {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}
It also equals the principal value of the argument of the complex number x + iy .
This limited version of the function above may also be defined using the tangent half-angle formulae as follows: atan2 ( y , x ) = 2 arctan ( y x 2 + y 2 + x ) {\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)} provided that either x > 0 or y ≠ 0 . However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order ( y , x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language , but a few authors may use the opposite convention ( x , y ) so some caution is warranted. (See variations at atan2 § Realizations of the function in common computer languages .)
In many applications [ 17 ] the solution y {\displaystyle y} of the equation x = tan ( y ) {\displaystyle x=\tan(y)} is to come as close as possible to a given value − ∞ < η < ∞ {\displaystyle -\infty <\eta <\infty } . The adequate solution is produced by the parameter modified arctangent function
The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer.
For angles near 0 and π , arccosine is ill-conditioned , and similarly with arcsine for angles near − π /2 and π /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. [ 18 ] | https://en.wikipedia.org/wiki/Inverse_secant |
In mathematics , the inverse trigonometric functions (occasionally also called antitrigonometric , [ 1 ] cyclometric , [ 2 ] or arcus functions [ 3 ] ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, [ 4 ] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. [ 1 ] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [ 5 ] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . [ 6 ]
The notations sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., as introduced by John Herschel in 1813, [ 7 ] [ 8 ] are often used as well in English-language sources, [ 1 ] much more than the also established sin [−1] ( x ) , cos [−1] ( x ) , tan [−1] ( x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin 2 ( x ) (although only sin 2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . [ 9 ]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) −1 = sec( x ) . Nevertheless, certain authors advise against using it, since it is ambiguous. [ 1 ] [ 10 ] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. [ 11 ] Although it is intended to avoid confusion with the reciprocal , which should be represented by sin −1 ( x ) , cos −1 ( x ) , etc., or, better, by sin −1 x , cos −1 x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for the standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), [ 12 ] because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
If x is allowed to be a complex number , then the range of y applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians .
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained.
Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } :
The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } :
The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :}
This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined .
Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} "
The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote [ note 1 ] for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ).
With the help of integer parity Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
And similarly for the secant function,
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same.
We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which is a shorthand way of saying that (at least) one of the following statement is true:
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ).
However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has.
An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
The equations above can be transformed by using the reflection and shift identities: [ 13 ]
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}
where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives the analogous equations for csc , sec , and cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively.
So for example, by using the equality sin ( π 2 − θ ) = cos θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,} the equation cos θ = x {\displaystyle \cos \theta =x} can be transformed into sin ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos θ = x {\displaystyle \;\cos \theta =x\;} is: θ = ( − 1 ) h + 1 arcsin ( x ) + π h + π 2 for some h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin x = π 2 − arccos x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express the right hand side of the above formula in terms of arccos x {\displaystyle \;\arccos x\;} instead of arcsin x . {\displaystyle \;\arcsin x.\;}
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .}
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
A useful form that follows directly from the table above is
It is obtained by recognizing that cos ( arctan ( x ) ) = 1 1 + x 2 = cos ( arccos ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} .
From the half-angle formula , tan ( θ 2 ) = sin ( θ ) 1 + cos ( θ ) {\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}} , we get:
This is derived from the tangent addition formula
by letting
The derivatives for complex values of z are as follows:
Only for real values of x :
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},} so
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals , but still well-defined.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series , as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition above (see Leibniz series ).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos ( x ) = π / 2 − arcsin ( x ) {\displaystyle \arccos(x)=\pi /2-\arcsin(x)} , arccsc ( x ) = arcsin ( 1 / x ) {\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)} , and so on. Another series is given by: [ 14 ]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series :
(The term in the sum for n = 0 is the empty product , so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit . [ 16 ]
Two alternatives to the power series for arctangent are these generalized continued fractions :
The second of these is valid in the cut complex plane. There are two cuts, from − i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2 , with each perfect square appearing once. The first was developed by Leonhard Euler ; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series .
For real and complex values of z :
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions :
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Using ∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du} (i.e. integration by parts ), set
Then
which by the simple substitution w = 1 − x 2 , d w = − 2 x d x {\displaystyle w=1-x^{2},\ dw=-2x\,dx} yields the final result:
Since the inverse trigonometric functions are analytic functions , they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points . One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
These functions may also be expressed using complex logarithms . This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } .
or
Simply taking the imaginary part works for any real-valued a {\displaystyle a} and b {\displaystyle b} , but if a {\displaystyle a} or b {\displaystyle b} is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of ln ( a + b i ) {\displaystyle \ln(a+bi)} also removes c {\displaystyle c} from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z {\displaystyle z} , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ {\displaystyle \theta } that result from plugging the values into the equations θ = − i ln ( a + i b c ) {\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)} above and simplifying.
The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the Im ( ln z ) ∈ ( − π , π ] {\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]} and Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0} principal branch for every function except arccotangent in the θ {\displaystyle \theta } column. Arccotangent in the θ {\displaystyle \theta } column will output on its usual principal branch by using the Im ( ln z ) ∈ [ 0 , 2 π ) {\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )} and Im ( z ) ≥ 0 {\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0} convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions . It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
Using the exponential definition of sine , and letting ξ = e i ϕ , {\displaystyle \xi =e^{i\phi },}
(the positive branch is chosen)
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem : a 2 + b 2 = h 2 {\displaystyle a^{2}+b^{2}=h^{2}} where h {\displaystyle h} is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
The two-argument atan2 function computes the arctangent of y / x given y and x , but with a range of (−π, π] . In other words, atan2( y , x ) is the angle between the positive x -axis of a plane and the point ( x , y ) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0 ), and negative sign for clockwise angles (lower half-plane, y < 0 ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (−π/2, π/2) , it can be expressed as follows:
atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y ≥ 0 , x < 0 arctan ( y x ) − π y < 0 , x < 0 π 2 y > 0 , x = 0 − π 2 y < 0 , x = 0 undefined y = 0 , x = 0 {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}
It also equals the principal value of the argument of the complex number x + iy .
This limited version of the function above may also be defined using the tangent half-angle formulae as follows: atan2 ( y , x ) = 2 arctan ( y x 2 + y 2 + x ) {\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)} provided that either x > 0 or y ≠ 0 . However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order ( y , x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language , but a few authors may use the opposite convention ( x , y ) so some caution is warranted. (See variations at atan2 § Realizations of the function in common computer languages .)
In many applications [ 17 ] the solution y {\displaystyle y} of the equation x = tan ( y ) {\displaystyle x=\tan(y)} is to come as close as possible to a given value − ∞ < η < ∞ {\displaystyle -\infty <\eta <\infty } . The adequate solution is produced by the parameter modified arctangent function
The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer.
For angles near 0 and π , arccosine is ill-conditioned , and similarly with arcsine for angles near − π /2 and π /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. [ 18 ] | https://en.wikipedia.org/wiki/Inverse_sine |
In mathematics , the inverse trigonometric functions (occasionally also called antitrigonometric , [ 1 ] cyclometric , [ 2 ] or arcus functions [ 3 ] ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, [ 4 ] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. [ 1 ] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [ 5 ] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . [ 6 ]
The notations sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., as introduced by John Herschel in 1813, [ 7 ] [ 8 ] are often used as well in English-language sources, [ 1 ] much more than the also established sin [−1] ( x ) , cos [−1] ( x ) , tan [−1] ( x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin 2 ( x ) (although only sin 2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . [ 9 ]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) −1 = sec( x ) . Nevertheless, certain authors advise against using it, since it is ambiguous. [ 1 ] [ 10 ] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. [ 11 ] Although it is intended to avoid confusion with the reciprocal , which should be represented by sin −1 ( x ) , cos −1 ( x ) , etc., or, better, by sin −1 x , cos −1 x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for the standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), [ 12 ] because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
If x is allowed to be a complex number , then the range of y applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians .
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained.
Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } :
The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } :
The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :}
This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined .
Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} "
The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote [ note 1 ] for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ).
With the help of integer parity Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
And similarly for the secant function,
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same.
We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which is a shorthand way of saying that (at least) one of the following statement is true:
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ).
However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has.
An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
The equations above can be transformed by using the reflection and shift identities: [ 13 ]
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}
where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives the analogous equations for csc , sec , and cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively.
So for example, by using the equality sin ( π 2 − θ ) = cos θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,} the equation cos θ = x {\displaystyle \cos \theta =x} can be transformed into sin ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos θ = x {\displaystyle \;\cos \theta =x\;} is: θ = ( − 1 ) h + 1 arcsin ( x ) + π h + π 2 for some h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin x = π 2 − arccos x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express the right hand side of the above formula in terms of arccos x {\displaystyle \;\arccos x\;} instead of arcsin x . {\displaystyle \;\arcsin x.\;}
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .}
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
A useful form that follows directly from the table above is
It is obtained by recognizing that cos ( arctan ( x ) ) = 1 1 + x 2 = cos ( arccos ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} .
From the half-angle formula , tan ( θ 2 ) = sin ( θ ) 1 + cos ( θ ) {\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}} , we get:
This is derived from the tangent addition formula
by letting
The derivatives for complex values of z are as follows:
Only for real values of x :
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},} so
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals , but still well-defined.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series , as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition above (see Leibniz series ).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos ( x ) = π / 2 − arcsin ( x ) {\displaystyle \arccos(x)=\pi /2-\arcsin(x)} , arccsc ( x ) = arcsin ( 1 / x ) {\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)} , and so on. Another series is given by: [ 14 ]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series :
(The term in the sum for n = 0 is the empty product , so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit . [ 16 ]
Two alternatives to the power series for arctangent are these generalized continued fractions :
The second of these is valid in the cut complex plane. There are two cuts, from − i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2 , with each perfect square appearing once. The first was developed by Leonhard Euler ; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series .
For real and complex values of z :
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions :
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Using ∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du} (i.e. integration by parts ), set
Then
which by the simple substitution w = 1 − x 2 , d w = − 2 x d x {\displaystyle w=1-x^{2},\ dw=-2x\,dx} yields the final result:
Since the inverse trigonometric functions are analytic functions , they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points . One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
These functions may also be expressed using complex logarithms . This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } .
or
Simply taking the imaginary part works for any real-valued a {\displaystyle a} and b {\displaystyle b} , but if a {\displaystyle a} or b {\displaystyle b} is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of ln ( a + b i ) {\displaystyle \ln(a+bi)} also removes c {\displaystyle c} from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z {\displaystyle z} , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ {\displaystyle \theta } that result from plugging the values into the equations θ = − i ln ( a + i b c ) {\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)} above and simplifying.
The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the Im ( ln z ) ∈ ( − π , π ] {\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]} and Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0} principal branch for every function except arccotangent in the θ {\displaystyle \theta } column. Arccotangent in the θ {\displaystyle \theta } column will output on its usual principal branch by using the Im ( ln z ) ∈ [ 0 , 2 π ) {\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )} and Im ( z ) ≥ 0 {\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0} convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions . It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
Using the exponential definition of sine , and letting ξ = e i ϕ , {\displaystyle \xi =e^{i\phi },}
(the positive branch is chosen)
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem : a 2 + b 2 = h 2 {\displaystyle a^{2}+b^{2}=h^{2}} where h {\displaystyle h} is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
The two-argument atan2 function computes the arctangent of y / x given y and x , but with a range of (−π, π] . In other words, atan2( y , x ) is the angle between the positive x -axis of a plane and the point ( x , y ) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0 ), and negative sign for clockwise angles (lower half-plane, y < 0 ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (−π/2, π/2) , it can be expressed as follows:
atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y ≥ 0 , x < 0 arctan ( y x ) − π y < 0 , x < 0 π 2 y > 0 , x = 0 − π 2 y < 0 , x = 0 undefined y = 0 , x = 0 {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}
It also equals the principal value of the argument of the complex number x + iy .
This limited version of the function above may also be defined using the tangent half-angle formulae as follows: atan2 ( y , x ) = 2 arctan ( y x 2 + y 2 + x ) {\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)} provided that either x > 0 or y ≠ 0 . However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order ( y , x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language , but a few authors may use the opposite convention ( x , y ) so some caution is warranted. (See variations at atan2 § Realizations of the function in common computer languages .)
In many applications [ 17 ] the solution y {\displaystyle y} of the equation x = tan ( y ) {\displaystyle x=\tan(y)} is to come as close as possible to a given value − ∞ < η < ∞ {\displaystyle -\infty <\eta <\infty } . The adequate solution is produced by the parameter modified arctangent function
The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer.
For angles near 0 and π , arccosine is ill-conditioned , and similarly with arcsine for angles near − π /2 and π /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. [ 18 ] | https://en.wikipedia.org/wiki/Inverse_tangent |
In mathematics , the inverse trigonometric functions (occasionally also called antitrigonometric , [ 1 ] cyclometric , [ 2 ] or arcus functions [ 3 ] ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, [ 4 ] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry .
Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. [ 1 ] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. [ 5 ] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . [ 6 ]
The notations sin −1 ( x ) , cos −1 ( x ) , tan −1 ( x ) , etc., as introduced by John Herschel in 1813, [ 7 ] [ 8 ] are often used as well in English-language sources, [ 1 ] much more than the also established sin [−1] ( x ) , cos [−1] ( x ) , tan [−1] ( x ) – conventions consistent with the notation of an inverse function , that is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ( x ) = { arctan ( x ) + π k ∣ k ∈ Z } . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with the common semantics for expressions such as sin 2 ( x ) (although only sin 2 x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . [ 9 ]
The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) −1 = sec( x ) . Nevertheless, certain authors advise against using it, since it is ambiguous. [ 1 ] [ 10 ] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin −1 ( x ) , Cos −1 ( x ) , Tan −1 ( x ) , etc. [ 11 ] Although it is intended to avoid confusion with the reciprocal , which should be represented by sin −1 ( x ) , cos −1 ( x ) , etc., or, better, by sin −1 x , cos −1 x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for the standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case.
Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.
Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.
For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ( x ) {\displaystyle y=\arcsin(x)} is defined so that sin ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π ≤ y < 3 π 2 {\textstyle \pi \leq y<{\frac {3\pi }{2}}} ), [ 12 ] because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ( arcsec ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with the range ( 0 ≤ y < π 2 {\textstyle 0\leq y<{\frac {\pi }{2}}} or π 2 < y ≤ π {\textstyle {\frac {\pi }{2}}<y\leq \pi } ), we would have to write tan ( arcsec ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).}
If x is allowed to be a complex number , then the range of y applies only to its real part.
The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians .
The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes the set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi } is denoted by
π Z := { π n : n ∈ Z } = { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}
The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )} is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).}
The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that is used above to concisely write the domains of cot , csc , tan , and sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained.
Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } :
The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which sin θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,}
π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}
Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } :
The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}
π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :}
This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer.
The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions.
It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined .
Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } " is just another way of saying "for some integer k . {\displaystyle k.} "
The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote [ note 1 ] for more details and an example illustrating this concept).
where the first four solutions can be written in expanded form as:
For example, if cos θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec θ = − 1 {\displaystyle \sec \theta =-1} and csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} have the same solutions as cos θ = − 1 {\displaystyle \cos \theta =-1} and sin θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec θ = − 1 {\displaystyle \sec \theta =-1} ), the integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ).
With the help of integer parity Parity ( h ) = { 0 if h is even 1 if h is odd {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol:
And similarly for the secant function,
where π h + π Parity ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when the integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd.
The solutions to cos θ = x {\displaystyle \cos \theta =x} and sec θ = x {\displaystyle \sec \theta =x} involve the "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec θ = x {\displaystyle \sec \theta =x} is the same.
We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there is an angle θ {\displaystyle \theta } in some interval that satisfies cos θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which is a shorthand way of saying that (at least) one of the following statement is true:
As mentioned above, if arccos x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ).
However, if x ≠ − 1 {\displaystyle x\neq -1} then the integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos x {\displaystyle \,\pm \arccos x\,} is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold).
Having considered the cases arccos x = 0 {\displaystyle \,\arccos x=0\,} and arccos x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on the case where arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos x + 2 π k for some k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more is known). Then arccos x = arccos 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it is not possible to determine which of these values θ {\displaystyle \theta } has.
An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ).
The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other.
The vertical double arrow ⇕ {\displaystyle \Updownarrow } in the last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin θ | = | sin φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos θ | = | cos φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}
Thus given a single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ( arcsin y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin y {\displaystyle \theta :=\arcsin y} is always a solution), the set of all solutions to it are:
The equations above can be transformed by using the reflection and shift identities: [ 13 ]
These formulas imply, in particular, that the following hold:
sin θ = − sin ( − θ ) = − sin ( π + θ ) = − sin ( π − θ ) = − cos ( π 2 + θ ) = − cos ( π 2 − θ ) = − cos ( − π 2 − θ ) = − cos ( − π 2 + θ ) = − cos ( 3 π 2 − θ ) = − cos ( − 3 π 2 + θ ) cos θ = − cos ( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − θ ) = − cot ( − π 2 + θ ) = − cot ( 3 π 2 − θ ) = − cot ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}
where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives the analogous equations for csc , sec , and cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively.
So for example, by using the equality sin ( π 2 − θ ) = cos θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,} the equation cos θ = x {\displaystyle \cos \theta =x} can be transformed into sin ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos θ = x {\displaystyle \;\cos \theta =x\;} is: θ = ( − 1 ) h + 1 arcsin ( x ) + π h + π 2 for some h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin x = π 2 − arccos x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express the right hand side of the above formula in terms of arccos x {\displaystyle \;\arccos x\;} instead of arcsin x . {\displaystyle \;\arcsin x.\;}
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.
Complementary angles:
Negative arguments:
Reciprocal arguments:
The identities above can be used with (and derived from) the fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .}
Useful identities if one only has a fragment of a sine table:
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
A useful form that follows directly from the table above is
It is obtained by recognizing that cos ( arctan ( x ) ) = 1 1 + x 2 = cos ( arccos ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} .
From the half-angle formula , tan ( θ 2 ) = sin ( θ ) 1 + cos ( θ ) {\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}} , we get:
This is derived from the tangent addition formula
by letting
The derivatives for complex values of z are as follows:
Only for real values of x :
These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},} so
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
When x equals 1, the integrals with limited domains are improper integrals , but still well-defined.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series , as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition above (see Leibniz series ).
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccos ( x ) = π / 2 − arcsin ( x ) {\displaystyle \arccos(x)=\pi /2-\arcsin(x)} , arccsc ( x ) = arcsin ( 1 / x ) {\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)} , and so on. Another series is given by: [ 14 ]
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series :
(The term in the sum for n = 0 is the empty product , so is 1.)
Alternatively, this can be expressed as
Another series for the arctangent function is given by
where i = − 1 {\displaystyle i={\sqrt {-1}}} is the imaginary unit . [ 16 ]
Two alternatives to the power series for arctangent are these generalized continued fractions :
The second of these is valid in the cut complex plane. There are two cuts, from − i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2 , with each perfect square appearing once. The first was developed by Leonhard Euler ; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series .
For real and complex values of z :
For real x ≥ 1:
For all real x not between -1 and 1:
The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions :
The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.
All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.
Using ∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du} (i.e. integration by parts ), set
Then
which by the simple substitution w = 1 − x 2 , d w = − 2 x d x {\displaystyle w=1-x^{2},\ dw=-2x\,dx} yields the final result:
Since the inverse trigonometric functions are analytic functions , they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points . One possible way of defining the extension is:
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.
The arcsine function may then be defined as:
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
which has the same cut as arcsin;
which has the same cut as arctan;
where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
which has the same cut as arcsec.
These functions may also be expressed using complex logarithms . This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:
or
where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } .
or
Simply taking the imaginary part works for any real-valued a {\displaystyle a} and b {\displaystyle b} , but if a {\displaystyle a} or b {\displaystyle b} is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of ln ( a + b i ) {\displaystyle \ln(a+bi)} also removes c {\displaystyle c} from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z {\displaystyle z} , we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ {\displaystyle \theta } that result from plugging the values into the equations θ = − i ln ( a + i b c ) {\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)} above and simplifying.
The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the Im ( ln z ) ∈ ( − π , π ] {\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]} and Re ( z ) ≥ 0 {\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0} principal branch for every function except arccotangent in the θ {\displaystyle \theta } column. Arccotangent in the θ {\displaystyle \theta } column will output on its usual principal branch by using the Im ( ln z ) ∈ [ 0 , 2 π ) {\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )} and Im ( z ) ≥ 0 {\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0} convention.
In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions . It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.
Using the exponential definition of sine , and letting ξ = e i ϕ , {\displaystyle \xi =e^{i\phi },}
(the positive branch is chosen)
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that
Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem : a 2 + b 2 = h 2 {\displaystyle a^{2}+b^{2}=h^{2}} where h {\displaystyle h} is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:
The two-argument atan2 function computes the arctangent of y / x given y and x , but with a range of (−π, π] . In other words, atan2( y , x ) is the angle between the positive x -axis of a plane and the point ( x , y ) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0 ), and negative sign for clockwise angles (lower half-plane, y < 0 ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.
In terms of the standard arctan function, that is with range of (−π/2, π/2) , it can be expressed as follows:
atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y ≥ 0 , x < 0 arctan ( y x ) − π y < 0 , x < 0 π 2 y > 0 , x = 0 − π 2 y < 0 , x = 0 undefined y = 0 , x = 0 {\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}
It also equals the principal value of the argument of the complex number x + iy .
This limited version of the function above may also be defined using the tangent half-angle formulae as follows: atan2 ( y , x ) = 2 arctan ( y x 2 + y 2 + x ) {\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)} provided that either x > 0 or y ≠ 0 . However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.
The above argument order ( y , x ) seems to be the most common, and in particular is used in ISO standards such as the C programming language , but a few authors may use the opposite convention ( x , y ) so some caution is warranted. (See variations at atan2 § Realizations of the function in common computer languages .)
In many applications [ 17 ] the solution y {\displaystyle y} of the equation x = tan ( y ) {\displaystyle x=\tan(y)} is to come as close as possible to a given value − ∞ < η < ∞ {\displaystyle -\infty <\eta <\infty } . The adequate solution is produced by the parameter modified arctangent function
The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer.
For angles near 0 and π , arccosine is ill-conditioned , and similarly with arcsine for angles near − π /2 and π /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. [ 18 ] | https://en.wikipedia.org/wiki/Inverse_trigonometric_functions |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_vercosine |
The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , [ 1 ] Section I) trigonometric tables . The versine of an angle is 1 minus its cosine .
There are several related functions, most notably the coversine and haversine . The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] or versed sine [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , [ 13 ] [ 14 ] vers , or siv . [ 15 ] [ 16 ] In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). [ 17 ]
Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
There are several related functions corresponding to the versine:
Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation .
hav θ = sin 2 ( θ 2 ) = 1 − cos θ 2 {\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}
The ordinary sine function ( see note on etymology ) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine ( sinus versus ). [ 31 ] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle :
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) (equal to the length of line OC ) and versin( θ ) (equal to the length of line CD ) is the radius OD (with length 1). Illustrated this way, the sine is vertical ( rectus , literally "straight") while the versine is horizontal ( versus , literally "turned against, out-of-place"); both are distances from C to the circle.
This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow . [ 17 ] [ 30 ] If the arc ADB of the double-angle Δ = 2 θ is viewed as a " bow " and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [ 30 ]
In 1821, Cauchy used the terms sinus versus ( siv ) for the versine and cosinus versus ( cosiv ) for the coversine. [ 15 ] [ 16 ] [ nb 1 ]
As θ goes to zero, versin( θ ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation , making separate tables for the latter convenient. [ 12 ] Even with a calculator or computer, round-off errors make it advisable to use the sin 2 formula for small θ .
Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle ( θ = 0, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half- chord ) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°). [ 31 ]
The versine appears as an intermediate step in the application of the half-angle formula sin 2 ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that was used to construct such tables.
The haversine, in particular, was important in navigation because it appears in the haversine formula , which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the Earth's radius vs. sphere ) given angular positions (e.g., longitude and latitude ). One could also use sin 2 ( θ / 2 ) directly, but having a table of the haversine removed the need to compute squares and square roots. [ 12 ]
An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. [ 14 ] [ 32 ]
The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". [ 33 ] [ 34 ] [ 17 ]
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. ) was coined [ 35 ] by James Inman [ 14 ] [ 36 ] [ 37 ] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth using spherical trigonometry for applications in navigation. [ 3 ] [ 35 ] Inman also used the terms nat. versine and nat. vers. for versines. [ 3 ]
Other high-regarded tables of haversines were those of Richard Farley in 1856 [ 33 ] [ 38 ] and John Caulfield Hannyngton in 1876. [ 33 ] [ 39 ]
The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 [ 40 ] [ 41 ] or in a more compact method for sight reduction since 2014. [ 29 ]
While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann , Hann–Poisson and Tukey windows ), because it smoothly ( continuous in value and slope ) "turns on" from zero to one (for haversine) and back to zero. [ nb 2 ] In these applications, it is named Hann function or raised-cosine filter .
The functions are circular rotations of each other.
Inverse functions like arcversine (arcversin, arcvers, [ 8 ] avers, [ 43 ] [ 44 ] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, [ 8 ] acovers, [ 43 ] [ 44 ] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin −1 , [ 45 ] invhav, [ 46 ] [ 47 ] [ 48 ] ahav, [ 43 ] [ 44 ] ahvs, ahv, hav −1 [ 49 ] [ 50 ] ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
These functions can be extended into the complex plane . [ 42 ] [ 19 ] [ 24 ]
Maclaurin series : [ 24 ]
When the versine v is small in comparison to the radius r , it may be approximated from the half-chord length L (the distance AC shown above) by the formula [ 51 ] v ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.}
Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle s\approx L+{\frac {v^{2}}{r}}} This formula was known to the Chinese mathematician Shen Kuo , and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing . [ 52 ]
A more accurate approximation used in engineering [ 53 ] is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of the straightness of the rail tracks [ 54 ] and it is the basis of the Hallade method for rail surveying .
The term sagitta (often abbreviated sag ) is used similarly in optics , for describing the surfaces of lenses and mirrors . | https://en.wikipedia.org/wiki/Inverse_versine |
Inverse vulcanization is a process that produces polysulfide polymers , which also contain some organic linkers. [ 1 ] In contrast, sulfur vulcanization produces material that is predominantly organic but has a small percentage of polysulfide crosslinks .
Like Thiokols and sulfur-vulcanization , inverse vulcanization uses the tendency of sulfur catenate . The polymers produced by inverse vulcanization consist of long sulfur linear chains interspersed with organic linkers. Traditional sulfur vulcanization produces a cross-linked material with short sulfur bridges, down to one or two sulfur atoms.
The polymerization process begins with the heating of elemental sulfur above its melting point (115.21 °C), to favor the ring-opening polymerization process (ROP) of the S 8 monomer , occurring at 159 °C. As a result, the liquid sulfur is constituted by linear polysulfide chains with diradical ends, which can be easily bridged together with small dienes , such as 1,3-Diisopropylbenzene (DIB), [ 1 ] 1,4-diphenylbutadiyne , [ 2 ] limonene , [ 3 ] divinylbenzene (DVB), [ 4 ] dicyclopentadiene , [ 5 ] styrene , [ 6 ] 4-vinylpyridine , [ 7 ] cycloalkene [ 8 ] and ethylidene norbornene , [ 9 ] or longer organic molecules as polybenzoxazines , [ 10 ] squalene [ 11 ] and triglyceride . [ 12 ] Chemically, the diene carbon-carbon double bond (C=C) of the substitutional group disappears, forming the carbon-sulfur single bond (C-S) which binds together the sulfur linear chains. The advantage of such a polymerization is the absence of a solvent; Sulphur acts as comonomer and solvent . This makes the process highly scalable at the industrial level, and kilogram-scale synthesis of the poly(S-r-DIB) has already been accomplished. [ 13 ]
Vibrational spectroscopy was performed to investigate the chemical structure of the copolymers, and the presence of the C-S bonds was detected through Infrared or Raman spectroscopies. [ 14 ] The high amount of S-S bonds makes the copolymer highly IR-inactive in the near and mid-infrared spectrum. As a consequence, sulfur-rich materials made via inverse vulcanization are characterized by a high refractive index (n~1.8), whose value depends again upon the composition and crosslinking species. [ 15 ] As shown by thermogravimetric analysis (TGA), the copolymer thermal stability increases with the amount of added crosslinker; however, all the tested compositions degrade above 222 °C. [ 2 ] [ 4 ]
Copolymer behavior included that, the glass-transition temperature depends upon the composition and crosslinking species. For given comonomers , the behavior of the copolymers as a function of the temperature depends on the chemical composition; for example, the poly (sulfur-random- divinylbenzene ) behaves as a plastomer for a diene content between 15 and 25%wt, and as a viscous resin with the 30–35%wt of DVB. On the other hand, the poly (sulfur-random- 1,3-diisopropenylbenzene ) acts as thermoplastic at 15–25%wt of DIB, while it becomes a thermoplastic - thermosetting polymer for a diene concentration of 30-35%wt. [ 16 ] The potential to break and reform the chemical bonds along the polysulfide chains (S-S) allows the repair of the copolymer by simply heating above 100 °C. This increases the ability to reform and recycle the high molecular weight copolymer. [ 17 ]
The sulfur-rich copolymers made via inverse vulcanization could in principle find diverse applications due to their simple synthesis process and thermoplasticity .
This new way of sulfur processing has been exploited for the cathode preparation of long-cycling lithium-sulfur batteries . Such electrochemical systems are characterized by a greater energy density than commercial Li-ion batteries , but they are not stable for long service life. Simmonds et al. first demonstrated improved capacity retention for over 500 cycles with an inverse vulcanization copolymer, suppressing the typical capacity fading of sulfur-polymer composites. [ 18 ] The poly (sulfur-random-1,3-diisopropenylbenzene), briefly defined as poly (S-r-DIB), showed a higher composition homogeneity compared with other cathodic materials, together with greater sulfur retention and an enhanced adjustment of the polysulfides' volume variations. These advantages made it possible to assemble a stable and durable Li-S cell. Subsequently, other copolymers were synthesized via inverse vulcanization and tested inside these electrochemical devices, again providing high stability over their cycles.
In order to overcome the disadvantages related to the materials' low electrical conductivity (10 15 –10 16 Ω·cm), [ 16 ] researchers have started to add special carbon-based particles to increase electron transport inside the copolymer. Furthermore, such carbonaceous additives improve the polysulfides' retention at the cathode through the polysulfides-capturing effect, increasing the battery performances. Examples of employed nanostructures are long carbon nanotubes , [ 22 ] graphene , [ 11 ] and carbon onions . [ 23 ]
The new materials could be used to remove toxic metals from soil or water. Pure sulfur cannot be employed to manufacture a functional filter because of its low mechanical properties; therefore, inverse vulcanization was investigated to produce porous materials, in particular for the mercury capturing process. The liquid metal binds together with the sulfur-rich copolymer , remaining mostly inside the filter. [ 3 ] [ 24 ] [ 25 ]
Sulfur-rich copolymers , made via inverse vulcanization, have advantages over traditional IR optical materials due to the simple manufacturing process, low cost reagents , and high refractive index . As mentioned before, the latter depends upon the S-S bonds concentration, leading to the ability to tune the optical properties of the material by modifying the chemical formulation. The ability to change the material's refractive index to fulfill the specific application requirements makes these copolymers applicable in military, civil or medical fields. [ 15 ] [ 26 ] [ 27 ] [ 28 ]
The inverse vulcanization process can also be employed for the synthesis of activated carbon with narrow pore-size distributions. The sulfur-rich copolymer acts as a template where the carbons are produced. The final material is doped with sulfur and exhibits a micro-porous network and high gas selectivity. Therefore, inverse vulcanization could also be used for gas separation applications. [ 29 ] | https://en.wikipedia.org/wiki/Inverse_vulcanization |
In computer science and discrete mathematics , an inversion in a sequence is a pair of elements that are out of their natural order .
Let π {\displaystyle \pi } be a permutation .
There is an inversion of π {\displaystyle \pi } between i {\displaystyle i} and j {\displaystyle j} if i < j {\displaystyle i<j} and π ( i ) > π ( j ) {\displaystyle \pi (i)>\pi (j)} . The inversion is indicated by an ordered pair containing either the places ( i , j ) {\displaystyle (i,j)} [ 1 ] [ 2 ] or the elements ( π ( i ) , π ( j ) ) {\displaystyle {\bigl (}\pi (i),\pi (j){\bigr )}} . [ 3 ] [ 4 ] [ 5 ]
The inversion set is the set of all inversions. A permutation's inversion set using place-based notation is the same as the inverse permutation's inversion set using element-based notation with the two components of each ordered pair exchanged. Likewise, a permutation's inversion set using element-based notation is the same as the inverse permutation's inversion set using place-based notation with the two components of each ordered pair exchanged. [ 6 ]
Inversions are usually defined for permutations, but may also be defined for sequences: Let S {\displaystyle S} be a sequence (or multiset permutation [ 7 ] ). If i < j {\displaystyle i<j} and S ( i ) > S ( j ) {\displaystyle S(i)>S(j)} , either the pair of places ( i , j ) {\displaystyle (i,j)} [ 7 ] [ 8 ] or the pair of elements ( S ( i ) , S ( j ) ) {\displaystyle {\bigl (}S(i),S(j){\bigr )}} [ 9 ] is called an inversion of S {\displaystyle S} .
For sequences, inversions according to the element-based definition are not unique, because different pairs of places may have the same pair of values.
The inversion number i n v ( X ) {\displaystyle {\mathtt {inv}}(X)} [ 10 ] of a sequence X = ⟨ x 1 , … , x n ⟩ {\displaystyle X=\langle x_{1},\dots ,x_{n}\rangle } , is the cardinality of the inversion set. It is a common measure of sortedness (sometimes called presortedness) of a permutation [ 5 ] or sequence. [ 9 ] The inversion number is between 0 and n ( n − 1 ) 2 {\displaystyle {\frac {n(n-1)}{2}}} inclusive. A permutation and its inverse have the same inversion number.
For example i n v ( ⟨ 1 , 2 , … , n ⟩ ) = 0 {\displaystyle {\mathtt {inv}}(\langle 1,2,\dots ,n\rangle )=0} since the sequence is ordered. Also, when n = 2 m {\displaystyle n=2m} is even, i n v ( ⟨ m + 1 , m + 2 , … , 2 m , 1 , 2 , … , m ⟩ ) = m 2 {\displaystyle {\mathtt {inv}}(\langle m+1,m+2,\dots ,2m,1,2,\dots ,m\rangle )=m^{2}} (because each pair ( 1 ≤ i ≤ m < j ≤ 2 m ) {\displaystyle (1\leq i\leq m<j\leq 2m)} is an inversion). This last example shows that a set that is intuitively "nearly sorted" can still have a quadratic number of inversions.
The inversion number is the number of crossings in the arrow diagram of the permutation, [ 6 ] the permutation's Kendall tau distance from the identity permutation, and the sum of each of the inversion related vectors defined below.
Other measures of sortedness include the minimum number of elements that can be deleted from the sequence to yield a fully sorted sequence, the number and lengths of sorted "runs" within the sequence, the Spearman footrule (sum of distances of each element from its sorted position), and the smallest number of exchanges needed to sort the sequence. [ 11 ] Standard comparison sorting algorithms can be adapted to compute the inversion number in time O( n log n ) . [ 12 ]
Three similar vectors are in use that condense the inversions of a permutation into a vector that uniquely determines it. They are often called inversion vector or Lehmer code . (A list of sources is found here .)
This article uses the term inversion vector ( v {\displaystyle v} ) like Wolfram . [ 13 ] The remaining two vectors are sometimes called left and right inversion vector , but to avoid confusion with the inversion vector this article calls them left inversion count ( l {\displaystyle l} ) and right inversion count ( r {\displaystyle r} ). Interpreted as a factorial number the left inversion count gives the permutations reverse colexicographic, [ 14 ] and the right inversion count gives the lexicographic index.
Inversion vector v {\displaystyle v} : With the element-based definition v ( i ) {\displaystyle v(i)} is the number of inversions whose smaller (right) component is i {\displaystyle i} . [ 3 ]
Left inversion count l {\displaystyle l} : With the place-based definition l ( i ) {\displaystyle l(i)} is the number of inversions whose bigger (right) component is i {\displaystyle i} .
Right inversion count r {\displaystyle r} , often called Lehmer code : With the place-based definition r ( i ) {\displaystyle r(i)} is the number of inversions whose smaller (left) component is i {\displaystyle i} .
Both v {\displaystyle v} and r {\displaystyle r} can be found with the help of a Rothe diagram , which is a permutation matrix with the 1s represented by dots, and an inversion (often represented by a cross) in every position that has a dot to the right and below it. r ( i ) {\displaystyle r(i)} is the sum of inversions in row i {\displaystyle i} of the Rothe diagram, while v ( i ) {\displaystyle v(i)} is the sum of inversions in column i {\displaystyle i} . The permutation matrix of the inverse is the transpose, therefore v {\displaystyle v} of a permutation is r {\displaystyle r} of its inverse, and vice versa.
The following sortable table shows the 24 permutations of four elements (in the π {\displaystyle \pi } column) with their place-based inversion sets (in the p-b column), inversion related vectors (in the v {\displaystyle v} , l {\displaystyle l} , and r {\displaystyle r} columns), and inversion numbers (in the # column). (The columns with smaller print and no heading are reflections of the columns next to them, and can be used to sort them in colexicographic order .)
It can be seen that v {\displaystyle v} and l {\displaystyle l} always have the same digits, and that l {\displaystyle l} and r {\displaystyle r} are both related to the place-based inversion set. The nontrivial elements of l {\displaystyle l} are the sums of the descending diagonals of the shown triangle, and those of r {\displaystyle r} are the sums of the ascending diagonals. (Pairs in descending diagonals have the right components 2, 3, 4 in common, while pairs in ascending diagonals have the left components 1, 2, 3 in common.)
The default order of the table is reverse colex order by π {\displaystyle \pi } , which is the same as colex order by l {\displaystyle l} . Lex order by π {\displaystyle \pi } is the same as lex order by r {\displaystyle r} .
The set of permutations on n items can be given the structure of a partial order , called the weak order of permutations , which forms a lattice .
The Hasse diagram of the inversion sets ordered by the subset relation forms the skeleton of a permutohedron .
If a permutation is assigned to each inversion set using the place-based definition, the resulting order of permutations is that of the permutohedron, where an edge corresponds to the swapping of two elements with consecutive values. This is the weak order of permutations. The identity is its minimum, and the permutation formed by reversing the identity is its maximum.
If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph , where an edge corresponds to the swapping of two elements on consecutive places. This Cayley graph of the symmetric group is similar to its permutohedron, but with each permutation replaced by its inverse.
Sequences in the OEIS : | https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics) |
In evolutionary developmental biology , inversion refers to the hypothesis that during the course of animal evolution, the structures along the dorsoventral (DV) axis have taken on an orientation opposite that of the ancestral form.
Inversion was first noted in 1822 by the French zoologist Étienne Geoffroy Saint-Hilaire , when he dissected a crayfish (an arthropod ) and compared it with the vertebrate body plan . The idea was heavily criticised, but periodically resurfaced, and is now supported by some molecular embryologists.
As early as 1822, the French zoologist Étienne Geoffroy Saint-Hilaire noted that the organization of dorsal and ventral structures in arthropods is opposite that of mammals . Five decades later, in light of Darwin 's theory of "descent with modification", German zoologist Anton Dohrn proposed that these groups arose from a common ancestor which possessed a body plan similar to that of modern annelids with a ventral nerve cord and dorsal heart. [ 1 ] Whereas this arrangement is retained in arthropods and other protostomes , in chordate deuterostomes , the nerve cord is located dorsally and the heart ventrally. The inversion hypothesis was met with scornful criticism each time it was proposed, and has periodically resurfaced and been rejected. [ 1 ] However, some modern molecular embryologists suggest that recent findings support the idea of inversion.
In addition to the simple observation that the dorsoventral axes of protostomes and chordates appear to be inverted with respect to each other, molecular biology provides some support for the inversion hypothesis. The most notable piece of evidence comes from analysis of the genes involved in establishing the DV axis in these two groups. [ 2 ] In the fruit fly Drosophila melanogaster , as well as in other protostomes, the β-type transforming growth factor ( TGF-β ) family member decapentaplegic ( dpp ) is expressed dorsally and is thought to suppress neural fate. On the ventral side of the embryo, a dpp inhibitor, short gastrulation ( sog ), is expressed, thus allowing nervous tissue to form ventrally. In chordates, the dpp homolog BMP-4 is expressed in the prospective ventral (non-neural) part of the embryo while several sog -like BMP inhibitors ( Chordin , Noggin , Follistatin ) are expressed dorsally. [ 1 ]
Other patterning genes also show conserved domains of expression. The neural patterning genes vnd , ind , msh , and netrin are expressed in the Drosophila ventral nerve cells and midline mesectoderm. The chordate homologs of these genes, NK2 , Gsh1/2, Msx1/3, and Netrin , are expressed in the dorsal neural tube . Furthermore, the tinman/Nkx2-5 gene is expressed very early in cells that will become the heart in both Drosophila (dorsally) and chordates (ventrally). [ 1 ]
Additional support comes from work on the development of the polychaete annelid Platynereis dumerilii , another protostome. Even more so than Drosophila , its pattern of central-nervous-system development is strikingly similar to that of vertebrates, but inverted. [ 3 ] There is also evidence from left-right asymmetry. Vertebrates have a highly conserved Nodal signaling pathway that acts on the left side of the body, determining left-right asymmetries of internal organs. Sea urchins have the same signaling pathway, but it acts on the right side of the body. [ 4 ] It was even shown that an opposing right-sided signal for regulating left-right asymmetry in vertebrates, i.e. BMP signaling pathway, is activated on the left side of the sea urchin larva, [ 5 ] suggesting an axial inversion during evolution from basal deuterostome to chordate such as amphioxus . The Nodal signaling pathway in amphioxus is on the left side of the embryo, which is the same situation as vertebrates. [ 6 ] Sea urchins, like other echinoderms, have radially-symmetric adults, but bilaterally -symmetric larvae. Since sea urchins are deuterostomes, this suggests that the ancestral deuterostome shared its orientation with protostomes, and that dorsoventral inversion originated in some ancestral chordate.
There is evidence that invertebrate chordates are also inverted. Ascidian larvae have a dorsal mouth, as one would expect from inversion. [ 7 ] The amphioxus has an odd feature: its mouth appears on the left side and migrates to the ventral side. [ 8 ] Biologist Thurston Lacalli speculates that this may be a recapitulation of the migration of the mouth from the dorsal to the ventral side in a protochordate. [ 9 ]
Some biologists have proposed that the Hemichordates (specifically the Enteropneusta ) may represent an intermediate body plan in the evolution of the "inverted" state of the chordates. [ 10 ] Though they are considered deuterostomes, the dorsoventral axis of hemichordates retains features of both protostomes and chordates. For example, enteropneusts have an ectodermally -derived dorsal nerve cord in the collar region which has been proposed to be homologous to the chordate neural tube. However, they also have a ventral nerve cord and a dorsal contractile vessel similar to protostomes. [ 10 ] Furthermore, the relative positions other "intermediate" structures in hemichordates, such as the hepatic organs and ventral pygochord, which has been proposed to be homologous to the chordate-defining notochord , [ 11 ] are retained but inverted. Nübler-Jung and Arendt argue that the principal innovation in the chordate lineage was the obliteration of the mouth on the neural side (as in hemichordates, arthropods, and annelids) and the development of a new mouth on the non-neural ventral side. [ 10 ]
While the idea of dorsoventral axis inversion appears to be supported by morphological and molecular data, others have proposed alternative plausible hypotheses (reviewed in Gerhart 2000). [ 1 ] One assumption of the inversion hypothesis is that the common ancestor of protostomes and chordates already possessed an organized central nervous system located at one pole of the dorsoventral axis. Alternatively, this ancestor may have possessed only a diffuse nerve net or several bundles of nervous tissue with no distinct dorsoventral localization. [ 1 ] [ 12 ] This would mean that the apparent inversion was simply a result of concentration of the central nervous system at opposite poles independently in the lineages leading to protostomes and chordates. Lacalli (1996) suggested a scenario in which the ancestor had a single opening to the digestive system, and that the neural and non-neural mouths arose independently in protostomes and chordates, respectively. [ 13 ] By this hypothesis, there is no need for inversion.
Martindale and Henry propose a ctenophore -like ancestor (biradial rather than bilateral) with a concentrated nerve cord and two anal pores on opposite sides of the animal in addition to a terminal gut opening. [ 1 ] [ 14 ] If one of these pores became the mouth in protostomes and the other became the mouth in deuterostomes, this would also preclude inversion. Another alternative, proposed by von Salvini-Plawen, states that the ancestor had a two-part nervous system - one part concentrated, the other diffuse. The nervous systems of protostome and deuterostome descendants of this ancestor may have arisen independently from these two distinct parts. [ 1 ] [ 15 ]
The Axial Twist theory proposes that not the whole body, but only the anterior region of the head is inverted. [ 16 ] [ 17 ] These theories by Kinsbourne [ 16 ] and de Lussanet & Osse [ 17 ] also explain the presence of an optic chiasm in vertebrates and the contralateral organization of the forebrain . One of these theories [ 17 ] is supported by developmental evidence and even explains the asymmetric organization of the heart and bowels . [ 18 ] | https://en.wikipedia.org/wiki/Inversion_(evolutionary_biology) |
Inversion recovery is a magnetic resonance imaging sequence that provides high contrast between tissue and lesion . It can be used to provide high T1 weighted image, high T2 weighted image, and to suppress the signals from fat , blood , or cerebrospinal fluid (CSF). [ 1 ]
Fluid-attenuated inversion recovery (FLAIR) [ 2 ] is an inversion-recovery pulse sequence used to nullify the signal from fluids. For example, it can be used in brain imaging to suppress cerebrospinal fluid so as to bring out periventricular hyperintense lesions, such as multiple sclerosis plaques. By carefully choosing the inversion time TI (the time between the inversion and excitation pulses), the signal from any particular tissue can be suppressed.
Turbo inversion recovery magnitude (TIRM) measures only the magnitude of a turbo spin echo after a preceding inversion pulse, thus is phase insensitive. [ 3 ]
TIRM is superior in the assessment of osteomyelitis and in suspected head and neck cancer . [ 4 ] [ 5 ] Osteomyelitis appears as high intensity areas. [ 6 ] In head and neck cancers, TIRM has been found to both give high signal in tumor mass, as well as low degree of overestimation of tumor size by reactive inflammatory changes in the surrounding tissues. [ 7 ]
Double inversion recovery is a sequence that suppresses both cerebrospinal fluid (CSF) and white matter , and samples the remaining transverse magnetisation in fast spin echo , where the majority of the signals are from the grey matter . Thus, this sequence is useful in detecting small changes on the brain cortex such as focal cortical dysplasia and hippocampal sclerosis in those with epilepsy . These lesions are difficult to detect in other MRI sequences. [ 8 ]
Erwin Hahn first used inversion recovery technique to determine the value of T1 (the time taken for longitudinal magnetisation to recover 63% of its maximum value) for water in 1949, 3 years after the nuclear magnetic resonance was discovered. [ 1 ] | https://en.wikipedia.org/wiki/Inversion_recovery |
The inversion temperature in thermodynamics and cryogenics is the critical temperature below which a non- ideal gas (all gases in reality) that is expanding at constant enthalpy will experience a temperature decrease, and above which will experience a temperature increase. This temperature change is known as the Joule–Thomson effect , and is exploited in the liquefaction of gases . Inversion temperature depends on the nature of the gas.
For a van der Waals gas we can calculate the enthalpy H {\displaystyle H} using statistical mechanics as
where N {\displaystyle N} is the number of molecules, V {\displaystyle V} is volume, T {\displaystyle T} is temperature (in the Kelvin scale ), k B {\displaystyle k_{\mathrm {B} }} is the Boltzmann constant , and a {\displaystyle a} and b {\displaystyle b} are constants depending on intermolecular forces and molecular volume, respectively.
From this equation, if enthalpy is kept constant and there is an increase of volume, temperature must change depending on the sign of b k B T − 2 a {\displaystyle bk_{\mathrm {B} }T-2a} . Therefore, our inversion temperature is given where the sign flips at zero, or
where T c {\displaystyle T_{\mathrm {c} }} is the critical temperature of the substance. So for T > T inv {\displaystyle T>T_{\text{inv}}} , an expansion at constant enthalpy increases temperature as the work done by the repulsive interactions of the gas is dominant, and so the change in kinetic energy is positive. But for T < T inv {\displaystyle T<T_{\text{inv}}} , expansion causes temperature to decrease because the work of attractive intermolecular forces dominates, giving a negative change in average molecular speed, and therefore kinetic energy. [ 1 ] | https://en.wikipedia.org/wiki/Inversion_temperature |
In mathematical physics , inversion transformations are a natural extension of Poincaré transformations to include all conformal , one-to-one transformations on coordinate space-time . [ 1 ] [ 2 ] They are less studied in physics because, unlike the rotations and translations of Poincaré symmetry, an object cannot be physically transformed by the inversion symmetry. Some physical theories are invariant under this symmetry, in these cases it is what is known as a 'hidden symmetry'. Other hidden symmetries of physics include gauge symmetry and general covariance .
In 1831 the mathematician Ludwig Immanuel Magnus began to publish on transformations of the plane generated by inversion in a circle of radius R . His work initiated a large body of publications, now called inversive geometry . The most prominently named mathematician became August Ferdinand Möbius once he reduced the planar transformations to complex number arithmetic. In the company of physicists employing the inversion transformation early on was Lord Kelvin , and the association with him leads it to be called the Kelvin transform .
In the following we shall use imaginary time ( t ′ = i t {\displaystyle t'=it} ) so that space-time is Euclidean and the equations are simpler. The Poincaré transformations are given by the coordinate transformation on space-time parametrized by the 4-vectors V
where O {\displaystyle O} is an orthogonal matrix and P {\displaystyle P} is a 4-vector. Applying this transformation twice on a 4-vector gives a third transformation of the same form. The basic invariant under this transformation is the space-time length given by the distance between two space-time points given by 4-vectors x and y :
These transformations are subgroups of general 1-1 conformal transformations on space-time. It is possible to extend these transformations to include all 1-1 conformal transformations on space-time
We must also have an equivalent condition to the orthogonality condition of the Poincaré transformations:
Because one can divide the top and bottom of the transformation by D , {\displaystyle D,} we lose no generality by setting D {\displaystyle D} to the unit matrix. We end up with
Applying this transformation twice on a 4-vector gives a transformation of the same form. The new symmetry of 'inversion' is given by the 3-tensor Q . {\displaystyle Q.} This symmetry becomes Poincaré symmetry if we set Q = 0. {\displaystyle Q=0.} When Q = 0 {\displaystyle Q=0} the second condition requires that O {\displaystyle O} is an orthogonal matrix. This transformation is 1-1 meaning that each point is mapped to a unique point only if we theoretically include the points at infinity.
The invariants for this symmetry in 4 dimensions is unknown however it is known that the invariant requires a minimum of 4 space-time points. In one dimension, the invariant is the well known cross-ratio from Möbius transformations :
Because the only invariants under this symmetry involve a minimum of 4 points, this symmetry cannot be a symmetry of point particle theory. Point particle theory relies on knowing the lengths of paths of particles through space-time (e.g., from x {\displaystyle x} to y {\displaystyle y} ). The symmetry can be a symmetry of a string theory in which the strings are uniquely determined by their endpoints. The propagator for this theory for a string starting at the endpoints ( x , X ) {\displaystyle (x,X)} and ending at the endpoints ( y , Y ) {\displaystyle (y,Y)} is a conformal function of the 4-dimensional invariant. A string field in endpoint-string theory is a function over the endpoints.
Although it is natural to generalize the Poincaré transformations in order to find hidden symmetries in physics and thus narrow down the number of possible theories of high-energy physics , it is difficult to experimentally examine this symmetry as it is not possible to transform an object under this symmetry. The indirect evidence of this symmetry is given by how accurately fundamental theories of physics that are invariant under this symmetry make predictions. Other indirect evidence is whether theories that are invariant under this symmetry lead to contradictions such as giving probabilities greater than 1. So far there has been no direct evidence that the fundamental constituents of the Universe are strings. The symmetry could also be a broken symmetry meaning that although it is a symmetry of physics, the Universe has 'frozen out' in one particular direction so this symmetry is no longer evident. | https://en.wikipedia.org/wiki/Inversion_transformation |
Invertebrate drift is the downstream transport of invertebrate organisms in lotic freshwater systems such as rivers and streams. The term lotic comes from the Latin word lotus , meaning "washing", and is used to describe moving freshwater systems . This is in contrast with lentic coming from the Latin word lentus , meaning slow or motionless that typically describe still or standing waters such as lakes , ponds , and swamps . [ 1 ]
Drift can service freshwater invertebrates by giving them an escape route from predation , or the use of a current to disperse progeny downstream. [ 2 ] On occasion, however, invertebrates will inadvertently lose their footing, and drift downstream. For that, invertebrates counter a stream's flow through physical and behavioral adaptations . [ 3 ] And just as invertebrates adapted to stabilize themselves in the water column , or use the stream's energy to their advantage, so too have predators adapted to catch invertebrates as they drift. Species of fish, commonly salmonids , catch drifting insects during the peak times after dusk, and before dawn. [ 4 ] Fishermen can exploit this relationship using fly fishing techniques and lures that mimic drifting insects to catch these fishes. [ 5 ]
Researchers have developed sampling techniques in lotic systems. From it, research as far back as 1928 has collected data on the phenomenon of drift. [ 6 ] The study of invertebrate drift has progressed the field of stream ecology. Drift has been documented to impact community structure, benthic production, and the energy flow through trophic levels . [ 7 ]
Invertebrate drift can be categorized by the conditions that caused the drift to occur.
Invertebrate species adapt to a stream's current through the organs or appendages that physically attach them to the substrate, or association with large boulders or thick plant growth to buffer the disturbances associated with flow. [ 9 ] An example of the former is the family Heptageniidae in the order Ephemeroptera . Larvae within the genus have modified gills forming a friction disc that allows them to cling to the substrate in rapid moving waters. [ 3 ] An example of the friction disk can be seen on the image to the right which shows the ventral side of a species within the genus epeorus .
Müller (1954) found that water mites ( Hydracarina ) and aquatic beetles ( Coleoptera ) made up a large portion of the benthos population in the stream Skravelbäcken of Sweden , but since they associated with boulders and thick plant growth, they avoid being dislodged by water currents into drift. [ 9 ]
Many predators of common insects and invertebrates found in streams feed off of those found in stream drift. Many of these predators have adapted or have become specialized to feeding on invertebrates found in stream drift. Predators that use this as their main source for food, typically fish, are called drift-feeders.
The most common example of drift-feeding predators are stream salmonids , especially trout . These fish catch a lot of their prey during dusk and dawn. This has led to studies concluding that many invertebrates have adapted to drifting at night, where they can avoid predation due to these fish being mainly visual hunters. [ 4 ]
Other fish, such as the sculpin , have evolved with highly developed lateral lines , allowing them to have better nocturnal predation skills. As such, sculpins were found to catch a majority of their prey at night, as well as during the day. [ 4 ]
Fish predation on invertebrate has been seen to alter prey densities in streams by individual feeding of insects or by effecting insect dispersal behavior. [ 4 ]
Common invertebrate species have adapted drifting behaviors to help avoid predation. The biggest example, as mentioned before, is to drift at night. However, invertebrate have adapted to change their drift behavior to avoid predation after receiving other certain signals and indicators. For example, the mayfly Baetis bicaudatis was shown to change its behavior based on odors chemically released in the water system similar to fish predators. [ 7 ]
Although fish are the main predators of invertebrates in stream drift, there are others as well, such as birds and large insects. For example, the white-throated dipper Cinclus cinclus , is an aquatic bird that feeds off of invertebrates in stream drift. [ 10 ] Another example is the stonefly, which is a large insect that has been found to prey on other small drifting insects, such as the mayfly. [ 11 ]
Changes to the environment as a result of abiotic factors can lead to both increases and decreases of invertebrate drift. Factors such as a reduction of stream flow can lead to an increase of invertebrate drift, as observed by Minshall and Winger in their 1968 study. [ 12 ] They found that as stream flow and the frequency had an inverse relationship over the course of July, August, and September in the Rocky Mountains of Idaho. [ 12 ]
Koetsier and Bryan sought to assess the effect of abiotic factors on invertebrate drift in the lower Mississippi river. Just as with Minshall and Winger, they found that there was a negative correlation between stream discharge and frequency of invertebrate drift. [ 13 ] According to their 1995 study, river discharge could be attributed to approximately 40% of variation in the taxa of invertebrates which are more prone to drift. [ 13 ]
Invertebrate drift is also affected by the day/night cycle. At night invertebrate drift can be up to 10 times higher than during the day. [ 14 ] Benke et al. found that all of the invertebrates that they sampled had a consistency to be more active in the drift at night especially during the summer. [ 14 ] They found that this pattern of drift happening at night continued all throughout the year, but that the extent of the difference in drift between day and night was not as exaggerated as during the summer. [ 14 ] Benke et al. also found that in southern states the invertebrate drift is more abundant and consistent which is largely attributed to the fact that there is not a sharp decline in temperature during winter like happens in more northern states. [ 14 ]
The concept of drift can be traced back to 1928 where an experiment conducted by P. R. Needham of Cornell University sought to quantify allochthonous animal material in various stream environments. Needham used drift and stop nets to collect available drifting material and organisms and calculated the collection based on the area length of stream reaches. The study served as a proof of concept for use of these methods for future quantitative and qualitative ecological studies of drifting organisms. [ 6 ]
After a period of dormancy, there was a resurgence on the research of drift in the 1950s and 1960s. [ 2 ] A prominent paper of the time was Müller's 1954 Investigations on the organic drift in North Sweden streams. Müller proposed the term "colonization cycle" after observing the upper stream reaches in Sweden were recolonizing quickly despite their progeny's physical inability to migrate against a current. To counter competition, immature organisms disperse downstream then migrated back upstream as adults to spawn, thus, replenishing populations. [ 9 ]
In the early 1960s, research done by Hikaru Tanaka, Thomas F. Walters, and Karl Müller discovered that invertebrate drift followed a distinct diel periodicity. [ 2 ] In Walter’ 1962 paper "Diurnal Periodicity in the Drift of Stream Invertebrates", Walters measured the high volume of drifting scuds ( Gammarus limnaeus ) over 24 hours within four different months spanning each of the seasons. Uniformly, across August, October, February, and May, there was a notable increase in drift 1 hour after sunset and a notable decrease an hour before sunrise. In August specifically, there was a significant spike in scud volume after sunset reaching numbers of 100-fold of that caught a few hours prior. Walters measured other species caught like mayflies ( Baetis vagans ), caddisflies ( Glossosoma intermedium ), and water boatman adults ( Hesperocorixa sp.) and they all displayed a similar pattern of diurnal periodicity. Walters hypothesized that the higher drift rates coincided with higher invertebrate activity during the night, and as the invertebrates moved around freely, they were swept downstream by the current. [ 15 ]
There are three well used methods for sampling invertebrate drift: samplers with flow meters, samplers without flow meters, and tube samplers. Invertebrate drift is observed to function on 24-hour intervals. [ 2 ]
Tube samplers are used to pass stream discharge ending in the air above a filtering net. Therefore, the tube will be extended out of the water allowing water to exit and flow through a net filtering all invertebrates. [ 18 ] Advantages to this method is there is a rarity of back flow. [ 18 ] Disadvantages to this method involve invertebrates that have the ability to survive in the tube without being transferred through the filter. This can be solved by cleaning the tube after gathering sufficient data. [ 18 ]
The efficiency of these methods has been confirmed. Although, there are many factors that control samplers, it is believed that samplers maintain "laminar flow and do not significantly affect the velocity of water at the mouth." [ 20 ] These models sample drift at close to maximum efficiency.
Fly fishing is a method of angling that uses lures composed of hair , feathers , and synthetic materials that mimics a fly , bug , or other prey items. [ 21 ] [ 22 ] Using a long rod , typically between 7 and 11 feet (2 to 3.5 meters), the angler snaps the rod back and forth allowing the lure to rest just above the water's surface before flicking back. [ 21 ] [ 23 ] The method described is referred to as dry-fly fishing as the lure is on or above the water. In contrast, there is wet-fly fishing where the lure sits on or beneath the water's surface. [ 23 ] In wet-fly fishing, the angler casts their lure upstream allowing the current to carry the fly, whether submerged or on the surface, downstream to the target trout. [ 23 ]
A wet fly-fishing technique known as nymph fishing (or nymphing) is used commonly to catch trout who feed on drifting nymphs in shallow riffles . Anglers take advantage of invertebrate drift and cast their lines with a mimic nymph fly upstream and allow the river's current to carry their submerged lure downstream to where the trout are waiting to catch their prey . [ 8 ] | https://en.wikipedia.org/wiki/Invertebrate_drift |
The invertebrate mitochondrial code ( translation table 5 ) is a genetic code used by the mitochondrial genome of invertebrates . Mitochondria contain their own DNA and reproduce independently from their host cell. Variation in translation of the mitochondrial genetic code occurs when DNA codons result in non-standard amino acids has been identified in invertebrates, most notably arthropods . [ 1 ] This variation has been helpful as a tool to improve upon the phylogenetic tree of invertebrates, like flatworms . [ 2 ]
Bases: adenine (A), cytosine (C), guanine (G) and thymine (T) or uracil (U).
Amino acids: Alanine (Ala, A), Arginine (Arg, R), Asparagine (Asn, N), Aspartic acid (Asp, D), Cysteine (Cys, C), Glutamic acid (Glu, E), Glutamine (Gln, Q), Glycine (Gly, G), Histidine (His, H), Isoleucine (Ile, I), Leucine (Leu, L), Lysine (Lys, K), Methionine (Met, M), Phenylalanine (Phe, F), Proline (Pro, P), Serine (Ser, S), Threonine (Thr, T), Tryptophan (Trp, W), Tyrosine (Tyr, Y), Valine (Val, V).
Note: The codon AGG is absent in Drosophila . [ 3 ]
This article incorporates text from the United States National Library of Medicine , which is in the public domain . [ 15 ]
This microbiology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Invertebrate_mitochondrial_code |
Invertebrate zoology is the subdiscipline of zoology that consists of the study of invertebrates , animals without a backbone (a structure which is found only in fish , amphibians , reptiles , birds and mammals ).
Invertebrates are a vast and very diverse group of animals that includes sponges , echinoderms , tunicates , numerous different phyla of worms , molluscs , arthropods and many additional phyla. Single-celled organisms or protists are usually not included within the same group as invertebrates.
Invertebrates represent 97% of all named animal species , [ 1 ] and because of that fact, this subdivision of zoology has many further
subdivisions, including but not limited to:
These divisions are sometimes further divided into more specific specialties. For example, within arachnology, acarology is the study of mites and ticks ; within entomology, lepidoptery is the study of butterflies and moths , myrmecology is the study of ants and so on. Marine invertebrates are all those invertebrates that exist in marine habitats .
In the early modern period starting in the late 16th century, invertebrate zoology saw growth in the number of publications made and improvement in the experimental practices associated with the field. (Insects are one of the most diverse groups of organisms on Earth. They play important roles in ecosystems, including pollination, natural enemies, saprophytes, and biological information transfer.)
One of the major works to be published in the area of zoology was Conrad Gessner 's Historia animalium , which was published in numerous editions from 1551 to 1587. Though it was a work more generally addressing zoology in the large sense, it did contain information on insect life. Much of the information came from older works; Gessner restated the work of Pliny the Elder and Aristotle while mixing old knowledge of the natural history of insects with his own observations. [ 2 ]
With the invention of the Microscope in 1599 came a new way of observing the small creatures that fall under the umbrella of invertebrate. Robert Hooke , who worked out of the Royal Society in England, conducted observation of insects—including some of their larval forms—and other invertebrates, such as ticks. His Micrographia , published in 1665, included illustrations and written descriptions of the things he saw under the microscope. [ 3 ]
Others also worked with the microscope following its acceptance as a scientific tool. Francesco Redi , an Italian physician and naturalist, used a microscope for observation of invertebrates, but is known for his work in disproving the theory of spontaneous generation . Redi managed to prove that flies did not spontaneously arise from rotting meat. He conducted controlled experiments and detailed observation of the fly life cycle in order to do so. Redi also worked in the description and illustration of parasites for both plants and animals. [ 4 ]
Other men were also conducting research into pests and parasites at this time. Felix Plater , a Swiss physician, worked to differentiate between two types of tape worm. He also wrote descriptions of both the worms he observed and the effects these worms had on their hosts. [ 4 ]
Following the publication of Francis Bacon 's ideas about the value of experimentation in the sciences came a shift toward true experimental efforts in the biological sciences, including invertebrate zoology. Jan Swammerdam , a Dutch microscopist, supported an effort to work for a 'modern' science over blind belief in the work of ancient philosophers. He worked—like Redi—to disprove spontaneous generation using experimental techniques. Swammerdam also made a number of advancements in the study of anatomy and physiology. In the field of entomology, he conducted a number of dissections of insects and made detailed observations of the internal structures of these specimens. [ 5 ] Swammerdam also worked on a classification of insects based on life histories; he managed to contribute to the literature proving that an egg, larva, pupa, and adult are indeed the same individual. [ 6 ]
In the 18th century, the study of invertebrates focused on the naming of species that were relevant to economic pursuits, such as agricultural pests. Entomology was changing in big ways very quickly, as many naturalists and zoologists were working with hexapods. [ 7 ]
Work was also being done in the realm of parasitology and the study of worms. A French physician named Nicolas Andry de Bois-Regard determined that worms were the cause of some diseases. He also declared that worms do not spontaneously form within the animal or human gut; de Bois-Regard stated that there must be some kind of 'seed' which enters the body and contains the worm in some form. [ 7 ] Antonio Vallisneri also worked with parasitic worms, specifically members of the genera Ascaris and Neoascaris . He found that these worms came from eggs. In addition, Vallisneri worked to elucidate the reproduction of insects, specifically the sawfly . [ 7 ]
In 1735, the first edition of Carl Linnaeus ' Systema Naturae was published; this work included information on both insects and intestinal worms. [ 7 ] However, the tenth edition is considered the true starting point for the modern classification scheme for living things today. [ 8 ] Linnaeus' universal system of classification made a system based on binomial nomenclature , but included higher levels of classification than simply the genus and species names. [ 9 ] Systema Naturae was an investigation into the biodiversity on Earth. [ 8 ] However, because it was based only on very few characters, the system developed by Linnaeus was an artificial one. [ 10 ] The book also included descriptions of the organisms named inside of it. [ 9 ]
In 1859, Charles Darwin 's On the Origin of Species was published. In this book, he described his theory of evolution by natural selection . Both the work of Darwin and his contemporary, Alfred Russel Wallace —who was also working on the theory of evolution—were informed by the careful study of insects. [ 11 ] In addition, Darwin collected many species of invertebrate during his time aboard HMS Beagle ; many of the specimens collected were insects. Using these collections, he was able to study sexual dimorphism , geographic distribution of species, and mimicry ; all of these concepts influenced Darwin's theory of evolution. Unfortunately, a firm popular belief in the immutability of species was a major hurdle in the acceptance of the theory. [ 12 ]
Classification in the twentieth century shifted toward a focus on evolutionary relationships over morphological description. The development of phylogenetics and systematics based on this study is credited to Willi Hennig , a German entomologist. In 1966, his Phylogenetic Systematics was published; inside, Hennig redefined the goals of systematic schemes for classifying living things. He proposed that the focus be on evolutionary relationships over similar morphological features. He also defined monophyly and included his ideas about hierarchical classification. Though Hennig did not include information on outgroup comparison, he was apparently aware of the practice, which is considered important to today's systematic research. [ 13 ] | https://en.wikipedia.org/wiki/Invertebrate_zoology |
Inverted ligand field theory (ILFT) describes a phenomenon in the bonding of coordination complexes where the lowest unoccupied molecular orbital is primarily of ligand character. [ 2 ] [ 1 ] This is contrary to the traditional ligand field theory or crystal field theory picture and arises from the breaking down of the assumption that in organometallic complexes, ligands are more electronegative and have frontier orbitals below those of the d orbitals of electropositive metals. [ 3 ] [ 4 ] Towards the right of the d-block, when approaching the transition-metal–main group boundary, the d orbitals become more core-like, making their cations more electronegative. This decreases their energies and eventually arrives at a point where they are lower in energy than the ligand frontier orbitals. [ 2 ] Here the ligand field inverts so that the bonding orbitals are more metal-based, and antibonding orbitals more ligand-based. The relative arrangement of the d orbitals are also inverted in complexes displaying this inverted ligand field. [ 2 ]
The first example of an inverted ligand field was demonstrated in paper form 1995 by James Snyder. [ 5 ] In this theoretical paper, Snyder proposed that the [Cu(CF 3 ) 4 ] − complexes reported by Naumann et al. and assigned a formal oxidation state of +3 at the copper [ 6 ] would be better thought of as Cu(I). By comparing the d-orbital occupation, calculated charges and orbital population of [Cu(CF 3 ) 4 ] − "Cu(III)" complex and the formally Cu(I) [Cu(CH 3 ) 2 ] − complex, they illustrated how the former could be better described as a d10 copper complex experiencing two electron donation from the CF − 3 ligands. [ 5 ] The phenomenon, termed an inverted ligand field by Roald Hoffman, began to be described by Aullón and Alvarez as they identified this phenonmenon as being a result of relative electronegativities. [ 7 ] Lancaster and co-workers later provided experimental evidence to support the assignment of this oxidation state. Using UV/visible/near IR spectroscopy, Cu K-edge X-ray absorption spectroscopy, and 1s2p resonant inelastic X-ray scattering in concert with density functional theory, multiplet theory, and multireference calculations, they were able to map the ground state electronic configuration. This showed that the lowest unoccupied orbital was of primarily trifluoromethyl character. This confirmed the presence of an inverted ligand field and started building experimental tools to probe this phenomenon. [ 8 ] Since the Snyder case, many other complexes of later transition metals have been shown to display inverted ligand field through both theoretical and experimental methods.
Computational and experimental techniques have been imperative for the study of inverted ligand fields, especially when used in cooperatively.
Computational methods have played a large role in understanding the nature of bonding in both molecular and solid-state systems displaying inverted ligand fields. The Hoffman group has completed many calculations to probe occurrence of inverted ligand fields in varying systems. [ 9 ] In a study of the absorption of CO on PtBi and PtBi2 surfaces, on an octahedral [Pt(BiH 3 ) 6 ] 4+ model with a Pt thought of having a formal +4 oxidation state, the team found that the t 2g metal orbitals were higher energy that the e g orbitals. This inversion of the d orbital ordering was attributed to the bismuth based ligands being higher in energy than the metal d orbitals. [ 10 ] In another study involving calculations on Ag(III) salt KAgF 4 , other Ag(II), and Ag(III) compounds, the Ag d orbitals were found to be below those of the fluoride ligand orbitals, [ 11 ] and was confirmed by Grochala and cowrokers by core and valence spectroscopies. [ 12 ]
The Mealli group developed the program Computer Aided Composition of Atomic Orbitals (CACAO) to provide visualised molecular orbitals analyses based on perturbation theory principles. [ 13 ] This program successfully displayed orbital energy inversion with organometallic complexes containing electronegative metals such as Ni or Cu bound to electropositive ligand atoms such as B, Si, or Sn. [ 13 ] In these cases the bonding was described as a ligand to metal dative bond or sigma backdonation. [ 14 ]
Alvarez and coworkers used computational methods to illustrate ligand field inversion in the band structures of solid state materials. The group found that, contrary to the classical bonding scheme, in calculated MoNiP 8 band structures the e g -type orbitals of the octahedral nickel atom were found to be the major component of an occupied band below the t 2g set. [ 16 ] Additionally, the band around the fermi level which included the Ni + antibonding orbitals were found to be mostly of phosphorus character, a clear example if an inverted ligand field. Similar observations were made in other solid state materials like the skutterudite CoP 3 structure. [ 17 ] [ 15 ] A consequence of the inverted ligand field in this case is that the conductivity in skutterudites is associated with the phosphorus rings rather than the metal atoms.
X-ray absorption spectroscopy (XAS) has been a powerful tool in deducing the oxidation states of transition metals. [ 18 ] Energy shifts in XAS are higher due to the higher effective nuclear charge of atoms in higher oxidations, presumably due to the higher binding energy for deeper, more core-like electrons. [ 2 ]
Despite this being a very powerful technique, competing effects on the rising edge positions can make assignment difficult. It was initially thought that the weak, quadrupole-allowed pre-edge peak assigned as the Cu 1s to 3d transition could be used to distinguish between Cu(II) and Cu(III) with the features appearing at 8979 ±0.3 eV and 8981 ±0.5 eV, respectively. [ 19 ] Ab initio calculations by Tomson, Wieghardt, and co-workers displayed that pre-peaks previously assigned as Cu(III) could be displayed by Cu(II) bearing complexes. [ 20 ] Many groups have displayed that metal K-edge XAS transitions involving ligand-localised acceptor orbitals, as well as spectral shifts from change in coordination environment, can make metal K-edge analysis less predictable. [ 21 ] [ 22 ] [ 23 ] [ 24 ]
The most successful use of K and L-edge XAS provide valuable information on the composition of molecular orbitals and display inverted ligand fields has been done in studies that made use of computational techniques in concert with experimental techniques. This was the case of the L 2 [Cu 2 (S 2 ) n ] 2+ complexes of York, Brown, and Tolman, [ 25 ] and the Cu(CF 3 ) 4− by various groups including Hoffman, [ 2 ] Overgaard, [ 26 ] and Lancaster. [ 1 ] [ 8 ]
Another experimental tool used to probe ligand field inversion includes Electron paramagnetic resonance (ESR/EPR), which can provide information regarding the metal electronic configuration, the nature of the SOMO, and high resolution information on the ligands. [ 27 ]
Changes in both charge and geometry of organometallic complexes can greatly vary the energies of molecular orbitals and can therefore dictate the likelihood of observing an inverted ligand field. Hoffman and coworkers explored the impact of these variables by calculating the atomic composition of molecular orbitals for mono- di- and trianion copper complexes. [ 2 ] The square planar monoanion displayed the reported ligand field inversion. The "Cu(II)" which has an intermediate square planar to tetrahedral geometry also displayed this feature with the antibonding t 2 -derived orbital being mostly of ligand character and the x 2 -y 2 orbital being the lowest molecular orbital of the d block. The tetrahedral trianion showed a return to the Werner-type ligand field. [ 2 ] By modulating the geometry of the "Cu(II)" species and displaying the change in energies of MO on walsh diagrams , the group was able to show how the complex could display both a classical and inverted ligand field when in T d and SP geometry respectively. [ 2 ] Additional calculations on the Cu(I) with non-tetrahedral geometry also displayed an inverted ligand field. This indicated the importance of not just oxidation state but geometry in determining the inversion of a ligand field.
The inversion of ligand fields has interesting implications on the nature of reactivity of organometallic complexes. This sigma non-innocence of ligands arising from inverted ligand fields could therefore be used to tune reactivity of complexes and open space in understanding the mechanisms of existing reactions.
In an analysis of the [ZnF 4 ] 2− , it was found that due to ligand field inversion displayed in this species, core ionization removes an electron from the metal-rich bonding t 2 orbital, lengthening the Zn–F bonds. This is contrary to the classical ligand field where ionization would remove an electron from the antibonding t 2 orbital shortening the Zn–F bonds. [ 2 ]
The presence of electron-deficient ligands also result in an inverted ligand field. Calculations have shown that the large O 2p contribution into the LUMO/LUMO +1 in [(L TEED Cu) 2 (O 2 )] 2+ should make the complex highly oxidizing as it contains electron deficient O 2− ligands. [ 1 ] Studies have corroborated this property as this complex has shown to be able to undergo C–H and C–F activation and aromatic hydroxylation. [ 28 ] [ 29 ] [ 30 ]
There is evidence showing that reductive elimination on species displaying ligand field inversion do not undergo a redox event at the metal center. The C-CF 3 bond formation by "Ni(IV)" complexes [ 31 ] was completed without redox participation of the Nickel. [ 32 ] The metal appears to remain Ni(II) throughout the reaction. The mechanism is thought to be through the attack of a masked electrophilic cation by anionic CF 3 . The electron deficiency here is due to the inverted ligand field. [ 32 ] | https://en.wikipedia.org/wiki/Inverted_ligand_field_theory |
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and falls over without additional help. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a cart and pole apparatus. [ 1 ] Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation . Whereas a normal pendulum is stable when hanging downward, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger.
A second type of inverted pendulum is a tiltmeter for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position.
A pendulum with its bob hanging directly below the support pivot is at a stable equilibrium point, where it remains motionless because there is no torque on the pendulum. If displaced from this position, it experiences a restoring torque that returns it toward the equilibrium position. A pendulum with its bob in an inverted position, supported on a rigid rod directly above the pivot, 180° from its stable equilibrium position, is at an unstable equilibrium point. At this point again there is no torque on the pendulum, but the slightest displacement away from this position causes a gravitation torque on the pendulum that accelerates it away from equilibrium, causing it to fall over.
In order to stabilize a pendulum in this inverted position, a feedback control system can be used, which monitors the pendulum's angle and moves the position of the pivot point sideways when the pendulum starts to fall over, to keep it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms ( PID controllers , state-space representation , neural networks , fuzzy control , genetic algorithms , etc.). Variations on this problem include multiple links, allowing the motion of the cart to be commanded while maintaining the pendulum, and balancing the cart-pendulum system on a see-saw. The inverted pendulum is related to rocket or missile guidance, where the center of gravity is located behind the center of drag causing aerodynamic instability. [ 2 ] The understanding of a similar problem can be shown by simple robotics in the form of a balancing cart. Balancing an upturned broomstick on the end of one's finger is a simple demonstration, and the problem is solved by self-balancing personal transporters such as the Segway PT , the self-balancing hoverboard and the self-balancing unicycle .
Another way that an inverted pendulum may be stabilized, without any feedback or control mechanism, is by oscillating the pivot rapidly up and down. This is called Kapitza's pendulum . If the oscillation is sufficiently strong (in terms of its acceleration and amplitude) then the inverted pendulum can recover from perturbations in a strikingly counterintuitive manner. If the driving point moves in simple harmonic motion , the pendulum's motion is described by the Mathieu equation . [ 3 ]
The equations of motion of inverted pendulums are dependent on what constraints are placed on the motion of the pendulum. Inverted pendulums can be created in various configurations resulting in a number of Equations of Motion describing the behavior of the pendulum.
In a configuration where the pivot point of the pendulum is fixed in space, the equation of motion is similar to that for an uninverted pendulum . The equation of motion below assumes no friction or any other resistance to movement, a rigid massless rod, and the restriction to 2-dimensional movement.
Where θ ¨ {\displaystyle {\ddot {\theta }}} is the angular acceleration of the pendulum, g {\displaystyle g} is the standard gravity on the surface of the Earth, ℓ {\displaystyle \ell } is the length of the pendulum, and θ {\displaystyle \theta } is the angular displacement measured from the equilibrium position.
When θ ¨ {\displaystyle {\ddot {\theta }}} added to both sides, it has the same sign as the angular acceleration term:
Thus, the inverted pendulum accelerates away from the vertical unstable equilibrium in the direction initially displaced, and the acceleration is inversely proportional to the length. Tall pendulums fall more slowly than short ones.
Derivation using torque and moment of inertia:
The pendulum is assumed to consist of a point mass, of mass m {\displaystyle m} , affixed to the end of a massless rigid rod, of length ℓ {\displaystyle \ell } , attached to a pivot point at the end opposite the point mass.
The net torque of the system must equal the moment of inertia times the angular acceleration:
The torque due to gravity providing the net torque:
Where θ {\displaystyle \theta \ } is the angle measured from the inverted equilibrium position.
The resulting equation:
The moment of inertia for a point mass:
In the case of the inverted pendulum the radius is the length of the rod, ℓ {\displaystyle \ell } .
Substituting in I = m ℓ 2 {\displaystyle I=m\ell ^{2}}
Mass and ℓ 2 {\displaystyle \ell ^{2}} is divided from each side resulting in:
An inverted pendulum on a cart consists of a mass m {\displaystyle m} at the top of a pole of length ℓ {\displaystyle \ell } pivoted on a horizontally moving base as shown in the adjacent image. The cart is restricted to linear motion and is subject to forces resulting in or hindering motion.
The essentials of stabilizing the inverted pendulum can be summarized qualitatively in three steps.
1. If the tilt angle θ {\displaystyle \theta } is to the right, the cart must accelerate to the right and vice versa.
2. The position of the cart x {\displaystyle x} relative to track center is stabilized by slightly modulating the null angle (the angle error that the control system tries to null) by the position of the cart, that is, null angle = θ + k x {\displaystyle =\theta +kx} where k {\displaystyle k} is small. This makes the pole want to lean slightly toward track center and stabilize at track center where the tilt angle is exactly vertical. Any offset in the tilt sensor or track slope that would otherwise cause instability translates into a stable position offset. A further added offset gives position control.
3. A normal pendulum subject to a moving pivot point such as a load lifted by a crane, has a peaked response at the pendulum radian frequency of ω p = g / ℓ {\displaystyle \omega _{p}={\sqrt {g/\ell }}} . To prevent uncontrolled swinging, the frequency spectrum of the pivot motion should be suppressed near ω p {\displaystyle \omega _{p}} . The inverted pendulum requires the same suppression filter to achieve stability.
As a consequence of the null angle modulation strategy, the position feedback is positive, that is, a sudden command to move right produces an initial cart motion to the left followed by a move right to rebalance the pendulum. The interaction of the pendulum instability and the positive position feedback instability to produce a stable system is a feature that makes the mathematical analysis an interesting and challenging problem.
The equations of motion can be derived using Lagrange's equations . We refer to the drawing to the right where θ ( t ) {\displaystyle \theta (t)} is the angle of the pendulum of length l {\displaystyle l} with respect to the vertical direction and the acting forces are gravity and an external force F in the x-direction. Define x ( t ) {\displaystyle x(t)} to be the position of the cart.
The kinetic energy T {\displaystyle T} of the system is:
where v 1 {\displaystyle v_{1}} is the velocity of the cart and v 2 {\displaystyle v_{2}} is the velocity of the point mass m {\displaystyle m} . v 1 {\displaystyle v_{1}} and v 2 {\displaystyle v_{2}} can be expressed in terms of x and θ {\displaystyle \theta } by writing the velocity as the first derivative of the position;
Simplifying the expression for v 2 {\displaystyle v_{2}} leads to:
The kinetic energy is now given by:
The generalized coordinates of the system are θ {\displaystyle \theta } and x {\displaystyle x} , each has a generalized force.
On the x {\displaystyle x} axis, the generalized force Q x {\displaystyle Q_{x}} can be calculated through its virtual work
on the θ {\displaystyle \theta } axis, the generalized force Q θ {\displaystyle Q_{\theta }} can be also calculated through its virtual work
According to the Lagrange's equations , the equations of motion are:
substituting T {\displaystyle T} in these equations and simplifying leads to the equations that describe the motion of the inverted pendulum:
These equations are nonlinear, but since the goal of a control system would be to keep the pendulum upright, the equations can be linearized around θ ≈ 0 {\displaystyle \theta \approx 0} .
The generalized forces can be both written as potential energy V x {\displaystyle V_{x}} and V θ {\displaystyle V_{\theta }} ,
According to the D'Alembert's principle , generalized forces and potential energy are connected:
However, under certain circumstances, the potential energy is not accessible, only generalized forces are available.
After getting the Lagrangian L = T − V {\displaystyle L=T-V} , we can also use Euler–Lagrange equation to solve for equations of motion:
The only difference is whether to incorporate the generalized forces into the potential energy V j {\displaystyle V_{j}} or write them explicitly as Q j {\displaystyle Q_{j}} on the right side, they all lead to the same equations in the final.
Oftentimes it is beneficial to use Newton's second law instead of Lagrange's equations because Newton's equations give the reaction forces at the joint between the pendulum and the cart. These equations give rise to two equations for each body; one in the x-direction and the other in the y-direction. The equations of motion of the cart are shown below where the LHS is the sum of the forces on the body and the RHS is the acceleration.
In the equations above R x {\displaystyle R_{x}} and R y {\displaystyle R_{y}} are reaction forces at the joint. F N {\displaystyle F_{N}} is the normal force applied to the cart. This second equation depends only on the vertical reaction force, thus the equation can be used to solve for the normal force. The first equation can be used to solve for the horizontal reaction force. In order to complete the equations of motion, the acceleration of the point mass attached to the pendulum must be computed. The position of the point mass can be given in inertial coordinates as
Taking two derivatives yields the acceleration vector in the inertial reference frame.
Then, using Newton's second law, two equations can be written in the x-direction and the y-direction. Note that the reaction forces are positive as applied to the pendulum and negative when applied to the cart. This is due to Newton's third law.
The first equation allows yet another way to compute the horizontal reaction force in the event the applied force F {\displaystyle F} is not known. The second equation can be used to solve for the vertical reaction force. The first equation of motion is derived by substituting F − R x = M x ¨ {\displaystyle F-R_{x}=M{\ddot {x}}} into R x = m ( x ¨ + ℓ θ ˙ 2 sin θ − ℓ θ ¨ cos θ ) {\displaystyle R_{x}=m({\ddot {x}}+\ell {\dot {\theta }}^{2}\sin \theta -\ell {\ddot {\theta }}\cos \theta )} , which yields
By inspection this equation is identical to the result from Lagrange's Method. In order to obtain the second equation, the pendulum equation of motion must be dotted with a unit vector that runs perpendicular to the pendulum at all times and is typically noted as the x-coordinate of the body frame. In inertial coordinates this vector can be written using a simple 2-D coordinate transformation
The pendulum equation of motion written in vector form is ∑ F → = m a → P / I {\displaystyle \sum {\vec {F}}=m{\vec {a}}_{P/I}} . Dotting x ^ B {\displaystyle {\hat {x}}_{B}} with both sides yields the following on the LHS (note that a transpose is the same as a dot product )
In the above equation the relationship between body frame components of the reaction forces and inertial frame components of reaction forces is used. The assumption that the bar connecting the point mass to the cart is massless implies that this bar cannot transfer any load perpendicular to the bar. Thus, the inertial frame components of the reaction forces can be written simply as R p y ^ B {\displaystyle R_{p}{\hat {y}}_{B}} , which signifies that the bar can transfer loads only along the axis of the bar itself. This gives rise to another equation that can be used to solve for the tension in the rod itself:
The RHS of the equation is computed similarly by dotting x ^ B {\displaystyle {\hat {x}}_{B}} with the acceleration of the pendulum. The result (after some simplification) is shown below.
Combining the LHS with the RHS and dividing through by m yields
which again is identical to the result of Lagrange's method. The benefit of using Newton's method is that all reaction forces are revealed to ensure that nothing is damaged.
For a derivation of the equations of motions from Newton's second law, as above, using the Symbolic Math Toolbox [ 4 ] and references therein.
Achieving stability of an inverted pendulum has become a common engineering challenge for researchers. [ 5 ] There are different variations of the inverted pendulum on a cart ranging from a rod on a cart to a multiple segmented inverted pendulum on a cart. Another variation places the inverted pendulum's rod or segmented rod on the end of a rotating assembly. In both, (the cart and rotating system) the inverted pendulum can fall only in a plane. The inverted pendulums in these projects can either be required to maintain balance only after an equilibrium position is achieved, or can achieve equilibrium by itself. Another platform is a two-wheeled balancing inverted pendulum. The two wheeled platform has the ability to spin on the spot offering a great deal of maneuverability. [ 6 ] Yet another variation balances on a single point. A spinning top , a unicycle , or an inverted pendulum atop a spherical ball all balance on a single point.
An inverted pendulum in which the pivot is oscillated rapidly up and down can be stable in the inverted position. This is called Kapitza's pendulum , after Russian physicist Pyotr Kapitza who first analysed it. The equation of motion for a pendulum connected to a massless, oscillating base is derived the same way as with the pendulum on the cart. The position of the point mass is now given by:
and the velocity is found by taking the first derivative of the position:
The Lagrangian for this system can be written as:
and the equation of motion follows from:
resulting in:
If y represents a simple harmonic motion , y = A sin ω t {\displaystyle y=A\sin \omega t} , the following differential equation is:
This equation does not have elementary closed-form solutions, but can be explored in a variety of ways. It is closely approximated by the Mathieu equation , for instance, when the amplitude of oscillations are small. Analyses show that the pendulum stays upright for fast oscillations. The first plot shows that when y {\displaystyle y} is a slow oscillation, the pendulum quickly falls over when disturbed from the upright position. The angle θ {\displaystyle \theta } exceeds 90° after a short time, which means the pendulum has fallen on the ground. If y {\displaystyle y} is a fast oscillation the pendulum can be kept stable around the vertical position. The second plot shows that when disturbed from the vertical position, the pendulum now starts an oscillation around the vertical position ( θ = 0 {\displaystyle \theta =0} ). The deviation from the vertical position stays small, and the pendulum doesn't fall over.
Arguably the most prevalent example of a stabilized inverted pendulum is a human being . A person standing upright acts as an inverted pendulum with their feet as the pivot, and without constant small muscular adjustments would fall over. The human nervous system contains an unconscious feedback control system , the sense of balance or righting reflex , that uses proprioceptive input from the eyes, muscles and joints, and orientation input from the vestibular system consisting of the three semicircular canals in the inner ear , and two otolith organs, to make continual small adjustments to the skeletal muscles to keep us standing upright. Walking, running, or balancing on one leg puts additional demands on this system. Certain diseases and alcohol or drug intoxication can interfere with this reflex, causing dizziness and disequilibration , an inability to stand upright. A field sobriety test used by police to test drivers for the influence of alcohol or drugs, tests this reflex for impairment.
Some simple examples include balancing brooms or meter sticks by hand.
The inverted pendulum has been employed in various devices and trying to balance an inverted pendulum presents a unique engineering problem for researchers. [ 7 ] The inverted pendulum was a central component in the design of several early seismometers due to its inherent instability resulting in a measurable response to any disturbance. [ 8 ]
The inverted pendulum model has been used in some recent personal transporters , such as the two-wheeled self-balancing scooters and single-wheeled electric unicycles . These devices are kinematically unstable and use an electronic feedback servo system to keep them upright.
Swinging a pendulum on a cart into its inverted pendulum state is considered a traditional optimal control toy problem/benchmark. [ 9 ] [ 10 ] | https://en.wikipedia.org/wiki/Inverted_pendulum |
An inverted repeat (or IR ) is a single stranded sequence of nucleotides followed downstream by its reverse complement . [ 1 ] The intervening sequence of nucleotides between the initial sequence and the reverse complement can be any length including zero. For example, 5'---TTACGnnnnnn CGTAA---3' is an inverted repeat sequence. When the intervening length is zero, the composite sequence is a palindromic sequence . [ 2 ]
Both inverted repeats and direct repeats constitute types of nucleotide sequences that occur repetitively. These repeated DNA sequences often range from a pair of nucleotides to a whole gene , while the proximity of the repeat sequences varies between widely dispersed and simple tandem arrays . [ 3 ] The short tandem repeat sequences may exist as just a few copies in a small region to thousands of copies dispersed all over the genome of most eukaryotes . [ 4 ] Repeat sequences with about 10–100 base pairs are known as minisatellites , while shorter repeat sequences having mostly 2–4 base pairs are known as microsatellites . [ 5 ] The most common repeats include the dinucleotide repeats, which have the bases AC on one DNA strand, and GT on the complementary strand. [ 3 ] Some elements of the genome with unique sequences function as exons , introns and regulatory DNA. [ 6 ] Though the most familiar loci of the repetitive sequences are the centromere and the telomere , [ 6 ] a large portion of the repeated sequences in the genome are found among the noncoding DNA . [ 5 ]
Inverted repeats have a number of important biological functions. They define the boundaries in transposons and indicate regions capable of self-complementary base pairing (regions within a single sequence which can base pair with each other). These properties play an important role in genome instability [ 7 ] and contribute not only to cellular evolution and genetic diversity [ 8 ] but also to mutation and disease . [ 9 ] In order to study these effects in detail, a number of programs and databases have been developed to assist in discovery and annotation of inverted repeats in various genomes.
Beginning with this initial sequence: 5'-TTACG-3'
The complement created by base pairing is: 3'-AATGC-5'
The reverse complement is: 5'-CGTAA-3'
And, the inverted repeat sequence is: 5'---TTACGnnnnnn CGTAA---3'
"nnnnnn" represents any number of intervening nucleotides.
A direct repeat occurs when a sequence is repeated with the same pattern downstream. [ 1 ] There is no inversion and no reverse complement associated with a direct repeat. The nucleotide sequence written in bold characters signifies the repeated sequence. It may or may not have intervening nucleotides.
Linguistically, a typical direct repeat is comparable to rhyming, as in "t ime on a d ime ".
A direct repeat with no intervening nucleotides between the initial sequence and its downstream copy is a Tandem repeat . The nucleotide sequence written in bold characters signifies the repeated sequence.
Linguistically, a typical tandem repeat is comparable to stuttering, or deliberately repeated words, as in "bye-bye".
An inverted repeat sequence with no intervening nucleotides between the initial sequence and its downstream reverse complement is a palindrome . [ 1 ] EXAMPLE: Step 1: start with an inverted repeat: 5' TTACGnnnnnnCGTAA 3' Step 2: remove intervening nucleotides: 5' TTACGCGTAA 3' This resulting sequence is palindromic because it is the reverse complement of itself. [ 1 ]
The diverse genome-wide repeats are derived from transposable elements , which are now understood to "jump" about different genomic locations, without transferring their original copies. [ 10 ] Subsequent shuttling of the same sequences over numerous generations ensures their multiplicity throughout the genome. [ 10 ] The limited recombination of the sequences between two distinct sequence elements known as conservative site-specific recombination (CSSR) results in inversions of the DNA segment, based on the arrangement of the recombination recognition sequences on the donor DNA and recipient DNA. [ 10 ] Again, the orientation of two of the recombining sites within the donor DNA molecule relative to the asymmetry of the intervening DNA cleavage sequences, known as the crossover region, is pivotal to the formation of either inverted repeats or direct repeats. [ 10 ] Thus, recombination occurring at a pair of inverted sites will invert the DNA sequence between the two sites. [ 10 ] Very stable chromosomes have been observed with comparatively fewer numbers of inverted repeats than direct repeats, suggesting a relationship between chromosome stability and the number of repeats. [ 11 ]
Terminal inverted repeats have been observed in the DNA of various eukaryotic transposons, even though their source remains unknown. [ 12 ] Inverted repeats are principally found at the origins of replication of cell organism and organelles that range from phage plasmids, mitochondria, and eukaryotic viruses to mammalian cells. [ 13 ] The replication origins of the phage G4 and other related phages comprise a segment of nearly 139 nucleotide bases that include three inverted repeats that are essential for replication priming. [ 13 ]
To a large extent, portions of nucleotide repeats are quite often observed as part of rare DNA combinations. [ 14 ] The three main repeats which are largely found in particular DNA constructs include the closely precise homopurine-homopyrimidine inverted repeats, which is otherwise referred to as H palindromes, a common occurrence in triple helical H conformations that may comprise either the TAT or CGC nucleotide triads. The others could be described as long inverted repeats having the tendency to produce hairpins and cruciform, and finally direct tandem repeats, which commonly exist in structures described as slipped-loop, cruciform and left-handed Z-DNA. [ 14 ]
Past studies suggest that repeats are a common feature of eukaryotes unlike the prokaryotes and archaea . [ 14 ] Other reports suggest that irrespective of the comparative shortage of repeat elements in prokaryotic genomes, they nevertheless contain hundreds or even thousands of large repeats. [ 15 ] Current genomic analysis seem to suggest the existence of a large excess of perfect inverted repeats in many prokaryotic genomes as compared to eukaryotic genomes. [ 16 ]
For quantification and comparison of inverted repeats between several species, namely on archaea, see [ 17 ]
Pseudoknots are common structural motifs found in RNA. They are formed by two nested stem-loops such that the stem of one structure is formed from the loop of the other. There are multiple folding topologies among pseudoknots and great variation in loop lengths, making them a structurally diverse group. [ 18 ]
Inverted repeats are a key component of pseudoknots as can be seen in the illustration of a naturally occurring pseudoknot found in the human telomerase RNA component . [ 19 ] Four different sets of inverted repeats are involved in this structure. Sets 1 and 2 are the stem of stem-loop A and are part of the loop for stem-loop B. Similarly, sets 3 and 4 are the stem for stem-loop B and are part of the loop for stem-loop A.
Pseudoknots play a number of different roles in biology. The telomerase pseudoknot in the illustration is critical to that enzyme's activity. [ 19 ] The ribozyme for the hepatitis delta virus (HDV) folds into a double-pseudoknot structure and self-cleaves its circular genome to produce a single-genome-length RNA. Pseudoknots also play a role in programmed ribosomal frameshifting found in some viruses and required in the replication of retroviruses . [ 18 ]
Inverted repeats play an important role in riboswitches , which are RNA regulatory elements that control the expression of genes that produce the mRNA, of which they are part. [ 10 ] A simplified example of the flavin mononucleotide (FMN) riboswitch is shown in the illustration. This riboswitch exists in the mRNA transcript and has several stem-loop structures upstream from the coding region . However, only the key stem-loops are shown in the illustration, which has been greatly simplified to help show the role of the inverted repeats. There are multiple inverted repeats in this riboswitch as indicated in green (yellow background) and blue (orange background).
In the absence of FMN, the Anti-termination structure is the preferred conformation for the mRNA transcript. It is created by base-pairing of the inverted repeat region circled in red. When FMN is present, it may bind to the loop and prevent formation of the Anti-termination structure. This allows two different sets of inverted repeats to base-pair and form the Termination structure. [ 20 ] The stem-loop on the 3' end is a transcriptional terminator because the sequence immediately following it is a string of uracils (U). If this stem-loop forms (due to the presence of FMN) as the growing RNA strand emerges from the RNA polymerase complex, it will create enough structural tension to cause the RNA strand to dissociate and thus terminate transcription. The dissociation occurs easily because the base-pairing between the U's in the RNA and the A's in the template strand are the weakest of all base-pairings. [ 10 ] Thus, at higher concentration levels, FMN down-regulates its own transcription by increasing the formation of the termination structure.
Inverted repeats are often described as "hotspots" of eukaryotic and prokaryotic genomic instability. [ 7 ] Long inverted repeats are deemed to greatly influence the stability of the genome of various organisms. [ 21 ] This is exemplified in E. coli , where genomic sequences with long inverted repeats are seldom replicated, but rather deleted with rapidity. [ 21 ] Again, the long inverted repeats observed in yeast greatly favor recombination within the same and adjacent chromosomes, resulting in an equally very high rate of deletion. [ 21 ] Finally, a very high rate of deletion and recombination were also observed in mammalian chromosomes regions with inverted repeats. [ 21 ] Reported differences in the stability of genomes of interrelated organisms are always an indication of a disparity in inverted repeats. [ 11 ] The instability results from the tendency of inverted repeats to fold into hairpin- or cruciform-like DNA structures. These special structures can hinder or confuse DNA replication and other genomic activities. [ 7 ] Thus, inverted repeats lead to special configurations in both RNA and DNA that can ultimately cause mutations and disease . [ 9 ]
The illustration shows an inverted repeat undergoing cruciform extrusion. DNA in the region of the inverted repeat unwinds and then recombines, forming a four-way junction with two stem-loop structures. The cruciform structure occurs because the inverted repeat sequences self-pair to each other on their own strand. [ 22 ]
Extruded cruciforms can lead to frameshift mutations when a DNA sequence has inverted repeats in the form of a palindrome combined with regions of direct repeats on either side. During transcription , slippage and partial dissociation of the polymerase from the template strand can lead to both deletion and insertion mutations. [ 9 ] Deletion occurs when a portion of the unwound template strand forms a stem-loop that gets "skipped" by the transcription machinery. Insertion occurs when a stem-loop forms in a dissociated portion of the nascent (newly synthesized) strand causing a portion of the template strand to be transcribed twice. [ 9 ]
Imperfect inverted repeats can lead to mutations through intrastrand and interstrand switching. [ 9 ] The antithrombin III gene's coding region is an example of an imperfect inverted repeat as shown in the figure on the right.
The stem-loop structure forms with a bump at the bottom because the G and T do not pair up. A strand switch event could result in the G (in the bump) being replaced by an A which removes the "imperfection" in the inverted repeat and provides a stronger stem-loop structure. However, the replacement also creates a point mutation converting the GCA codon to ACA. If the strand switch event is followed by a second round of DNA replication , the mutation may become fixed in the genome and lead to disease. Specifically, the missense mutation would lead to a defective gene and a deficiency in antithrombin which could result in the development of venous thromboembolism (blood clots within a vein). [ 9 ]
Mutations in the collagen gene can lead to the disease Osteogenesis Imperfecta , which is characterized by brittle bones. [ 9 ] In the illustration, a stem-loop formed from an imperfect inverted repeat is mutated with a thymine (T) nucleotide insertion as a result of an inter- or intrastrand switch. The addition of the T creates a base-pairing "match up" with the adenine (A) that was previously a "bump" on the left side of the stem. While this addition makes the stem stronger and perfects the inverted repeat, it also creates a frameshift mutation in the nucleotide sequence which alters the reading frame and will result in an incorrect expression of the gene. [ 9 ]
The following list provides information and external links to various programs and databases for inverted repeats: | https://en.wikipedia.org/wiki/Inverted_repeat |
An inverter-based resource ( IBR ) is a source of electricity that is asynchronously connected to the electrical grid via an electronic power converter (" inverter "). The devices in this category, also known as converter interfaced generation ( CIG ) and power electronic interface source, include the variable renewable energy generators (wind, solar) and battery storage power stations . [ 1 ] [ 2 ] These devices lack the intrinsic behaviors (like the inertial response of a synchronous generator ) and their features are almost entirely defined by the control algorithms, presenting specific challenges to system stability as their penetration increases, [ 1 ] for example, a single software fault can affect all devices of a certain type in a contingency (cf. section on Blue Cut fire below). IBRs are sometimes called non-synchronous generators . [ 3 ] The design of inverters for the IBR generally follows the IEEE 1547 and NERC PRC-024-2 standards. [ 4 ]
A grid-following ( GFL ) device is synchronized to the local grid voltage and injects an electric current vector aligned with the voltage (in other words, behaves like a current source [ 5 ] ). The GFL inverters are built into an overwhelming majority of installed IBR devices. [ 1 ] Due to their following nature, the GFL device will shut down if a large voltage/frequency disturbance is observed. [ 6 ] The GFL devices cannot contribute to the grid strength , dampen active power oscillations, or provide inertia . [ 7 ]
A grid-forming ( GFM ) device partially mimics the behavior of a synchronous generator: its voltage is controlled by a free-running oscillator that slows down when more energy is withdrawn from the device. Unlike a conventional generator, the GFM device has no overcurrent capacity and thus will react very differently in the short-circuit situation. [ 1 ] Adding the GFM capability to a GFL device is not expensive in terms of components, but affects the revenues: in order to support the grid stability by providing extra power when needed, the power semiconductors need to be oversized and energy storage added. Modeling demonstrates, however, that it is possible to run a power system that almost entirely is based on the GFL devices. [ 8 ] A combination of GFM battery storage power station and synchronous condensers ( SuperFACTS ) is being researched. [ 9 ]
European Network of Transmission System Operators for Electricity (ENTSO-E) groups the GFM devices into three classes from 1 to 3, with Class 1 being at the lowest level of contribution to the grid stability (the original classification had the numbers in reverse, with class 1 being the highest). Class 2 is further subdivided in to 2A, 2B, 2C, with 2A being the most basic of the three: [ 10 ]
Compliance with IEEE 1547 standard makes the IBR to support safety features: [ 11 ]
Once an IBR ceases to provide power, it can come back only gradually, ramping its output from zero to full power. [ 12 ]
The electronic nature of IBRs limits their overload capability: the thermal stress causes their components to even temporarily be able to function at no more than 1-2 times the nameplate capacity , while the synchronous machines can briefly tolerate an overload as high as 5-6 times their rated power. [ 13 ]
A typical failure of a conventional synchronous generator (like a loss of prime mover ) is slow (seconds), while the IBR has to disconnect quickly due to low margin for overload. [ 14 ]
North American Electric Reliability Corporation (NERC) notes that IBR, like conventional generators, can provide essential reliability services , and summarizes the differences as follows: [ 15 ]
The IBR devices come with many protection functions built into the inverters. Experience of the late 2010s and early 2020s had shown that some of these protections are unnecessary, as they were designed with an expectation of a strong grid with little IBR penetration. NERC 2018 guidelines suggested removing some of these checks in order to avoid unnecessary disconnections ("trips") of the IBRs, and newer devices might not have them. The remaining checks are essential for the self-protection of the inverters that, compared to a synchronous generator, have relatively little tolerance for overvoltage and overcurrent. [ 14 ] The typical protections include: [ 16 ]
Once tripped, the IBRs will restart based on a timer or through manual intervention. A typical timer setting is in the seconds to minutes range (the IEEE-1547 default is 300 seconds). [ 17 ]
New challenges to the system stability came with the increased penetration of IBRs. Incidences of disconnections during contingency events where the fault ride through was expected, and poor damping of subsynchronous oscillations in weak grids were reported. [ 1 ]
One of the most studied major power contingencies that involved IBRs is the Blue Cut Fire of 2016 in Southern California , with a temporary loss of more than a gigawatt of photovoltaic power in a very short time. [ 12 ]
The Blue Cut fire in the Cajon Pass on August 16, 2016, has affected multiple high-voltage (500 kV and 287 kV) power transmission lines passing through the canyon. Throughout the day thirteen 500 kV line faults and two 287 kV faults were recorded. [ 18 ] The faults themselves were transitory and self-cleared in a short time (2-3.5 cycles , less than 60 milliseconds ), but the unexpected features of the algorithms in the photovoltaic inverter software triggered multiple massive losses of power, with the largest one of almost 1,200 megawatts [ 19 ] at 11:45:16 AM, persisting for multiple minutes. [ 20 ]
The analysis performed by the North American Electric Reliability Corporation (NERC) had shown that:
As a result of the incident, NERC had issued multiple recommendations, involving the changes in inverter design and amendments to the standards. [ 4 ] | https://en.wikipedia.org/wiki/Inverter-based_resource |
In linear algebra , an invertible matrix ( non-singular , non-degenarate or regular ) is a square matrix that has an inverse . In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix . Invertible matrices are the same size as their inverse.
An n -by- n square matrix A is called invertible if there exists an n -by- n square matrix B such that A B = B A = I n , {\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},} where I n denotes the n -by- n identity matrix and the multiplication used is ordinary matrix multiplication . [ 1 ] If this is the case, then the matrix B is uniquely determined by A , and is called the (multiplicative) inverse of A , denoted by A −1 . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [ 2 ]
Over a field , a square matrix that is not invertible is called singular or degenerate . A square matrix with entries in a field is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the number line or complex plane , the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices, i.e. m -by- n matrices for which m ≠ n , do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse . If A is m -by- n and the rank of A is equal to n , ( n ≤ m ), then A has a left inverse, an n -by- m matrix B such that BA = I n . If A has rank m ( m ≤ n ), then it has a right inverse, an n -by- m matrix B such that AB = I m .
While the most common case is that of matrices over the real or complex numbers, all of those definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings ). However, in the case of a ring being commutative , the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring , the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group , the general linear group of degree n , denoted GL n ( R ) .
Let A be a square n -by- n matrix over a field K (e.g., the field R {\displaystyle \mathbb {R} } of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix: [ 3 ]
Furthermore, the following properties hold for an invertible matrix A :
The rows of the inverse matrix V of a matrix U are orthonormal to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where the rows of V are denoted as v i T {\displaystyle v_{i}^{\mathrm {T} }} and the columns of U as u j {\displaystyle u_{j}} for 1 ≤ i , j ≤ n . {\displaystyle 1\leq i,j\leq n.} Then clearly, the Euclidean inner product of any two v i T u j = δ i , j . {\displaystyle v_{i}^{\mathrm {T} }u_{j}=\delta _{i,j}.} This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of orthogonal vectors (but not necessarily orthonormal vectors) to the columns of U are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse V .
A matrix that is its own inverse (i.e., a matrix A such that A = A −1 and consequently A 2 = I ) is called an involutory matrix .
The adjugate of a matrix A can be used to find the inverse of A as follows:
If A is an invertible matrix, then
It follows from the associativity of matrix multiplication that if
for finite square matrices A and B , then also
Over the field of real numbers, the set of singular n -by- n matrices, considered as a subset of R n × n , {\displaystyle \mathbb {R} ^{n\times n},} is a null set , that is, has Lebesgue measure zero. That is true because singular matrices are the roots of the determinant function. It is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of measure theory , almost all n -by- n matrices are invertible.
Furthermore, the set of n -by- n invertible matrices is open and dense in the topological space of all n -by- n matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of n -by- n matrices.
In practice, however, non-invertible matrices may be encountered. In numerical calculations , matrices that are invertible but close to a non-invertible matrix may still be problematic and are said to be ill-conditioned .
This example with rank of n − 1 is a non-invertible matrix:
We can see the rank of this 2-by-2 matrix is 1, which is n − 1 ≠ n , so it is non-invertible.
Consider the following 2-by-2 matrix:
The matrix B {\displaystyle \mathbf {B} } is invertible. To check this, one can compute that det B = − 1 2 {\textstyle \det \mathbf {B} =-{\frac {1}{2}}} , which is non-zero.
As an example of a non-invertible, or singular, matrix, consider:
The determinant of C {\displaystyle \mathbf {C} } is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.
Gaussian elimination is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix is first created with the left side being the matrix to invert and the right side being the identity matrix . Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix.
For example, take the following matrix: A = ( − 1 3 2 1 − 1 ) . {\displaystyle \mathbf {A} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.}
The first step to compute its inverse is to create the augmented matrix ( − 1 3 2 1 0 1 − 1 0 1 ) . {\displaystyle \left(\!\!{\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\1&-1&0&1\end{array}}\!\!\right).}
Call the first row of this matrix R 1 {\displaystyle R_{1}} and the second row R 2 {\displaystyle R_{2}} . Then, add row 1 to row 2 ( R 1 + R 2 → R 2 ) . {\displaystyle (R_{1}+R_{2}\to R_{2}).} This yields ( − 1 3 2 1 0 0 1 2 1 1 ) . {\displaystyle \left(\!\!{\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\0&{\tfrac {1}{2}}&1&1\end{array}}\!\!\right).}
Next, subtract row 2, multiplied by 3, from row 1 ( R 1 − 3 R 2 → R 1 ) , {\displaystyle (R_{1}-3\,R_{2}\to R_{1}),} which yields ( − 1 0 − 2 − 3 0 1 2 1 1 ) . {\displaystyle \left(\!\!{\begin{array}{cc|cc}-1&0&-2&-3\\0&{\tfrac {1}{2}}&1&1\end{array}}\!\!\right).}
Finally, multiply row 1 by −1 ( − R 1 → R 1 ) {\displaystyle (-R_{1}\to R_{1})} and row 2 by 2 ( 2 R 2 → R 2 ) . {\displaystyle (2\,R_{2}\to R_{2}).} This yields the identity matrix on the left side and the inverse matrix on the right: ( 1 0 2 3 0 1 2 2 ) . {\displaystyle \left(\!\!{\begin{array}{cc|cc}1&0&2&3\\0&1&2&2\end{array}}\!\!\right).}
Thus, A − 1 = ( 2 3 2 2 ) . {\displaystyle \mathbf {A} ^{-1}={\begin{pmatrix}2&3\\2&2\end{pmatrix}}.} It works because the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using elementary matrices ( E n {\displaystyle \mathbf {E} _{n}} ), such as E n E n − 1 ⋯ E 2 E 1 A = I . {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {A} =\mathbf {I} .}
Applying right-multiplication using A − 1 , {\displaystyle \mathbf {A} ^{-1},} we get E n E n − 1 ⋯ E 2 E 1 I = I A − 1 . {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} =\mathbf {I} \mathbf {A} ^{-1}.} And the right side I A − 1 = A − 1 , {\displaystyle \mathbf {I} \mathbf {A} ^{-1}=\mathbf {A} ^{-1},} which is the inverse we want.
To obtain E n E n − 1 ⋯ E 2 E 1 I , {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} ,} we create the augumented matrix by combining A with I and applying Gaussian elimination . The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes I , the right portion applied the same elementary row operation sequence will become A −1 .
A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient if it is convenient to find a suitable starting seed:
Victor Pan and John Reif have done work that includes ways of generating a starting seed. [ 5 ] [ 6 ]
Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix. For example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration . That may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors from imperfect computer arithmetic .
The Cayley–Hamilton theorem allows the inverse of A to be expressed in terms of det( A ) , traces and powers of A : [ 7 ]
where n is size of A , and tr( A ) is the trace of matrix A given by the sum of the main diagonal . The sum is taken over s and the sets of all k l ≥ 0 {\displaystyle k_{l}\geq 0} satisfying the linear Diophantine equation
The formula can be rewritten in terms of complete Bell polynomials of arguments t l = − ( l − 1 ) ! tr ( A l ) {\displaystyle t_{l}=-(l-1)!\operatorname {tr} \left(A^{l}\right)} as
That is described in more detail under Cayley–Hamilton method .
If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A is invertible and its inverse is given by
where Q is the square ( N × N ) matrix whose i th column is the eigenvector q i {\displaystyle q_{i}} of A , and Λ is the diagonal matrix whose diagonal entries are the corresponding eigenvalues, that is, Λ i i = λ i . {\displaystyle \Lambda _{ii}=\lambda _{i}.} If A is symmetric, Q is guaranteed to be an orthogonal matrix , therefore Q − 1 = Q T . {\displaystyle \mathbf {Q} ^{-1}=\mathbf {Q} ^{\mathrm {T} }.} Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate:
If matrix A is positive definite , then its inverse can be obtained as
where L is the lower triangular Cholesky decomposition of A , and L * denotes the conjugate transpose of L .
Writing the transpose of the matrix of cofactors , known as an adjugate matrix , may also be an efficient way to calculate the inverse of small matrices, but the recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors:
so that
where | A | is the determinant of A , C is the matrix of cofactors, and C T represents the matrix transpose .
The cofactor equation listed above yields the following result for 2 × 2 matrices. Inversion of these matrices can be done as follows: [ 8 ]
This is possible because 1/( ad − bc ) is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.
The Cayley–Hamilton method gives
A computationally efficient 3 × 3 matrix inversion is given by
(where the scalar A is not to be confused with the matrix A ).
If the determinant is non-zero, the matrix is invertible, with the entries of the intermediary matrix on the right side above given by
The determinant of A can be computed by applying the rule of Sarrus as follows:
The Cayley–Hamilton decomposition gives
The general 3 × 3 inverse can be expressed concisely in terms of the cross product and triple product . If a matrix A = [ x 0 x 1 x 2 ] {\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {x} _{0}&\mathbf {x} _{1}&\mathbf {x} _{2}\end{bmatrix}}} (consisting of three column vectors, x 0 {\displaystyle \mathbf {x} _{0}} , x 1 {\displaystyle \mathbf {x} _{1}} , and x 2 {\displaystyle \mathbf {x} _{2}} ) is invertible, its inverse is given by
The determinant of A , det( A ) , is equal to the triple product of x 0 , x 1 , and x 2 —the volume of the parallelepiped formed by the rows or columns:
The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of A –1 is orthogonal to the non-corresponding two columns of A (causing the off-diagonal terms of I = A − 1 A {\displaystyle \mathbf {I} =\mathbf {A} ^{-1}\mathbf {A} } be zero). Dividing by
causes the diagonal entries of I = A −1 A to be unity. For example, the first diagonal is:
With increasing dimension, expressions for the inverse of A get complicated. For n = 4 , the Cayley–Hamilton method leads to an expression that is still tractable:
Let
M = [ A B C D ] {\displaystyle \mathbf {M} ={\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}}
where A , B , C and D are matrix sub-blocks of arbitrary size and M / A := D − C A − 1 B {\displaystyle \mathbf {M} /\mathbf {A} :=\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {B} } is the Schur complement of A . ( A must be square, so that it can be inverted. Furthermore, A and D − CA −1 B must be nonsingular. [ 9 ] )
Matrices can also be inverted blockwise by using the analytic inversion formula: [ 10 ]
The strategy is particularly advantageous if A is diagonal and M / A is a small matrix, since they are the only matrices requiring inversion.
This technique was reinvented several times by Hans Boltz (1923), [ citation needed ] who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness.
The nullity theorem says that the nullity of A equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of B equals the nullity of the sub-block in the upper right of the inverse matrix.
The inversion procedure that led to Equation ( 1 ) performed matrix block operations that operated on C and D first. Instead, if A and B are operated on first, and provided D and M / D := A − BD −1 C are nonsingular, [ 11 ] the result is
Equating the upper-left sub-matrices of Equations ( 1 ) and ( 2 ) leads to
where Equation ( 3 ) is the Woodbury matrix identity , which is equivalent to the binomial inverse theorem .
If A and D are both invertible, then the above two block matrix inverses can be combined to provide the simple factorization
By the Weinstein–Aronszajn identity , one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.
This formula simplifies significantly when the upper right block matrix B is the zero matrix . This formulation is useful when the matrices A and D have relatively simple inverse formulas (or pseudo inverses in the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above becomes
If the given invertible matrix is a symmetric matrix with invertible block A the following block inverse formula holds [ 12 ]
where S = D − C A − 1 C T {\displaystyle \mathbf {S} =\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {C} ^{T}} . This requires 2 inversions of the half-sized matrices A and S and only 4 multiplications of half-sized matrices, if organized properly W 1 = C A − 1 , W 2 = W 1 C T = C A − 1 C T , W 3 = S − 1 W 1 = S − 1 C A − 1 , W 4 = W 1 T W 3 = A − 1 C T S − 1 C A − 1 , {\displaystyle {\begin{aligned}\mathbf {W} _{1}&=\mathbf {C} \mathbf {A} ^{-1},\\[3mu]\mathbf {W} _{2}&=\mathbf {W} _{1}\mathbf {C} ^{T}=\mathbf {C} \mathbf {A} ^{-1}\mathbf {C} ^{T},\\[3mu]\mathbf {W} _{3}&=\mathbf {S} ^{-1}\mathbf {W} _{1}=\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1},\\[3mu]\mathbf {W} _{4}&=\mathbf {W} _{1}^{T}\mathbf {W} _{3}=\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1},\end{aligned}}} together with some additions, subtractions, negations and transpositions of negligible complexity. Any matrix M {\displaystyle \mathbf {M} } has an associated positive semidefinite, symmetric matrix M T M {\displaystyle \mathbf {M} ^{T}\mathbf {M} } , which is exactly invertible (and positive definite), if and only if M {\displaystyle \mathbf {M} } is invertible. By writing M − 1 = ( M T M ) − 1 M T {\displaystyle \mathbf {M} ^{-1}=\left(\mathbf {M} ^{T}\mathbf {M} \right)^{-1}\mathbf {M} ^{T}} matrix inversion can be reduced to inverting symmetric matrices and 2 additional matrix multiplications, because the positive definite matrix M T M {\displaystyle \mathbf {M} ^{T}\mathbf {M} } satisfies the invertibility condition for its left upper block A .
Those formulas together allow to construct a divide and conquer algorithm that uses blockwise inversion of associated symmetric matrices to invert a matrix with the same time complexity as the matrix multiplication algorithm that is used internally. [ 12 ] Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of O ( n 2.371552 ) operations, while the best proven lower bound is Ω ( n 2 log n ) . [ 13 ]
If a matrix A has the property that
then A is nonsingular and its inverse may be expressed by a Neumann series : [ 14 ]
Truncating the sum results in an "approximate" inverse which may be useful as a preconditioner . Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a geometric sum . As such, it satisfies
Therefore, only 2 L − 2 matrix multiplications are needed to compute 2 L terms of the sum.
More generally, if A is "near" the invertible matrix X in the sense that
then A is nonsingular and its inverse is
If it is also the case that A − X has rank 1 then this simplifies to
If A is a matrix with integer or rational entries, and we seek a solution in arbitrary-precision rationals, a p -adic approximation method converges to an exact solution in O( n 4 log 2 n ) , assuming standard O( n 3 ) matrix multiplication is used. [ 15 ] The method relies on solving n linear systems via Dixon's method of p -adic approximation (each in O( n 3 log 2 n ) ) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML. [ 16 ]
Given an n × n square matrix X = [ x i j ] {\displaystyle \mathbf {X} =\left[x^{ij}\right]} , 1 ≤ i , j ≤ n {\displaystyle 1\leq i,j\leq n} , with n rows interpreted as n vectors x i = x i j e j {\displaystyle \mathbf {x} _{i}=x^{ij}\mathbf {e} _{j}} ( Einstein summation assumed) where the e j {\displaystyle \mathbf {e} _{j}} are a standard orthonormal basis of Euclidean space R n {\displaystyle \mathbb {R} ^{n}} ( e i = e i , e i ⋅ e j = δ i j {\displaystyle \mathbf {e} _{i}=\mathbf {e} ^{i},\mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}} ), then using Clifford algebra (or geometric algebra ) we compute the reciprocal (sometimes called dual ) column vectors:
as the columns of the inverse matrix X − 1 = [ x j i ] . {\displaystyle \mathbf {X} ^{-1}=[x_{ji}].} Note that, the place " ( ) i {\displaystyle ()_{i}} " indicates that " x i {\displaystyle \mathbf {x} _{i}} " is removed from that place in the above expression for x i {\displaystyle \mathbf {x} ^{i}} . We then have X X − 1 = [ x i ⋅ x j ] = [ δ i j ] = I n {\displaystyle \mathbf {X} \mathbf {X} ^{-1}=\left[\mathbf {x} _{i}\cdot \mathbf {x} ^{j}\right]=\left[\delta _{i}^{j}\right]=\mathbf {I} _{n}} , where δ i j {\displaystyle \delta _{i}^{j}} is the Kronecker delta . We also have X − 1 X = [ ( e i ⋅ x k ) ( e j ⋅ x k ) ] = [ e i ⋅ e j ] = [ δ i j ] = I n {\displaystyle \mathbf {X} ^{-1}\mathbf {X} =\left[\left(\mathbf {e} _{i}\cdot \mathbf {x} ^{k}\right)\left(\mathbf {e} ^{j}\cdot \mathbf {x} _{k}\right)\right]=\left[\mathbf {e} _{i}\cdot \mathbf {e} ^{j}\right]=\left[\delta _{i}^{j}\right]=\mathbf {I} _{n}} , as required. If the vectors x i {\displaystyle \mathbf {x} _{i}} are not linearly independent, then ( x 1 ∧ x 2 ∧ ⋯ ∧ x n ) = 0 {\displaystyle (\mathbf {x} _{1}\wedge \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})=0} and the matrix X {\displaystyle \mathbf {X} } is not invertible (has no inverse).
Suppose that the invertible matrix A depends on a parameter t . Then the derivative of the inverse of A with respect to t is given by [ 17 ]
To derive the above expression for the derivative of the inverse of A , one can differentiate the definition of the matrix inverse A − 1 A = I {\displaystyle \mathbf {A} ^{-1}\mathbf {A} =\mathbf {I} } and then solve for the inverse of A :
Subtracting A − 1 d A d t {\displaystyle \mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}} from both sides of the above and multiplying on the right by A − 1 {\displaystyle \mathbf {A} ^{-1}} gives the correct expression for the derivative of the inverse:
Similarly, if ε {\displaystyle \varepsilon } is a small number then
More generally, if
then,
Given a positive integer n {\displaystyle n} ,
Therefore,
Some of the properties of inverse matrices are shared by generalized inverses (such as the Moore–Penrose inverse ), which can be defined for any m -by- n matrix. [ 18 ]
For most practical applications, it is not necessary to invert a matrix to solve a system of linear equations ; however, for a unique solution, it is necessary for the matrix involved to be invertible.
Decomposition techniques like LU decomposition are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.
Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy and is found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases. [ 19 ]
Matrix inversion plays a significant role in computer graphics , particularly in 3D graphics rendering and 3D simulations . Examples include screen-to-world ray casting , world-to-subspace-to-world object transformations, and physical simulations.
Matrix inversion also plays a significant role in the MIMO (Multiple-Input, Multiple-Output) technology in wireless communications . The MIMO system consists of N transmit and M receive antennas. Unique signals, occupying the same frequency band , are sent via N transmit antennas and are received via M receive antennas. The signal arriving at each receive antenna will be a linear combination of the N transmitted signals forming an N × M transmission matrix H . It is crucial for the matrix H to be invertible so that the receiver can figure out the transmitted information. | https://en.wikipedia.org/wiki/Invertible_matrix |
In mathematics , particularly commutative algebra , an invertible module is intuitively a module that has an inverse with respect to the tensor product . Invertible modules form the foundation for the definition of invertible sheaves in algebraic geometry .
Formally, a finitely generated module M over a ring R is said to be invertible if it is locally a free module of rank 1. In other words, M P ≅ R P {\displaystyle M_{P}\cong R_{P}} for all primes P of R . Now, if M is an invertible R -module, then its dual M * = Hom( M , R ) is its inverse with respect to the tensor product, i.e. M ⊗ R M ∗ ≅ R {\displaystyle M\otimes _{R}M^{*}\cong R} .
The theory of invertible modules is closely related to the theory of codimension one varieties including the theory of divisors . | https://en.wikipedia.org/wiki/Invertible_module |
The United States Food and Drug Administration 's Investigational New Drug ( IND ) program is the means by which a pharmaceutical company obtains permission to start human clinical trials and to ship an experimental drug across state lines (usually to clinical investigators) before a marketing application for the drug has been approved. Regulations are primarily at 21 CFR 312 . Similar procedures are followed in the European Union, Japan, and Canada due to regulatory harmonization efforts by the International Council for Harmonisation . [ 1 ]
The IND application may be divided into the following categories: [ 2 ]
An IND application must also include an Investigator's Brochure intended to educate the trial investigators of the significant facts about the trial drug they need to know to conduct their clinical trial with the least hazard to the subjects or patients. [ citation needed ]
Once an IND application is submitted, the FDA has 30 days to object to the IND or it automatically becomes effective and clinical trials may begin. If the FDA detects a problem, it may place a clinical hold on the IND, prohibiting the start of the clinical studies until the problem is resolved, as outlined in 21 CFR 312.42 .
An IND must be labeled "Caution: New Drug – Limited by Federal (or United States) law to investigational use," per 21 CFR 312.6
Approximately two-thirds of both INDs and new drug applications (NDAs) are small-molecule drugs . The rest is biopharmaceuticals . About half of the INDs fail in preclinical and clinical phases of drug development. [ citation needed ]
The FDA runs a medical marijuana IND program (the Compassionate Investigational New Drug program ). It stopped accepting new patients in 1992 after public health authorities concluded there was no scientific value to it, and due to President George H. W. Bush administration's desire to "get tough on crime and drugs." As of 2011, four patients continue to receive cannabis from the government under the program. [ 3 ]
Sanctioned by Executive Order 13139 , the US Department of Defense employed an anthrax vaccine classified as an investigational new drug (IND) in its Anthrax Vaccine Immunization Program (AVIP). [ citation needed ] | https://en.wikipedia.org/wiki/Investigational_New_Drug |
In fluid dynamics , inviscid flow is the flow of an inviscid fluid which is a fluid with zero viscosity . [ 1 ]
The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, such as the case of inviscid flow, the Navier–Stokes equation can be simplified to a form known as the Euler equation . This simplified equation is applicable to inviscid flow as well as flow with low viscosity and a Reynolds number much greater than one. Using the Euler equation, many fluid dynamics problems involving low viscosity are easily solved, however, the assumed negligible viscosity is no longer valid in the region of fluid near a solid boundary (the boundary layer ) or, more generally in regions with large velocity gradients which are evidently accompanied by viscous forces. [ 1 ] [ 2 ] [ 3 ]
The flow of a superfluid is inviscid. [ 4 ]
Inviscid flows are broadly classified into potential flows (or, irrotational flows) and rotational inviscid flows.
Ludwig Prandtl developed the modern concept of the boundary layer . His hypothesis establishes that for fluids of low viscosity, shear forces due to viscosity are evident only in thin regions at the boundary of the fluid, adjacent to solid surfaces. Outside these regions, and in regions of favorable pressure gradient, viscous shear forces are absent so the fluid flow field can be assumed to be the same as the flow of an inviscid fluid. By employing the Prandtl hypothesis it is possible to estimate the flow of a real fluid in regions of favorable pressure gradient by assuming inviscid flow and investigating the irrotational flow pattern around the solid body. [ 5 ]
Real fluids experience separation of the boundary layer and resulting turbulent wakes but these phenomena cannot be modelled using inviscid flow. Separation of the boundary layer usually occurs where the pressure gradient reverses from favorable to adverse so it is inaccurate to use inviscid flow to estimate the flow of a real fluid in regions of unfavorable pressure gradient . [ 5 ]
The Reynolds number (Re) is a dimensionless quantity that is commonly used in fluid dynamics and engineering. [ 6 ] [ 7 ] Originally described by George Gabriel Stokes in 1850, it became popularized by Osborne Reynolds after whom the concept was named by Arnold Sommerfeld in 1908. [ 7 ] [ 8 ] [ 9 ] The Reynolds number is calculated as:
The value represents the ratio of inertial forces to viscous forces in a fluid, and is useful in determining the relative importance of viscosity. [ 6 ] In inviscid flow, since the viscous forces are zero, the Reynolds number approaches infinity. [ 1 ] When viscous forces are negligible, the Reynolds number is much greater than one. [ 1 ] In such cases (Re>>1), assuming inviscid flow can be useful in simplifying many fluid dynamics problems.
In a 1757 publication, Leonhard Euler described a set of equations governing inviscid flow: [ 10 ]
Assuming inviscid flow allows the Euler equation to be applied to flows in which viscous forces are insignificant. [ 1 ] Some examples include flow around an airplane wing, upstream flow around bridge supports in a river, and ocean currents. [ 1 ]
In 1845, George Gabriel Stokes published another important set of equations, today known as the Navier-Stokes equations . [ 1 ] [ 11 ] Claude-Louis Navier developed the equations first using molecular theory, which was further confirmed by Stokes using continuum theory. [ 1 ] The Navier-Stokes equations describe the motion of fluids: [ 1 ]
When the fluid is inviscid, or the viscosity can be assumed to be negligible, the Navier-Stokes equation simplifies to the Euler equation: [ 1 ] This simplification is much easier to solve, and can apply to many types of flow in which viscosity is negligible. [ 1 ] Some examples include flow around an airplane wing, upstream flow around bridge supports in a river, and ocean currents. [ 1 ]
The Navier-Stokes equation reduces to the Euler equation when μ = 0 {\displaystyle \mu =0} . Another condition that leads to the elimination of viscous force is ∇ 2 v = 0 {\displaystyle \nabla ^{2}\mathbf {v} =0} , and this results in an "inviscid flow arrangement". [ 12 ] Such flows are found to be vortex-like.
It is important to note, that negligible viscosity can no longer be assumed near solid boundaries, such as the case of the airplane wing. [ 1 ] In turbulent flow regimes (Re >> 1), viscosity can typically be neglected, however this is only valid at distances far from solid interfaces. [ 1 ] When considering flow in the vicinity of a solid surface, such as flow through a pipe or around a wing, it is convenient to categorize four distinct regions of flow near the surface: [ 1 ]
Although these distinctions can be a useful tool in illustrating the significance of viscous forces near solid interfaces, it is important to note that these regions are fairly arbitrary. [ 1 ] Assuming inviscid flow can be a useful tool in solving many fluid dynamics problems, however, this assumption requires careful consideration of the fluid sub layers when solid boundaries are involved.
Superfluid is the state of matter that exhibits frictionless flow, zero viscosity, also known as inviscid flow. [ 4 ]
To date, helium is the only fluid to exhibit superfluidity that has been discovered. Helium-4 becomes a superfluid once it is cooled to below 2.2K, a point known as the lambda point . [ 13 ] At temperatures above the lambda point, helium exists as a liquid exhibiting normal fluid dynamic behavior. Once it is cooled to below 2.2K it begins to exhibit quantum behavior. For example, at the lambda point there is a sharp increase in heat capacity, as it is continued to be cooled, the heat capacity begins to decrease with temperature. [ 14 ] In addition, the thermal conductivity is very large, contributing to the excellent coolant properties of superfluid helium. [ 15 ] Similarly, Helium-3 is found become a superfluid at 2.491mK.
Spectrometers are kept at a very low temperature using helium as the coolant. This allows for minimal background flux in far-infrared readings. Some of the designs for the spectrometers may be simple, but even the frame is at its warmest less than 20 Kelvin. These devices are not commonly used as it is very expensive to use superfluid helium over other coolants. [ 16 ]
Superfluid helium has a very high thermal conductivity, which makes it very useful for cooling superconductors. Superconductors such as the ones used at the Large Hadron Collider (LHC) are cooled to temperatures of approximately 1.9 Kelvin. This temperature allows the niobium-titanium magnets to reach a superconductor state. Without the use of the superfluid helium, this temperature would not be possible. Using helium to cool to these temperatures is very expensive and cooling systems that use alternative fluids are more numerous. [ 17 ]
Another application of the superfluid helium is its uses in understanding quantum mechanics. Using lasers to look at small droplets allows scientists to view behaviors that may not normally be viewable. This is due to all the helium in each droplet being at the same quantum state. This application does not have any practical uses by itself, but it helps us better understand quantum mechanics which has its own applications. | https://en.wikipedia.org/wiki/Inviscid_flow |
Invitae Corp. is a biotechnology company that was created as a subsidiary of Genomic Health in 2010 and then spun-off in 2012. [ 2 ]
In 2017, Invitae acquired Good Start Genetics and CombiMatrix. [ 3 ] [ 4 ] In 2020, Invitae announced the acquisition of ArcherDX for $1.4 billion. [ 5 ] In 2021, Invitae announced the acquisition of health care AI startup Ciitizen for $325 million. [ 6 ]
In early 2024, Invitae filed for Chapter 11 bankruptcy protection , and later announced an agreement for an acquisition by Labcorp . [ 7 ]
CombiMatrix Corp. ( Nasdaq : CBMX ) [ 8 ] was a clinical diagnostic laboratory specializing in pre-implantation genetic screening, miscarriage analysis, prenatal and pediatric diagnostics, offering DNA-based testing for the detection of genetic abnormalities beyond what can be identified through traditional methodologies. As a full-scale cytogenetic and cytogenomic laboratory, CombiMatrix performs genetic testing utilizing a variety of advanced cytogenomic techniques, including chromosomal microarray analysis, standardized and customized fluorescence in situ hybridization (FISH) and high-resolution karyotyping. [ 9 ] CombiMatrix is dedicated to providing high-level clinical support for healthcare professionals in order to help them incorporate the results of complex genetic testing into patient-centered medical decision making. [ citation needed ]
In 2012 CombiMatrix shifted its focus from providing oncology genetic testing to developmental testing. Their focus is cytogenomic miscarriage analysis , prenatal analysis and postnatal / pediatric analysis. [ 10 ] [ 11 ]
In the mid-1990s, a PhD from Caltech invented a method of analyzing and immobilizing genetic material on the surface of a modified semiconductor wafer. The CombiMatrix microarray ("CombiMatrix" refers to combinatorial chemistry on a matrix array) was born and received US patents in the late '90s. Located in Mukilteo, Washington , the company received several rounds of private financings and filed to go public in November 2000. Unfortunately, CombiMatrix missed its "IPO window" by a few months due to the turmoil that was happening in the financial markets in late 2000 and early 2001, and then the events of Sept. 11 2001 put the final "nail in the coffin" to the company's fundraising plans at the time. So, CombiMatrix shifted gears and began pursuing several funded R&D projects to further develop its proprietary microarray technology with outside entities such as Roche Applied Sciences in Europe, biotech researchers in Japan as well as the U.S. Department of Defense . In late 2002, CombiMatrix merged with and became a wholly owned subsidiary of Acacia Research Corporation of Newport Beach, California. [ 12 ]
From 2002 to 2007, CombiMatrix continued its commercial development of "CustomArray" as it was called and began selling its DNA synthesis instruments and arrays to various R&D facilities throughout the United States, Europe and Asia. In an effort to diversify the company from the "tools" R&D sector, the company created CombiMatrix Molecular Diagnostics, Inc. and opened a diagnostics lab in Irvine, California, which was originally led by former US Labs employees with the intent of entering the much larger clinical diagnostics market using its proprietary microarray platform for oncology-related diagnostic services. CMD then branched into pediatric diagnostic testing by introducing other array technologies beyond the CustomArray platform. In 2007, CombiMatrix spun out from Acacia Research Corp. and became an independent public company in August of that year. [ 12 ]
By 2010, the clinical lab business was showing promise, while the legacy CustomArray tools business was not achieving the commercial success that was needed to grow the company. As a result, the CombiMatrix Board made the difficult decision to scuttle the CustomArray tools business in Washington state and instead focus all commercial and operational resources on the diagnostic services business in California. By the end of 2010, the Irvine CLIA lab was the sole remaining commercial operations of CombiMatrix.
Since that time, the company has launched several different genetic diagnostic tests in oncology, pediatric disorders and reproductive health, with the latter becoming the primary growth engine of the Company in recent years. As our Board has often said over the past three years in particular, we have become a "real, commercial company" at CombiMatrix. The company's recent success over the past 16 quarters of consistently increasing revenue and collections while decreasing costs and cash burn led to the successful sale of CombiMatrix to Invitae in November 2017.
On February 5, 2024, The Wall Street Journal reported that the company was preparing to file for bankruptcy. [ 13 ] The following day, the New York Stock Exchange announced it was delisting the company's stock and trading halted. [ 14 ] On February 13, 2024, Invitae declared Chapter 11 bankruptcy in New Jersey. [ 15 ]
CombiMatrix uses the CombiSNP Array as their technology platform. SNP stands for Single nucleotide polymorphism probes. These probes allow for increased precision and greater diagnostic yield. This array contains more than 845,000 SNP markers covering both coding and non-coding human genome sequences . The median spatial resolution between probes is 1 Kb within gene rich regions and 5 Kb outside of gene-rich regions. [ 16 ] | https://en.wikipedia.org/wiki/Invitae |
Involution is the shrinking or return of an organ to a former size. At a cellular level, involution is characterized by the process of proteolysis of the basement membrane (basal lamina), leading to epithelial regression and apoptosis , with accompanying stromal fibrosis . The consequent reduction in cell number and reorganization of stromal tissue leads to the reduction in the size of the organ.
The thymus continues to grow between birth and sexual maturity and then begins to atrophy , a process directed by the high levels of circulating sex hormones . Proportional to thymic size, thymic activity ( T cell output) is most active before maturity. Upon atrophy, the size and activity are dramatically reduced, and the organ is primarily replaced with fat . The atrophy is due to the increased circulating level of sex hormones , and chemical or physical castration of an adult results in the thymus increasing in size and activity. [ 1 ]
Involution is the process by which the uterus is transformed from pregnant to non-pregnant state. This period is characterized by the restoration of ovarian function in order to prepare the body for a new pregnancy. It is a physiological process occurring after parturition ; the hypertrophy of the uterus has to be undone since it does not need to house the fetus anymore. This process is primarily due to the hormone oxytocin . The completion of this period is defined as when the diameter of the uterus returns to the size it is normally during a woman's menstrual cycle. [ citation needed ]
During pregnancy until after birth, mammary glands grow steadily to a size required for optimal milk production. At the end of nursing , the number of cells in the mammary gland becomes reduced until approximately the same number is reached as before the start of pregnancy. | https://en.wikipedia.org/wiki/Involution_(medicine) |
Invoxia is a French consumer electronics company known for the design and development of innovative smart devices that use artificial intelligence , such as the first GPS tracker on the market to use LoRa [ 1 ] technology (introduced in 2017), the first connected speaker outside the Amazon ecosystem to use the Alexa [ 2 ] voice system and a line of GPS trackers for preventing bike theft [ 3 ] and monitoring pet activity. [ 4 ] For the B2B market, it provides fleet tracking and asset management services. [ 5 ] It also provides industrial IoT services [ 6 ] including hardware design and development and the training and integration of neural networks .
Invoxia was founded in 2010 by the French serial entrepreneur Éric Careel (also Withings , Sculpteo and Zoov ) and Serge Renouard.
Invoxia is backed by Newfund since 2012. In 2013, Invoxia took control of the ancestral telephone manufacturer Swissvoice. [ 7 ]
In October 2015, Amazon [ 8 ] announces Invoxia as a recipient of the Alexa Fund [ 9 ] to integrate Alexa voice services [ 10 ] into Triby. | https://en.wikipedia.org/wiki/Invoxia_Triby |
Ioannis Kontoyiannis (born January 1972) is a Greek mathematician and information theorist. He is the Churchill Professor of Mathematics of Information with the Statistical Laboratory, in the Department of Pure Mathematics and Mathematical Statistics , of the University of Cambridge . He is also a Fellow of Darwin College, Cambridge , and Chairman of the Rollo Davidson Trust . [ 1 ]
He is an affiliated member of the Division of Information Engineering, Cambridge, [ 2 ] a Research Fellow of the Foundation for Research and Technology - Hellas, [ 3 ] and a Senior Member of Robinson College, Cambridge .
His research interests are in information theory, probability and statistics , including their applications in data compression, bioinformatics, neuroscience , machine learning , and the connections between core information-theoretic ideas and results in probability theory and additive combinatorics .
Kontoyiannis earned a B.S. in mathematics from Imperial College, University of London (1992), he obtained a distinction in Part III of the Cambridge University Pure Mathematics Tripos (1993), and he earned an M.S. in statistics (1997) and a Ph.D. in electrical engineering (1998), both from Stanford University . Between 1998 and 2018 he taught at Purdue University , Brown University , Columbia University , and at the Athens University of Economics and Business . In January 2018 he joined the Information Engineering Division at Cambridge University, as Professor of Information and Communications, and Head of the Signal Processing and Communications Laboratory. Since June 2020 he has been with the Statistical Laboratory , in the Department of Pure Mathematics and Mathematical Statistics , University of Cambridge , where he holds the Churchill Chair in Mathematics. | https://en.wikipedia.org/wiki/Ioannis_Kontoyiannis |
Iodic acid is a white water-soluble solid with the chemical formula HIO 3 . Its robustness contrasts with the instability of chloric acid and bromic acid . Iodic acid features iodine in the oxidation state +5 and is one of the most stable oxo-acids of the halogens . When heated, samples dehydrate to give iodine pentoxide . On further heating, the iodine pentoxide further decomposes, giving a mix of iodine, oxygen and lower oxides of iodine.
Iodic acid can be produced by oxidizing iodine with strong oxidizers such as nitric acid , chlorine, chloric acid or hydrogen peroxide, [ 3 ] for example:
Iodic acid is also produced by the reaction of iodine monochloride with water:
Iodic acid crystallises from acidic solution as orthorhombic α- HIO 3 in space group P 2 1 2 1 2 1 . The structure consists of pyramidal molecules linked by hydrogen bonding and intermolecular iodine-oxygen interactions. The I=O bond lengths are 1.81 Å while the I–OH distance is 1.89 Å. [ 4 ] [ 5 ] [ 6 ] Several other polymorphs have been reported, including an orthorhombic γ form in space group Pbca [ 7 ] and an orthorhombic δ form in space group P 2 1 2 1 2 1 . [ 8 ] All of the polymorphs contain pyramidal molecules, hydrogen bonding and I···O interactions, but differ in packing arrangement.
Iodic acid is a relatively strong acid with a p K a of 0.75. It is strongly oxidizing in acidic solution, less so in basic solution. When iodic acid acts as oxidizer, then the product of the reaction is either iodine, or iodide ion. Under some special conditions (very low pH and high concentration of chloride ions, such as in concentrated hydrochloric acid), iodic acid is reduced to iodine trichloride , a golden yellow compound in solution and no further reduction occurs. In the absence of chloride ions, when there is an excess amount of reductant, then all iodate is converted to iodide ion. When there is an excess amount of iodate, then part of the iodate is converted to iodine. [ citation needed ]
Iodic acid is used as a strong acid (though it is not truly a strong acid, but a weak acid that is very close to being a strong acid) in analytical chemistry . It may be used to standardize solutions of both weak and strong bases , using methyl red or methyl orange as the indicator .
Iodic acid can be used to synthesize sodium or potassium iodate for increasing iodine content of salt. [ citation needed ]
Iodic acid is part of a series of oxyacids in which iodine can assume oxidation states of −1, +1, +3, +5, or +7. A number of neutral iodine oxides are also known. | https://en.wikipedia.org/wiki/Iodic_acid |
Iodine-125 ( 125 I) is a radioisotope of iodine which has uses in biological assays , nuclear medicine imaging and in radiation therapy as brachytherapy to treat a number of conditions, including prostate cancer , uveal melanomas , and brain tumors . It is the second longest-lived radioisotope of iodine, after iodine-129 .
Its half-life is 59.392 days and it decays by electron capture to an excited state of tellurium-125 . This state is not the metastable 125m Te, but rather a lower energy state. The excited 125 Te may (7% chance) undergo gamma decay with a maximum energy of 35 keV . More often (93% chance), the excited 125 Te undergoes internally conversion and ejects an electron (< 35 keV). The resulting electron vacancy leads to emission of characteristic X-rays (27–32 keV) and a total of 21 Auger electrons (50 to 500 eV). [ 3 ] Eventually, stable ground state 125 Te is produced as the final decay product.
In medical applications, the internal conversion and Auger electrons cause little damage outside the cell which contains the isotope atom. The X-rays and gamma rays are of low enough energy to deliver a higher radiation dose selectively to nearby tissues, in "permanent" brachytherapy where the isotope capsules are left in place ( 125 I competes with palladium-103 in such uses). [ 4 ]
Because of its relatively long half-life and emission of low-energy photons which can be detected by gamma-counter crystal detectors , 125 I is a preferred isotope for tagging antibodies in radioimmunoassay and other gamma-counting procedures involving proteins outside the body. The same properties of the isotope make it useful for brachytherapy, and for certain nuclear medicine scanning procedures, in which it is attached to proteins ( albumin or fibrinogen ), and where a half-life longer than that provided by 123 I is required for diagnostic or lab tests lasting several days.
Iodine-125 can be used in scanning/imaging the thyroid , but iodine-123 is preferred for this purpose, due to better radiation penetration and shorter half-life (13 hours). 125 I is useful for glomerular filtration rate (GFR) testing in the diagnosis or monitoring of patients with kidney disease . Iodine-125 is used therapeutically in brachytherapy treatments of tumors . For radiotherapy ablation of tissues that absorb iodine (such as the thyroid), or that absorb an iodine-containing radiopharmaceutical , the beta-emitter iodine-131 is the preferred isotope.
When studying plant immunity , 125 I is used as the radiolabel in tracking ligands to determine which plant pattern recognition receptors (PRRs) they bind to. [ 5 ]
125 I is produced by the electron capture decay of 125 Xe , which is an artificial isotope of xenon , itself created by neutron capture of near-stable 124 Xe (it undergoes double electron capture with a half life orders of magnitude larger than the age of the universe), which makes up around 0.1% of naturally occurring xenon. Because of the artificial production route of 125 I and its short half-life, its natural abundance on Earth is effectively zero.
125 I is a reactor-produced radionuclide and is available in large quantities. Its production involves the two following nuclear reactions :
The irradiation target is the primordial nuclide 124 Xe, which is the target isotope for making 125 I by neutron capture . It is loaded into irradiation capsules of the zirconium alloy zircaloy-2 (a corrosion resisting alloy transparent to neutrons ) to a pressure of about 100 bar (~ 100 atm ) . Upon irradiation with slow neutrons in a nuclear reactor , several radioisotopes of xenon are produced. However, only the decay of 125 Xe leads to a radioiodine: 125 I. The other xenon radioisotopes decay either to stable xenon , or to various caesium isotopes , some of them radioactive (a.o., the long-lived 135 Cs ( t ½ = 1.33 Ma) and 137 Cs ( t ½ = 30 a)).
Long irradiation times are disadvantageous. Iodine-125 itself has a neutron capture cross section of 900 barns , and consequently during a long irradiation, part of the 125 I formed will be converted to 126 I, a beta-emitter and positron-emitter with a half-life of 12.93 days, [ 1 ] which is not medically useful. In practice, the most useful irradiation time in the reactor amounts to a few days. Thereafter, the irradiated gas is allowed to decay for three or four days to eliminate short-lived unwanted radioisotopes, and to allow the newly produced xenon-125 ( t ½ = 17 hours) to decay to iodine-125.
To isolate radio-iodine, the irradiated capsule is first cooled at low temperature (to condense the free iodine gas onto the capsule inner wall) and the remaining Xe gas is vented in a controlled way and recovered for further use. The inner walls of the capsule are then rinsed with a dilute NaOH solution to collect iodine as soluble iodide (I − ) and hypoiodite (IO − ), according to the standard disproportionation reaction of halogens in alkaline solution. Any caesium atom present immediately oxidizes and passes into the water as Cs + . In order to eliminate any long-lived 135 Cs and 137 Cs which may be present in small amounts, the solution is passed through a cation-exchange column, which exchanges Cs + for another non-radioactive cation (e.g., Na + ). The radioiodine (as anion I − or IO − ) remains in solution as a mixture iodide/hypoiodite.
Iodine-125 is commercially available in dilute NaOH solution as 125 I-iodide (or the hypohalite sodium hypoiodite , NaIO). The radioactive concentration lies at 4 to 11 GBq/mL and the specific radioactivity is > 75 GBq/μmol (7.5 × 10 16 Bq/mol) . The chemical and radiochemical purity is high. The radionuclidic purity is also high; some 126 I ( t 1/2 = 12.93 d) [ 1 ] is unavoidable due to the neutron capture noted above. The 126 I tolerable content (which is set by the unwanted isotope interfering with dose calculations in brachytherapy ) lies at about 0.2 atom % ( atom fraction ) of the total iodine (the rest being 125 I).
As of October 2019, there were two producers of iodine-125, the McMaster Nuclear Reactor in Hamilton , Ontario , Canada; and a VVR-SM research reactor in Uzbekistan. [ 6 ] The McMaster reactor is presently the largest producer of iodine-125, producing approximately 60 per cent of the global supply in 2018; [ 7 ] with the remaining global supply produced at the reactor based in Uzbekistan. Annually, the McMaster reactor produces enough iodine-125 to treat approximately 70,000 patients. [ 8 ]
In November 2019, the research reactor in Uzbekistan shut down temporarily in order to facilitate repairs. The temporary shutdown threatened the global supply of the radioisotope by leaving the McMaster reactor as the sole producer of iodine-125 during the period. [ 6 ] [ 8 ]
Prior to 2018, the National Research Universal (NRU) reactor at Chalk River Laboratories in Deep River , Ontario, was one of three reactors to produce iodine-125. [ 9 ] However, on March 31, 2018, the NRU reactor was permanently shut down ahead of its scheduled decommissioning in 2028, as a result of a government order. [ 10 ] [ 11 ] The Russian nuclear reactor equipped to produce iodine-125, was offline as of December 2019. [ 6 ]
The detailed decay mechanism to form the stable daughter nuclide tellurium-125 is a multi-step process that begins with electron capture , which produces a tellurium-125 nucleus in an excited state with a half-life of 1.6 ns. The excited tellurium-125 nucleus may undergo gamma decay , emitting a gamma photon at 35.5 keV, or undergo internal conversion to emit an electron . The electron vacancy from internal conversion results in a cascade of electron relaxation as the core electron hole moves toward the valence orbitals . The cascade involves many characteristic X-rays and Auger transitions . In the case the excited tellurium-125 nucleus undergoes gamma decay, a different electron relaxation cascade follows before the nuclide comes to rest. Throughout the entire process an average of 13.3 electrons are emitted (10.3 of which are Auger electrons ), most with energies less than 400 eV (79% of yield). [ 12 ] The internal conversion and Auger electrons from the radioisotope have been found in one study to do little cellular damage, unless the radionuclide is directly incorporated chemically into cellular DNA , which is not the case for present radiopharmaceuticals which use 125 I as the radioactive label nuclide. [ 13 ] Rather, cellular damage results from the gamma and characteristic X-ray photons.
As with other radioisotopes of iodine, accidental iodine-125 uptake in the body (mostly by the thyroid gland) can be blocked by the prompt administration of stable iodine-127 in the form of an iodide salt. [ 14 ] [ 15 ] Potassium iodide (KI) is typically used for this purpose. [ 16 ]
However, unjustified self-medicated preventive administration of stable KI is not recommended in order to avoid disturbing the normal thyroid function . Such a treatment must be carefully dosed and requires an appropriate KI amount prescribed by a specialised physician. | https://en.wikipedia.org/wiki/Iodine-125 |
Iodine-129 ( 129 I) is a long-lived radioisotope of iodine that occurs naturally but is also of special interest in the monitoring and effects of man-made nuclear fission products , where it serves as both a tracer and a potential radiological contaminant.
129 I is one of seven long-lived fission products . It is primarily formed from the fission of uranium and plutonium in nuclear reactors . Significant amounts were released into the atmosphere by nuclear weapons testing in the 1950s and 1960s, by nuclear reactor accidents and by both military and civil reprocessing of spent nuclear fuel. [ 3 ]
It is also naturally produced in small quantities, due to the spontaneous fission of natural uranium , by cosmic ray spallation of trace levels of xenon in the atmosphere, and by cosmic ray muons striking tellurium -130. [ 4 ] [ 5 ]
129 I decays with a half-life of 16.14 million years, with low-energy beta and gamma emissions, to stable xenon-129 ( 129 Xe). [ 6 ]
129 I is one of the seven long-lived fission products that are produced in significant amounts. Its yield is 0.706% per fission of 235 U . [ 7 ] Larger proportions of other iodine isotopes such as 131 I are produced, but because these all have short half-lives, iodine in cooled spent nuclear fuel consists of about 5/6 129 I and 1/6 the only stable iodine isotope, 127 I.
Because 129 I is long-lived and relatively mobile in the environment, it is of particular importance in long-term management of spent nuclear fuel. In a deep geological repository for unreprocessed used fuel, 129 I is likely to be the radionuclide of most potential impact at long times.
Since 129 I has a modest neutron absorption cross-section of 30 barns , [ 8 ] and is relatively undiluted by other isotopes of the same element, it is being studied for disposal by nuclear transmutation by re-irradiation with neutrons [ 9 ] or gamma irradiation. [ 10 ]
A large fraction of the 129 I contained in spent fuel is released into the gas phase, when spent fuel is first chopped and then dissolved in boiling nitric acid during reprocessing. [ 3 ] At least for civil reprocessing plants, special scrubbers are supposed to withhold 99.5% (or more) of the Iodine by adsorption, [ 3 ] before exhaust air is released into the environment. However, the Northeastern Radiological Health Laboratory (NERHL) found, during their measurements at the first US civil reprocessing plant, which was operated by Nuclear Fuel Services, Inc. (NFS) in Western New York, that "between 5 and 10% of the total 129 I available from the dissolved fuel" was released into the exhaust stack. [ 3 ] They further wrote that "these values are greater than predicted output (Table 1). This was expected since the iodine scrubbers were not operating during the dissolution cycles monitored." [ 3 ]
The Northeastern Radiological Health Laboratory further states that, due to limitations of their measuring systems, the actual release of 129 I may have even been higher, "since [ 129 I] losses [by adsorption] probably occurred in the piping and ductwork between the stack and the sampler". [ 3 ] Furthermore, the sample taking system used by the NERHL had a bubbler trap for measuring the tritium content of the gas samples before the iodine trap. The NERHL found out only after taking the samples that "the bubbler trap retained 60 to 90% of the 129 I sampled". [ 3 ] NERHL concluded: "The bubblers located upstream of the ion exchangers removed a major portion of the gaseous 129 I before it reached the ion exchange sampler. The iodine removal ability of the bubbler was anticipated, but not in the magnitude that it occurred." The documented release of "between 5 and 10% of the total 129 I available from the dissolved fuel" [ 3 ] is not corrected for those two measurement deficiencies.
Military isolation of plutonium from spent fuel has also released 129 I to the atmosphere: "More than 685,000 curies of iodine 131 spewed from the stacks of Hanford's separation plants in the first three years of operation." [ 11 ] As 129 I and 131 I have very similar physical and chemical properties, and no isotope separation was performed at Hanford, 129 I must have also been released there in large quantities during the Manhattan project. As Hanford reprocessed "hot" fuel, that had been irradiated in a reactor only a few months earlier, the activity of the released short-lived 131 I, with a half-life time of just 8 days, was much higher than that of the long-lived 129 I. However, while all of the 131 I released during the times of the Manhattan project has decayed by now, over 99.999% of the 129 I is still in the environment.
Ice borehole data obtained from the university of Bern at the Fiescherhorn glacier in the Alpian mountains at a height of 3950 m show a somewhat steady increase in the 129 I deposit rate (shown in the image as a solid line) with time. In particular, the highest values obtained in 1983 and 1984 are about six times as high as the maximum that was measured during the period of the atmospheric bomb testing in 1961. This strong increase following the conclusion of the atmospheric bomb testing indicates that nuclear fuel reprocessing has been the primary source of atmospheric iodine-129 since then. These measurements lasted until 1986. [ 12 ]
129 I is not deliberately produced for any practical purposes. However, its long half-life and its relative mobility in the environment have made it useful for a variety of dating applications. These include identifying older groundwaters based on the amount of natural 129 I (or its 129 Xe decay product) present, as well as identifying younger groundwaters by the increased anthropogenic 129 I levels since the 1960s. [ 13 ] [ 14 ] [ 15 ]
In 1960, physicist John H. Reynolds discovered that certain meteorites contained an isotopic anomaly in the form of an overabundance of 129 Xe. He inferred that this must be a decay product of long-decayed radioactive 129 I. This isotope is produced in quantity in nature only in supernova explosions. As the half-life of 129 I is comparatively short in astronomical terms, this demonstrated that only a short time had passed between the supernova and the time the meteorites had solidified and trapped the 129 I. These two events (supernova and solidification of gas cloud) were inferred to have happened during the early history of the Solar System , as the 129 I isotope was likely generated before the Solar System was formed, but not long before, and seeded the solar gas cloud isotopes with isotopes from a second source. This supernova source may also have caused collapse of the solar gas cloud. [ 16 ] [ 17 ] | https://en.wikipedia.org/wiki/Iodine-129 |
Iodine-131 ( 131 I , I-131 ) is an important radioisotope of iodine discovered by Glenn Seaborg and John Livingood in 1938 at the University of California, Berkeley. [ 3 ] It has a radioactive decay half-life of about eight days. It is associated with nuclear energy, medical diagnostic and treatment procedures, and natural gas production. It also plays a major role as a radioactive isotope present in nuclear fission products, and was a significant contributor to the health hazards from open-air atomic bomb testing in the 1950s, and from the Chernobyl disaster , as well as being a large fraction of the contamination hazard in the first weeks in the Fukushima nuclear crisis . This is because 131 I is a major fission product of uranium and plutonium , comprising nearly 3% of the total products of fission (by weight). See fission product yield for a comparison with other radioactive fission products. 131 I is also a major fission product of uranium-233 , produced from thorium .
Due to its mode of beta decay , iodine-131 causes mutation and death in cells that it penetrates, and other cells up to several millimeters away. For this reason, high doses of the isotope are sometimes less dangerous than low doses, since they tend to kill thyroid tissues that would otherwise become cancerous as a result of the radiation. For example, children treated with moderate dose of 131 I for thyroid adenomas had a detectable increase in thyroid cancer , but children treated with a much higher dose did not. [ 4 ] Likewise, most studies of very-high-dose 131 I for treatment of Graves' disease have failed to find any increase in thyroid cancer, even though there is linear increase in thyroid cancer risk with 131 I absorption at moderate doses. [ 5 ] Thus, iodine-131 is increasingly less employed in small doses in medical use (especially in children), but increasingly is used only in large and maximal treatment doses, as a way of killing targeted tissues. This is known as "therapeutic use".
Iodine-131 can be "seen" by nuclear medicine imaging techniques (e.g., gamma cameras ) whenever it is given for therapeutic use, since about 10% of its energy and radiation dose is via gamma radiation. However, since the other 90% of radiation (beta radiation) causes tissue damage without contributing to any ability to see or "image" the isotope, other less-damaging radioisotopes of iodine such as iodine-123 (see isotopes of iodine ) are preferred in situations when only nuclear imaging is required. The isotope 131 I is still occasionally used for purely diagnostic (i.e., imaging) work, due to its low expense compared to other iodine radioisotopes. Very small medical imaging doses of 131 I have not shown any increase in thyroid cancer. The low-cost availability of 131 I, in turn, is due to the relative ease of creating 131 I by neutron bombardment of natural tellurium in a nuclear reactor, then separating 131 I out by various simple methods (i.e., heating to drive off the volatile iodine). By contrast, other iodine radioisotopes are usually created by far more expensive techniques, starting with cyclotron radiation of capsules of pressurized xenon gas. [ 6 ]
Iodine-131 is also one of the most commonly used gamma-emitting radioactive industrial tracer . Radioactive tracer isotopes are injected with hydraulic fracturing fluid to determine the injection profile and location of fractures created by hydraulic fracturing. [ 7 ]
Much smaller incidental doses of iodine-131 than those used in medical therapeutic procedures, are supposed by some studies to be the major cause of increased thyroid cancers after accidental nuclear contamination. These studies suppose that cancers happen from residual tissue radiation damage caused by the 131 I, and should appear mostly years after exposure, long after the 131 I has decayed. [ 8 ] [ 9 ] Other studies did not find a correlation. [ 10 ] [ 11 ]
Most 131 I production is from neutron irradiation of a natural tellurium target in a nuclear reactor. Irradiation of natural tellurium produces almost entirely 131 I as the only radionuclide with a half-life longer than hours, since most lighter isotopes of tellurium become heavier stable isotopes, or else stable iodine or xenon. However, the heaviest naturally occurring tellurium nuclide, 130 Te (34% of natural tellurium) absorbs a neutron to become tellurium-131, which beta decays with a half-life of 25 minutes to 131 I.
A tellurium compound can be irradiated while bound as an oxide to an ion exchange column, with evolved 131 I then eluted into an alkaline solution. [ 12 ] More commonly, powdered elemental tellurium is irradiated and then 131 I separated from it by dry distillation of the iodine, which has a far higher vapor pressure . The element is then dissolved in a mildly alkaline solution in the standard manner, to produce 131 I as iodide and hypoiodate (which is soon reduced to iodide). [ 13 ]
131 I is a fission product with a yield of 2.878% from uranium-235 , [ 14 ] and can be released in nuclear weapons tests and nuclear accidents . However, the short half-life means it is not present in significant quantities in cooled spent nuclear fuel , unlike iodine-129 whose half-life is nearly a billion times that of 131 I.
It is discharged to the atmosphere in small quantities by some nuclear power plants. [ 15 ]
131 I decays with a half-life of 8.0249(6) days [ 1 ] with beta minus and gamma emissions. This isotope of iodine has 78 neutrons in its nucleus, while the only stable nuclide, 127 I, has 74. On decaying, 131 I most often (89% of the time) expends its 971 keV of decay energy by transforming into stable xenon-131 in two steps, with gamma decay following rapidly after beta decay:
The primary emissions of 131 I decay are thus electrons with a maximal energy of 606 keV (89% abundance, others 248–807 keV) and 364 keV gamma rays (81% abundance, others 723 keV). [ 16 ] Beta decay also produces an antineutrino , which carries off variable amounts of the beta decay energy. The electrons, due to their high mean energy (190 keV, with typical beta-decay spectra present) have a tissue penetration of 0.6 to 2 mm . [ 17 ]
Iodine in food is absorbed by the body and preferentially concentrated in the thyroid where it is needed for the functioning of that gland. When 131 I is present in high levels in the environment from radioactive fallout , it can be absorbed through contaminated food, and will also accumulate in the thyroid. As it decays, it may cause damage to the thyroid. The primary risk from exposure to 131 I is an increased risk of radiation-induced cancer in later life. Other risks include the possibility of non-cancerous growths and thyroiditis . [ 5 ]
The risk of thyroid cancer in later life appears to diminish with increasing age at time of exposure. Most risk estimates are based on studies in which radiation exposures occurred in children or teenagers. When adults are exposed, it has been difficult for epidemiologists to detect a statistically significant difference in the rates of thyroid disease above that of a similar but otherwise-unexposed group. [ 5 ] [ 19 ]
The risk can be mitigated by taking iodine supplements, raising the total amount of iodine in the body and, therefore, reducing uptake and retention in the face and chest and lowering the relative proportion of radioactive iodine. However, such supplements were not consistently distributed to the population living nearest to the Chernobyl nuclear power plant after the disaster, [ 20 ] though they were widely distributed to children in Poland.
Within the US, the highest 131 I fallout doses occurred during the 1950s and early 1960s to children having consumed fresh milk from sources contaminated as the result of above-ground testing of nuclear weapons. [ 8 ] The National Cancer Institute provides additional information on the health effects from exposure to 131 I in fallout, [ 21 ] as well as individualized estimates, for those born before 1971, for each of the 3070 counties in the US. The calculations are taken from data collected regarding fallout from the nuclear weapons tests conducted at the Nevada Test Site . [ 22 ]
On 27 March 2011, the Massachusetts Department of Public Health reported that 131 I was detected in very low concentrations in rainwater from samples collected in Massachusetts, and that this likely originated from the Fukushima power plant. [ 23 ] Farmers near the plant dumped raw milk, while testing in the United States found 0.8 pico- curies per liter of iodine-131 in a milk sample, but the radiation levels were 5,000 times lower than the FDA's "defined intervention level".
The levels were expected to drop relatively quickly [ 24 ]
A common treatment method for preventing iodine-131 exposure is by saturating the thyroid with regular, stable iodine-127 , as an iodide or iodate salt.
Iodine-131 is used for unsealed source radiotherapy in nuclear medicine to treat several conditions. It can also be detected by gamma cameras for diagnostic imaging , however it is rarely administered for diagnostic purposes only, imaging will normally be done following a therapeutic dose. [ 26 ] Use of the 131 I as iodide salt exploits the mechanism of absorption of iodine by the normal cells of the thyroid gland.
Major uses of 131 I include the treatment of thyrotoxicosis (hyperthyroidism) due to Graves' disease , and sometimes hyperactive thyroid nodules (abnormally active thyroid tissue that is not malignant). The therapeutic use of radioiodine to treat hyperthyroidism from Graves' disease was first reported by Saul Hertz in 1941. The dose is typically administered orally (either as a liquid or capsule), in an outpatient setting, and is usually 400–600 megabecquerels (MBq). [ 27 ] Radioactive iodine (iodine-131) alone can potentially worsen thyrotoxicosis in the first few days after treatment. One side effect of treatment is an initial period of a few days of increased hyperthyroid symptoms. This occurs because when the radioactive iodine destroys the thyroid cells, they can release thyroid hormone into the blood stream. For this reason, sometimes patients are pre-treated with thyrostatic medications such as methimazole, and/or they are given symptomatic treatment such as propranolol. Radioactive iodine treatment is contraindicated in breast-feeding and pregnancy [ 28 ]
Iodine-131, in higher doses than for thyrotoxicosis, is used for ablation of remnant thyroid tissue following a complete thyroidectomy to treat thyroid cancer . [ 29 ] [ 27 ]
Typical therapeutic doses of I-131 are between 2220 and 7400 megabecquerels (MBq). [ 30 ] Because of this high radioactivity and because the exposure of stomach tissue to beta radiation would be high near an undissolved capsule, I-131 is sometimes administered to human patients in a small amount of liquid. Administration of this liquid form is usually by straw which is used to slowly and carefully suck up the liquid from a shielded container. [ 31 ] For administration to animals (for example, cats with hyperthyroidism), for practical reasons the isotope must be administered by injection. European guidelines recommend administration of a capsule, due to "greater ease to the patient and the superior radiation protection for caregivers". [ 32 ]
Ablation doses are usually administered on an inpatient basis, and IAEA International Basic Safety Standards recommend that patients are not discharged until the activity falls below 1100 MBq. [ 33 ] ICRP advice states that "comforters and carers" of patients undergoing radionuclide therapy should be treated as members of the public for dose constraint purposes and any restrictions on the patient should be designed based on this principle. [ 34 ]
Patients receiving I-131 radioiodine treatment may be warned not to have sexual intercourse for one month (or shorter, depending on dose given), and women told not to become pregnant for six months afterwards. "This is because a theoretical risk to a developing fetus exists, even though the amount of radioactivity retained may be small and there is no medical proof of an actual risk from radioiodine treatment. Such a precaution would essentially eliminate direct fetal exposure to radioactivity and markedly reduce the possibility of conception with sperm that might theoretically have been damaged by exposure to radioiodine." [ 35 ] These guidelines vary from hospital to hospital and will depend on national legislation and guidance, as well as the dose of radiation given. Some also advise not to hug or hold children when the radiation is still high, and a one- or two- metre distance to others may be recommended. [ 36 ]
I-131 will be eliminated from the body over the next several weeks after it is given. The majority of I-131 will be eliminated from the human body in 3–5 days, through natural decay, and through excretion in sweat and urine. Smaller amounts will continue to be released over the next several weeks, as the body processes thyroid hormones created with the I-131. For this reason, it is advised to regularly clean toilets, sinks, bed sheets and clothing used by the person who received the treatment. Patients may also be advised to wear slippers or socks at all times, and avoid prolonged close contact with others. This minimizes accidental exposure by family members, especially children. [ 37 ] Use of a decontaminant specially made for radioactive iodine removal may be advised. The use of chlorine bleach solutions, or cleaners that contain chlorine bleach for cleanup, are not advised, since radioactive elemental iodine gas may be released. [ 38 ] Airborne I-131 may cause a greater risk of second-hand exposure, spreading contamination over a wide area. Patient is advised if possible to stay in a room with a bathroom connected to it to limit unintended exposure to family members.
Many airports have radiation detectors to detect the smuggling of radioactive materials. Patients should be warned that if they travel by air, they may trigger radiation detectors at airports up to 95 days after their treatment with 131 I. [ 39 ]
The 131 I isotope is also used as a radioactive label for certain radiopharmaceuticals that can be used for therapy, e.g. 131 I- metaiodobenzylguanidine ( 131 I-MIBG) for imaging and treating pheochromocytoma and neuroblastoma . In all of these therapeutic uses, 131 I destroys tissue by short-range beta radiation . About 90% of its radiation damage to tissue is via beta radiation, and the rest occurs via its gamma radiation (at a longer distance from the radioisotope). It can be seen in diagnostic scans after its use as therapy, because 131 I is also a gamma-emitter.
Because of the carcinogenicity of its beta radiation in the thyroid in small doses, I-131 is rarely used primarily or solely for diagnosis (although in the past this was more common due to this isotope's relative ease of production and low expense). Instead the more purely gamma-emitting radioiodine iodine-123 is used in diagnostic testing ( nuclear medicine scan of the thyroid). The longer half-lived iodine-125 is also occasionally used when a longer half-life radioiodine is needed for diagnosis, and in brachytherapy treatment (isotope confined in small seed-like metal capsules), where the low-energy gamma radiation without a beta component makes iodine-125 useful. The other radioisotopes of iodine are never used in brachytherapy.
The use of 131 I as a medical isotope has been blamed for a routine shipment of biosolids being rejected from crossing the Canada—U.S. border. [ 40 ] Such material can enter the sewers directly from the medical facilities, or by being excreted by patients after a treatment
Used for the first time in 1951 to localize leaks in a drinking water supply system of Munich , Germany, iodine-131 became one of the most commonly used gamma-emitting industrial radioactive tracers , with applications in isotope hydrology and leak detection. [ 41 ] [ 42 ] [ 43 ] [ 44 ]
Since the late 1940s, radioactive tracers have been used by the oil industry. Tagged at the surface, water is then tracked downhole, using the appropriated gamma detector, to determine flows and detect underground leaks. I-131 has been the most widely used tagging isotope in an aqueous solution of sodium iodide . [ 45 ] [ 46 ] [ 47 ] It is used to characterize the hydraulic fracturing fluid to help determine the injection profile and location of fractures created by hydraulic fracturing . [ 48 ] [ 49 ] [ 50 ] | https://en.wikipedia.org/wiki/Iodine-131 |
Iodine ( 131 I) derlotuximab biotin is a monoclonal antibody designed for the treatment of recurrent glioblastoma multiforme . [ 1 ]
This drug was developed by Peregrine Pharmaceuticals, Inc.
This nuclear medicine article is a stub . You can help Wikipedia by expanding it .
This monoclonal antibody –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Iodine_(131_I)_derlotuximab_biotin |
The iodine clock reaction is a classical chemical clock demonstration experiment to display chemical kinetics in action; it was discovered by Hans Heinrich Landolt in 1886. [ 1 ] The iodine clock reaction exists in several variations, which each involve iodine species ( iodide ion, free iodine, or iodate ion) and redox reagents in the presence of starch . Two colourless solutions are mixed and at first there is no visible reaction. After a short time delay, the liquid suddenly turns to a shade of dark blue due to the formation of a triiodide–starch complex . In some variations, the solution will repeatedly cycle from colorless to blue and back to colorless, until the reagents are depleted.
This method starts with a solution of hydrogen peroxide and sulfuric acid . To this a solution containing potassium iodide , sodium thiosulfate , and starch is added. There are two reactions occurring simultaneously in the solution.
In the first, slow reaction, iodine is produced:
In the second, fast reaction, iodine is reconverted to two iodide ions by the thiosulfate:
After some time the solution changes color to a very dark blue, almost black.
When the solutions are mixed, the second reaction causes the iodine to be consumed much faster than it is generated , and only a small amount of iodine is present in the dynamic equilibrium . Once the thiosulfate ion has been exhausted, this reaction stops and the blue colour caused by the iodine – starch complex appears.
Anything that accelerates the first reaction will shorten the time until the solution changes color. Decreasing the pH (increasing H + concentration), or increasing the concentration of iodide or hydrogen peroxide will shorten the time. Adding more thiosulfate will have the opposite effect; it will take longer for the blue colour to appear.
Aside from using sodium thiosulfate as a substrate, cysteine can also be used. [ 2 ]
Iodide from potassium iodide is converted to iodine in the first reaction:
2 I − + 2 H + + H 2 O 2 → I 2 + 2 H 2 O
The iodine produced in the first reaction is reduced back to iodide by the reducing agent , cysteine. At the same time, cysteine is oxidized into cystine.
2 C 3 H 7 NO 2 S + I 2 → C 6 H 12 N 2 O 4 S 2 + 2 I − + 2 H +
Similar to the thiosulfate case, when cysteine is exhausted, the blue color appears.
An alternative protocol uses a solution of iodate ion (for instance potassium iodate) to which an acidified solution (again with sulfuric acid ) of sodium bisulfite is added. [ 3 ]
In this protocol, iodide ion is generated by the following slow reaction between the iodate and bisulfite:
This first step is the rate determining step. Next, the iodate in excess will oxidize the iodide generated above to form iodine:
However, the iodine is reduced immediately back to iodide by the bisulfite:
When the bisulfite is fully consumed, the iodine will survive (i.e., no reduction by the bisulfite) to form the dark blue complex with starch.
This clock reaction uses sodium , potassium or ammonium persulfate to oxidize iodide ions to iodine . Sodium thiosulfate is used to reduce iodine back to iodide before the iodine can complex with the starch to form the characteristic blue-black color.
Iodine is generated:
And is then removed:
Once all the thiosulfate is consumed the iodine may form a complex with the starch. Potassium persulfate is less soluble (cfr. Salters website) while ammonium persulfate has a higher solubility and is used instead in the reaction described in examples from Oxford University. [ 4 ]
An experimental iodine clock sequence has also been established for a system consisting of iodine potassium-iodide , sodium chlorate and perchloric acid that takes place through the following reactions. [ 5 ]
Triiodide is present in equilibrium with iodide anion and molecular iodine :
Chlorate ion oxidizes iodide ion to hypoiodous acid and chlorous acid in the slow and rate-determining step :
Chlorate consumption is accelerated by reaction of hypoiodous acid to iodous acid and more chlorous acid:
More autocatalysis when newly generated iodous acid also converts chlorate in the fastest reaction step:
In this clock the induction period is the time it takes for the autocatalytic process to start after which the concentration of free iodine falls rapidly as observed by UV–visible spectroscopy . | https://en.wikipedia.org/wiki/Iodine_clock_reaction |
Iodine heptafluoride is an interhalogen compound with the chemical formula I F 7 . [ 2 ] [ 3 ] It has an unusual pentagonal bipyramidal structure, with D 5h symmetry , as predicted by VSEPR theory . [ 4 ] The molecule can undergo a pseudorotational rearrangement called the Bartell mechanism , which is like the Berry mechanism but for a heptacoordinated system. [ 5 ]
Below 4.5 °C, IF 7 forms a snow-white powder of colorless crystals, melting at 5-6 °C. However, this melting is difficult to observe, as the liquid form is thermodynamically unstable at 760 mmHg : instead, the compound begins to sublime at 4.77 °C. The dense vapor has a mouldy, acrid odour. [ 6 ] [ 7 ]
IF 7 is prepared by passing F 2 through liquid IF 5 at 90 °C, then heating the vapours to 270 °C. Alternatively, this compound can be prepared from fluorine and dried palladium or potassium iodide to minimize the formation of IOF 5 , an impurity arising by hydrolysis. [ 8 ] [ 9 ] Iodine heptafluoride is also produced as a by-product when dioxygenyl hexafluoroplatinate is used to prepare other platinum(V) compounds such as potassium hexafluoroplatinate(V) , using potassium fluoride in iodine pentafluoride solution: [ 10 ]
Iodine heptafluoride decomposes at 200 °C to fluorine gas and iodine pentafluoride . [ 11 ]
IF 7 is highly irritating to both the skin and the mucous membranes . It also is a strong oxidizer and can cause fire on contact with organic material. | https://en.wikipedia.org/wiki/Iodine_heptafluoride |
Iodine is an essential trace element in biological systems. It has the distinction of being the heaviest element commonly needed by living organisms as well as the second-heaviest known to be used by any form of life (only tungsten , a component of a few bacterial enzymes, has a higher atomic number and atomic weight ). It is a component of biochemical pathways in organisms from all biological kingdoms, suggesting its fundamental significance throughout the evolutionary history of life. [ 1 ]
Iodine is critical to the proper functioning of the vertebrate endocrine system, and plays smaller roles in numerous other organs, including those of the digestive and reproductive systems. An adequate intake of iodine-containing compounds is important at all stages of development, especially during the fetal and neonatal periods, and diets deficient in iodine can present serious consequences for growth and metabolism.
In vertebrate biology, iodine's primary function is as a constituent of the thyroid hormones , thyroxine (T4) and triiodothyronine (T3). These molecules are made from addition-condensation products of the amino acid tyrosine , and are stored prior to release in an iodine-containing protein called thyroglobulin . T4 and T3 contain four and three atoms of iodine per molecule, respectively; iodine accounts for 65% of the molecular weight of T4 and 59% of T3. The thyroid gland actively absorbs iodine from the blood to produce and release these hormones into the blood, actions which are regulated by a second hormone, called thyroid-stimulating hormone (TSH), which is produced by the pituitary gland . Thyroid hormones are phylogenetically very old molecules which are synthesized by most multicellular organisms , and which even have some effect on unicellular organisms.
Thyroid hormones play a fundamental role in biology, acting upon gene transcription mechanisms to regulate the basal metabolic rate . T3 acts on small intestine cells and adipocytes to increase carbohydrate absorption and fatty acid release, respectively. [ 2 ] A deficiency of thyroid hormones can reduce basal metabolic rate up to 50%, while an excessive production of thyroid hormones can increase the basal metabolic rate by 100%. [ 3 ] T4 acts largely as a precursor to T3, which is (with minor exceptions) the biologically active hormone.
Via the thyroid hormones, iodine has a nutritional relationship with selenium . A family of selenium-dependent enzymes called deiodinases converts T4 to T3 (the active hormone) by removing an iodine atom from the outer tyrosine ring. These enzymes also convert T4 to reverse T3 (rT3) by removing an inner ring iodine atom, and also convert T3 to 3,3'-Diiodothyronine (T2) by removing an inner ring atom. Both of the latter products are inactivated hormones which have essentially no biological effects and are quickly prepared for disposal. A family of non-selenium-dependent enzymes then further deiodinates the products of these reactions.
The total amount of iodine in the human body is still controversial, and in 2001, M.T. Hays published in Thyroid that "it is surprising that the total iodine content of the human body remains uncertain after many years of interest in iodine metabolism. Only the iodine content of the thyroid gland has been measured accurately by fluorescent scanning, and it is now well estimate of 5–15 mg in the normal human thyroid. But similar methods are not available for other tissues and for the extrathyroidal organs. Many researchers reported different numbers of 10–50 mg of the total iodine content in human body". [ 4 ] [ 5 ] Selenium also plays a very important role in the production of glutathione , the body's most powerful antioxidant . During the production of the thyroid hormones, hydrogen peroxide is produced in large quantities, and therefore high iodine in the absence of selenium can destroy the thyroid gland (often described as a sore throat feeling); the peroxides are neutralized through the production of glutathione from selenium. In turn, an excess of selenium increases demand for iodine, and deficiency will result when a diet is high in selenium and low in iodine. [ citation needed ]
Extra-thyroidal iodine exists in several other organs, including the mammary glands , eyes, gastric mucosa , cervix , cerebrospinal fluid , arterial walls, ovary and salivary glands . [ 6 ] In the cells of these tissues the iodide ion (I − ) enters directly by the sodium-iodide symporter (NIS). Different tissue responses for iodine and iodide occur in the mammary glands and the thyroid gland of rats. [ 7 ] The role of iodine in mammary tissue is related to fetal and neonatal development, but its role in the other tissues is not well known. [ 8 ] It has been shown to act as an antioxidant [ 8 ] and antiproliferant [ 9 ] in various tissues that can uptake iodine. Molecular iodine (I 2 ) has been shown to have a suppressive effect on benign and cancerous neoplasias . [ 9 ]
The U.S. Food and Nutrition Board and Institute of Medicine recommended daily allowance of iodine ranges from 150 micrograms per day for adult humans to 290 micrograms per day for lactating mothers. However, the thyroid gland needs no more than 70 micrograms per day to synthesize the requisite daily amounts of T4 and T3. The higher recommended daily allowance levels of iodine seem necessary for optimal function of a number of other body systems, including lactating breasts, gastric mucosa, salivary glands, oral mucosa, arterial walls, thymus , epidermis, choroid plexus and cerebrospinal fluid , among others. [ 10 ] [ 11 ] [ 12 ]
Iodine and thyroxine have also been shown to stimulate the spectacular apoptosis of the cells of the larval gills, tail and fins during metamorphosis in amphibians , as well as the transformation of their nervous system from that of the aquatic, herbivorous tadpole into that of the terrestrial, carnivorous adult. The frog species Xenopus laevis has proven to be an ideal model organism for experimental study of the mechanisms of apoptosis and the role of iodine in developmental biology. [ 13 ] [ 1 ] [ 14 ] [ 15 ]
It is believed that thyroid hormones evolved in the Urbilaterian well before the development of the thyroid itself and molluscs, echinoderms, cephalochordates and ascidians all use such hormones. [ 16 ] Cnidarians also respond to Thyroid hormone despite being parahoxozoans rather than bilaterians . [ 16 ] [ 17 ]
Insects use hormones similar to thyroid hormone using iodine. [ 18 ] [ 19 ] [ 20 ]
Phosphorylated tyrosines created with tyrosine kinases are fundamental signalling molecules in all animals and in choanoflagellates . [ 21 ] [ 22 ]
Iodine is known to be crucial for life in many unicellular organisms [ 23 ] Phosphorylated tyrosines created with tyrosine kinases are fundamental signalling molecules in all animals and in Choanoflagellates [ 21 ] [ 22 ] and may be linked to the usage of tyrosine iodine compounds for similar roles. [ 23 ] Crockford proposes that iodine was originally used in protecting cell membranes from oxidative damage in photosynthesis and later moved into cytoplasm and became involved with balancing cytoplasmic composition of ions, and later the non enzymatic synthesis of tyrosine in early life. [ 23 ]
It is common across all domains of life and uses tyrosine bonded to iodine. [ 23 ]
Plants, insects, zooplankton and algae store iodine as mono-iodotyrosine (MIT), di-iodotyrosine (DIT), iodocarbons, or iodoproteins. [ 24 ] [ 25 ] [ 26 ]
Many plants use thyroid like hormones for regulating growth. [ 24 ] [ 27 ]
Gut-inhabiting bacteria use iodine from host thyroid hormone. [ 28 ]
Thyroid-like hormones may be linked to the development of multicellularity. [ 29 ] [ 30 ] Iodotyrosines are highly reactive with other molecules [ 31 ] which may have made them important cell signalling molecules early in evolutionary history. [ 23 ] They form spontaneously without need for enzymatic catalysts which may have contributed to their early adoption by organisms, [ 32 ] [ 33 ] although enzymes make the yields significantly higher. [ 34 ]
The ease of reaction with water may explain why iodine is so common across cell signalling in all domains of life. [ 35 ]
Many photosynthetic microbes are able to reduce inorganic iodate to iodide in their cell walls [ 36 ] [ 37 ] [ 38 ] [ 39 ] [ 40 ] but much of it gets released into the environment rather than cytoplasm in compounds such as methyl iodide . [ 41 ] [ 36 ] [ 42 ] Many sulfate-reducing microorganisms and Iron-oxidizing bacteria also reduce iodate to iodide [ 43 ] [ 40 ] as well as many facultative anaerobic organisms [ 44 ] suggesting this may be ancestral among anaerobic organisms. [ 23 ]
Kelp store large quantities of iodide primarily as iodotyrosines for unknown reasons. [ 45 ] [ 46 ]
Molecular iodine (I 2 ) is toxic to most single-celled organisms by disrupting the cell membrane [ 47 ] however Alphaproteobacteria and Choanoflagellates are resistant. [ 48 ] Organisms such as Escherichia coli are killed by molecular iodine but require iodine from host thyroid hormone, [ 28 ] indicating that not all organisms that need iodine are resistant to the toxic effects of pure iodine. [ 23 ]
The U.S. Institute of Medicine (IOM) updated Estimated Average Requirements (EARs) and Recommended Dietary Allowances (RDAs) for iodine in 2000. For people age 14 and up, the iodine RDA is 150 μg/day; the RDA for pregnant women is 220 μg/day and the RDA during lactation is 290 μg/day. For children aged 1–8 years, the RDA is 90 μg/day; for children aged 8–13 years, it is 130 μg/day. [ 49 ] As a safety consideration, the IOM sets tolerable upper intake levels (ULs) for vitamins and minerals when evidence is sufficient. The UL for iodine for adults is 1,100 μg/day. This UL was assessed by analyzing the effect of supplementation on thyroid-stimulating hormone . [ 8 ] Collectively, the EARs, RDAs, AIs and ULs are referred to as Dietary Reference Intakes (DRIs). [ 49 ]
The European Food Safety Authority (EFSA) refers to the collective set of information as Dietary Reference Values, with Population Reference Intake (PRI) instead of RDA, and Average Requirement instead of EAR; AI and UL are defined the same as in the United States. For women and men ages 18 and older, the PRI for iodine is set at 150 μg/day; the PRI during pregnancy or lactation is 200 μg/day. For children aged 1–17 years, the PRI increases with age from 90 to 130 μg/day. These PRIs are comparable to the U.S. RDAs with the exception of that for lactation. [ 50 ] The EFSA reviewed the same safety question and set its adult UL at 600 μg/day, which is a bit more than half the U.S. value. [ 51 ] Notably, Japan reduced its adult iodine UL from 3,000 to 2,200 μg/day in 2010, but then increased it back to 3,000 μg/day in 2015. [ 52 ]
As of 2000, the median observed intake of iodine from food in the United States was 240 to 300 μg/day for men and 190 to 210 μg/day for women. [ 49 ] In Japan, consumption is much higher due to the frequent consumption of seaweed or kombu kelp. [ 8 ] The average daily intake in Japan ranges from 1,000 to 3,000 μg/day; previous estimates suggested an average intake as high as 13,000 μg/day. [ 53 ]
For U.S. food and dietary supplement labeling purposes, the amount in a serving is expressed as a percent of Daily Value (%DV). For iodine specifically, 100% of the Daily Value is considered 150 μg, and this figure remained at 150 μg in the May 27, 2016 revision. [ 54 ] [ 55 ] A table of the old and new adult daily values is provided at Reference Daily Intake .
Natural sources of iodine include many marine organisms, such as kelp and certain seafood products, as well as plants grown on iodine-rich soil. [ 56 ] [ 57 ] Iodized salt is fortified with iodine. [ 57 ] According to a Food Fortification Initiative 2016 report, 130 countries have mandatory iodine fortification of salt and an additional 10 have voluntary fortification. [ citation needed ]
Worldwide, iodine deficiency affects two billion people and is the leading preventable cause of intellectual disability . [ 58 ] Mental disability is a result which occurs primarily when babies or small children are rendered hypothyroidic by a lack of dietary iodine (new hypothyroidism in adults may cause temporary mental slowing, but not permanent damage).
In areas where there is little iodine in the diet, typically remote inland areas and semi-arid equatorial climates where no marine foods are eaten, iodine deficiency also gives rise to hypothyroidism , the most serious symptoms of which are epidemic goitre (swelling of the thyroid gland), extreme fatigue, mental slowing, depression, weight gain, and low basal body temperatures. [ 59 ]
The addition of iodine to table salt (so-called iodized salt ) has largely eliminated the most severe consequences of iodine deficiency in wealthier nations, but deficiency remains a serious public health problem in the developing world. [ 60 ] Iodine deficiency is also a problem in certain areas of Europe; in Germany, an estimated one billion dollars in healthcare costs is spent each year in combating and treating iodine deficiency. [ 8 ]
Source: [ 61 ]
Elemental iodine is an oxidizing irritant, and direct contact with skin can cause lesions , so iodine crystals should be handled with care. Solutions with high elemental iodine concentration such as tincture of iodine are capable of causing tissue damage if use for cleaning and antisepsis is prolonged. Although elemental iodine is used in the formulation of Lugol's solution , a common medical disinfectant, it becomes triiodide upon reacting with the potassium iodide used in the solution and is therefore non-toxic. [ citation needed ] Only a small amount of elemental iodine will dissolve in water, but triiodides are highly soluble; potassium iodide thus serves as a phase transfer catalyst in the tincture. This allows Lugol's iodine to be produced in strengths varying from 2% to 15% iodine.
Elemental iodine (I 2 ) is poisonous if taken orally in large amounts; 2–3 grams is a lethal dose for an adult human. [ 71 ] [ 72 ]
Iodine vapor is very irritating to the eye , to mucous membranes, and in the respiratory tract. Concentration of iodine in the air should not exceed 1 mg/m 3 (eight-hour time-weighted average).
When mixed with ammonia and water, elemental iodine forms nitrogen triiodide , which is extremely shock-sensitive and can explode unexpectedly.
Compared to the elemental form, potassium iodide has a median lethal dose (LD 50 ) that is relatively high in several animals: in rabbits, it is 10 g/kg; in rats, 14 g/kg, and in mice, 22 g/kg. [ 73 ] The tolerable upper intake level for iodine as established by the Food and Nutrition Board is 1,100 μg/day for adults. The safe upper limit of consumption set by the Ministry of Health, Labor and Welfare in Japan is 3,000 μg/day. [ 74 ]
The biological half-life of iodine differs between the various organs of the body, from 100 days in the thyroid, to 14 days in the kidneys and spleen, to 7 days in the reproductive organs. Typically the daily urinary elimination rate ranges from 100 to 200 μg/L in humans. [ 75 ] However, the Japanese diet, high in iodine-rich kelp , contains 1,000 to 3,000 μg of iodine per day, and research indicates the body can readily eliminate excess iodine that is not needed for thyroid hormone production. [ 74 ] The literature reports as much as 30,000 μg/L (30 mg/L) of iodine being safely excreted in the urine in a single day, with levels returning to the standard range in a couple of days, depending on seaweed intake. [ 76 ] One study concluded the range of total body iodine content in males was 12.1 mg to 25.3 mg, with a mean of 14.6 mg. [ 77 ] It is presumed that once thyroid-stimulating hormone is suppressed, the body simply eliminates excess iodine, and as a result, long-term supplementation with high doses of iodine has no additional effect once the body is replete with enough iodine. It is unknown if the thyroid gland is the rate-limiting factor in generating thyroid hormone from iodine and tyrosine, but assuming it is not, a short-term loading dose of one or two weeks at the tolerable upper intake level may quickly restore thyroid function in iodine-deficient patients. [ citation needed ]
Excessive iodine intake presents symptoms similar to those of iodine deficiency. Commonly encountered symptoms are abnormal growth of the thyroid gland and disorders in functioning, [ 78 ] as well as in growth of the organism as a whole. Iodide toxicity is similar to (but not the same as) toxicity to ions of the other halogens , such as bromides or fluorides . Excess bromine and fluorine can prevent successful iodine uptake, storage and use in organisms, as both elements can selectively replace iodine biochemically.
Excess iodine may also be more cytotoxic in combination with selenium deficiency . [ 79 ] Iodine supplementation in selenium-deficient populations is theoretically problematic, partly for this reason. [ 8 ] Selenocysteine (abbreviated as Sec or U , in older publications also as Se-Cys ) [ 80 ] is the 21st proteinogenic amino acid , and is the root of iodide ion toxicity when there is a simultaneous insufficiency of biologically available selenium. Selenocysteine exists naturally in all kingdoms of life as a building block of selenoproteins . [ 81 ]
Some people develop a hypersensitivity to compounds of iodine but there are no known cases of people being directly allergic to elemental iodine itself. [ 82 ] Notable sensitivity reactions that have been observed in humans include:
Medical use of iodine compounds (i.e. as a contrast agent ) can cause anaphylactic shock in highly sensitive patients, presumably due to sensitivity to the chemical carrier. Cases of sensitivity to iodine compounds should not be formally classified as iodine allergies, as this perpetuates the erroneous belief that it is the iodine to which patients react, rather than to the specific allergen. Sensitivity to iodine-containing compounds is rare but has a considerable effect given the extremely widespread use of iodine-based contrast media ; however, the only adverse effect of contrast material that can convincingly be ascribed to free iodide is iodide mumps and other manifestations of iodism. [ 84 ] | https://en.wikipedia.org/wiki/Iodine_in_biology |
Iodine monobromide is an interhalogen compound with the formula IBr. It is a dark red solid that melts near room temperature. [ 1 ] Like iodine monochloride , IBr is used in some types of iodometry . It serves as a source of I + . Its Lewis acid properties are compared with those of ICl and I 2 in the ECW model . It can form CT adducts with Lewis donors. [ 2 ]
Iodine monobromide is formed when iodine and bromine are combined in a chemical reaction:. [ 3 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Iodine_monobromide |
Iodine monochloride is an interhalogen compound with the formula ICl . It is a red-brown chemical compound that melts near room temperature . Because of the difference in the electronegativity of iodine and chlorine , this molecule is highly polar and behaves as a source of I + . Discovered in 1814 by Gay-Lussac , iodine monochloride is the first interhalogen compound discovered. [ 1 ]
Iodine monochloride is produced simply by combining the halogens in a 1:1 molar ratio, according to the equation
When chlorine gas is passed through iodine crystals, one observes the brown vapor of iodine monochloride. Dark brown iodine monochloride liquid is collected. Excess chlorine converts iodine monochloride into iodine trichloride in a reversible reaction:
ICl has two polymorphs ; α-ICl, which exists as black needles (red by transmitted light) with a melting point of 27.2 °C, and β-ICl, which exists as black platelets (red-brown by transmitted light) with a melting point 13.9 °C. [ 2 ]
In the crystal structures of both polymorphs the molecules are arranged in zigzag chains. β-ICl is monoclinic with the space group P2 1 /c. [ 3 ]
Iodine monochloride is soluble in acids such as HF and HCl but reacts with pure water to form HCl, iodine, and iodic acid :
ICl is a useful reagent in organic synthesis . [ 2 ] It is used as a source of electrophilic iodine in the synthesis of certain aromatic iodides. [ 4 ] It also cleaves C–Si bonds.
ICl will also add to the double bond in alkenes to give chloro-iodo alkanes .
When such reactions are conducted in the presence of sodium azide , the iodo-azide RCH(I)–CH(N 3 )R′ is obtained. [ 5 ]
The Wijs solution, iodine monochloride dissolved in acetic acid, is used to determine the iodine value of a substance.
It can also be used to prepare iodates, by reaction with a chlorate. Chlorine is released as a byproduct.
Iodine monochloride is a Lewis acid that forms 1:1 adducts with Lewis bases such as dimethylacetamide and benzene . | https://en.wikipedia.org/wiki/Iodine_monochloride |
Iodine monofluoride is an interhalogen compound of iodine and fluorine with formula IF. It is a chocolate-brown solid that decomposes at 0 °C, [ 1 ] disproportionating to elemental iodine and iodine pentafluoride :
However, its molecular properties can still be precisely determined by spectroscopy : the iodine-fluorine distance is 190.9 pm and the I−F bond dissociation energy is around 277 kJ mol −1 . At 298 K , its standard enthalpy change of formation is Δ f H ° = −95.4 kJ mol −1 , and its Gibbs free energy is Δ f G ° = −117.6 kJ mol −1 .
It can be generated, albeit only fleetingly, by the reaction of the elements at −45 °C in CCl 3 F :
It can also be generated by the reaction of iodine with iodine trifluoride at −78 °C in CCl 3 F:
The reaction of iodine with silver(I) fluoride at 0 °C also yields iodine monofluoride:
Iodine monofluoride is used to produce pure nitrogen triiodide : [ 2 ] | https://en.wikipedia.org/wiki/Iodine_monofluoride |
Iodine monoxide is a binary inorganic compound of iodine and oxygen with the chemical formula IO•. A free radical , this compound is the simplest of many iodine oxides . [ 1 ] [ 2 ] [ 3 ] It is similar to the oxygen monofluoride , chlorine monoxide and bromine monoxide radicals.
Iodine monoxide can be obtained by the reaction between iodine and oxygen : [ 4 ]
Iodine monoxide decomposes to its prime elements: [ citation needed ]
Iodine monoxide reacts with nitric oxide: [ 5 ]
Atmospheric iodine atoms (e.g. from iodomethane ) can react with ozone to produce the iodine monoxide radical: [ 6 ] [ 5 ]
This process can contribute to ozone depletion . [ citation needed ] | https://en.wikipedia.org/wiki/Iodine_monoxide |
In chemistry , the iodine value ( IV ; also iodine absorption value , iodine number or iodine index ) is the mass of iodine in grams that is consumed by 100 grams of a chemical substance . Iodine numbers are often used to determine the degree of unsaturation in fats , oils and waxes . In fatty acids , unsaturation occurs mainly as double bonds which are very reactive towards halogens , the iodine in this case. Thus, the higher the iodine value, the more unsaturations are present in the fat. [ 1 ] It can be seen from the table that coconut oil is very saturated, which means it is good for making soap . On the other hand, linseed oil is highly unsaturated , which makes it a drying oil , well suited for making oil paints .
The determination of iodine value is a particular example of iodometry . A solution of iodine I 2 is yellow/brown in color. When this is added to a solution to be tested, however, any chemical group (usually in this test −C=C− double bonds) that react with iodine effectively reduce the strength, or magnitude of the color (by taking I 2 out of solution). Thus the amount of iodine required to make a solution retain the characteristic yellow/brown color can effectively be used to determine the amount of iodine sensitive groups present in the solution.
The chemical reaction associated with this method of analysis involves formation of the diiodo alkane (R and R' symbolize alkyl or other organic groups):
The precursor alkene ( RCH=CHR’ ) is colorless and so is the organoiodine product ( RCHI−CHIR’ ).
In a typical procedure, the fatty acid is treated with an excess of the Hanuš or Wijs solution , which are, respectively, solutions of iodine monobromide (IBr) and iodine monochloride (ICl) in glacial acetic acid . Unreacted iodine monobromide (or monochloride) is then allowed to react with potassium iodide , converting it to iodine I 2 , whose concentration can be determined by back-titration with sodium thiosulfate ( Na 2 S 2 O 3 ) standard solution. [ 2 ] [ 3 ]
The basic principle of iodine value was originally introduced in 1884 by A. V. Hübl as “ Jodzahl ”. He used iodine alcoholic solution in presence of mercuric chloride ( HgCl 2 ) and carbon tetrachloride ( CCl 4 ) as fat solubilizer. [Note 1] The residual iodine is titrated against sodium thiosulfate solution with starch used as endpoint indicator. [ 4 ] This method is now considered as obsolete.
J. J. A. Wijs modified the Hübl method by using iodine monochloride (ICl) in glacial acetic acid, which became known as Wijs's solution , dropping the HgCl 2 reagent. [ 4 ] Alternatively, J. Hanuš used iodine monobromide (IBr), which is more stable than ICl when protected from light. Typically, fat is dissolved in chloroform [Note 2] and treated with excess ICl/IBr. Some of the halogen reacts with the double bonds in the unsaturated fat while the rest remains.
Then, saturated solution of potassium iodide (KI) is added to this mixture, which reacts with remaining free ICl/IBr to form potassium chloride (KCl) and diiodide ( I 2 ).
Afterward, the liberated I 2 is titrated against sodium thiosulfate, in presence of starch, to indirectly determine the concentration of the reacted iodine. [ 5 ]
IV (g I/ 100 g) is calculated from the formula :
The determination of IV according to Wijs is the official method currently accepted by international standards such as DIN 53241-1:1995-05, AOCS Method Cd 1-25, EN 14111 and ISO 3961:2018. One of the major limitations of is that halogens does not react stoichiometrically with conjugated double bonds (particularly abundant in some drying oils ). Therefore, Rosenmund-Kuhnhenn method makes more accurate measurement in this situation. [ 6 ]
Proposed by H. P. Kaufmann in 1935, it consists in the bromination of the double bonds using an excess of bromine and anhydrous sodium bromide dissolved in methanol . The reaction involves the formation of a bromonium intermediate as follows: [ 7 ]
Then the unused bromine is reduced to bromide with iodide ( I − ).
Now, the amount of iodine formed is determined by back-titration with sodium thiosulfate solution.
The reactions must be carried out in the dark, since the formation of bromine radicals is stimulated by light. This would lead to undesirable side reactions, and thus falsifying a result consumption of bromine. [ 8 ]
For educational purposes, Simurdiak et al. (2016) [ 3 ] suggested the use of pyridinium tribromide as bromination reagent which is more safer in chemistry class and reduces drastically the reaction time.
This method is suitable for the determination of iodine value in conjugated systems ( ASTM D1541). It has been observed that Wijs/ Hanuš method gives erratic values of IV for some sterols (i.e. cholesterol ) and other unsaturated components of insaponifible fraction. [ 9 ] The original method uses pyridine dibromide sulfate solution as halogenating agent and an incubation time of 5 min. [ 10 ]
Measurement of iodine value with the official method is time-consuming (incubation time of 30 min with Wijs solution) and uses hazardous reagents and solvents. [ 3 ] Several non-wet methods have been proposed for determining the iodine value. For example, IV of pure fatty acids and acylglycerols can be theoretically calculated as follows: [ 11 ]
Accordingly, the IVs of oleic , linoleic , and linolenic acids are respectively 90, 181, and 273. Therefore, the IV of the mixture can be approximated by the following equation : [ 12 ]
For fats and oils, the IV of the mixture can be calculated from the fatty acid composition profile as determined by gas chromatography ( AOAC Cd 1c-85; ISO 3961:2018). However this formula does not take into consideration the olefinic substances in the unsaponifiable fraction . Therefore, this method is not applicable for fish oils as they may contain appreciable amounts of squalene . [ 13 ]
IV can be also predicted from near-infrared , FTIR and Raman spectroscopy data using the ratio between the intensities of ν (C=C) and ν (CH 2 ) bands. [ 14 ] [ 15 ] High resolution proton-NMR provides also fast and reasonably accurate estimation of this parameter. [ 16 ]
Although modern analytical methods (such as GC ) provides more detailed molecular information including unsaturation degree, the iodine value still widely considered as an important quality parameter for oils and fats. Moreover, IV generally indicates oxidative stability of the fats which directly depend on unsaturation amount. Such a parameter has a direct impact on the processing, the shelf-life and the suitable applications for fat-based products. It is also of a crucial interest for lubricants and fuel industries. In biodiesel specifications, the required limit for IV is 120 g I 2 /100 g, according to standard EN 14214 . [ 17 ]
IV is extensively used to monitor the industrial processes of hydrogenation and frying . However it must be completed by additional analyses as it does not differentiate cis / trans isomers.
G. Knothe (2002) [ 12 ] criticized the use of IV as oxidative stability specification for fats esterification products. He noticed that not only the number but the position of double bonds is involved in oxidation susceptibility. For instance, linolenic acid with two bis - allylic positions (at the carbons no. 11 and 14 between the double bonds Δ9, Δ12 and Δ15) is more prone to autoxidation than linoleic acid exhibiting one bis - allylic position (at C-11 between Δ9 and Δ12). Therefore, Knothe introduced alternative indices termed allylic position and bis -allylic position equivalents (APE and BAPE), which can be calculated directly from the integration resultas of chromatographic analysis.
Iodine value helps to classify oils according to the degree of unsaturation into drying oils , having IV > 150 (i.e. linseed , tung ), semi-drying oils IV : 125 – 150 ( soybean , sunflower ) and non-drying oils with IV < 125 ( canola , olive , coconut ). The IV ranges of several common oils and fats is provided by the table below. | https://en.wikipedia.org/wiki/Iodine_number |
Iodine pentafluoride is an interhalogen compound with chemical formula IF 5 . It is one of the fluorides of iodine . It is a colorless liquid, although impure samples appear yellow. It is used as a fluorination reagent and even a solvent in specialized syntheses. [ 3 ]
It was first synthesized by Henri Moissan in 1891 by burning solid iodine in fluorine gas. [ 4 ] This exothermic reaction is still used to produce iodine pentafluoride, although the reaction conditions have been improved. [ 5 ]
IF 5 reacts vigorously with water forming hydrofluoric acid and iodic acid :
Upon treatment with fluorine, it converts to iodine heptafluoride : [ 6 ]
It has been used as a solvent for handling metal fluorides. For example, the reduction of osmium hexafluoride to osmium pentafluoride with iodine is conducted in a solution in iodine pentafluoride: [ 7 ]
Primary amines react with iodine pentafluoride forming nitriles after hydrolysis . [ 8 ] | https://en.wikipedia.org/wiki/Iodine_pentafluoride |
Iodine pentoxide is the chemical compound with the formula I 2 O 5 . This iodine oxide is the anhydride of iodic acid , and one of the few iodine oxides that is stable. It is produced by dehydrating iodic acid at 200 °C in a stream of dry air: [ 1 ]
I 2 O 5 is bent with an I–O–I angle of 139.2°, but the molecule has no mirror plane so its symmetry is C 2 rather than C 2v . The terminal I–O distances are around 1.80 Å and the bridging I–O distances are around 1.95 Å. [ 3 ]
Iodine pentoxide easily oxidises carbon monoxide to carbon dioxide at room temperature:
This reaction can be used to analyze the concentration of CO in a gaseous sample.
I 2 O 5 forms iodyl salts, [IO 2 + ], with SO 3 and S 2 O 6 F 2 , but iodosyl salts, [IO + ], with concentrated sulfuric acid .
Iodine pentoxide decomposes to iodine (vapor) and oxygen when heated to about 350 °C. [ 4 ] | https://en.wikipedia.org/wiki/Iodine_pentoxide |
Iodine trichloride is an interhalogen compound of iodine and chlorine . It is bright yellow but upon time and exposure to light it turns red due to the presence of elemental iodine. In the solid state is present as a planar dimer I 2 Cl 6 , with two bridging Cl atoms. [ 1 ]
It can be prepared by reacting iodine with an excess of liquid chlorine at −70 °C, [ 2 ] or heating a mixture of liquid iodine and chlorine gas to 105 °C. [ citation needed ] In the molten state it is conductive, which may indicate dissociation: [ 2 ]
It is an oxidizing agent , capable of causing fire on contact with organic materials. [ citation needed ] That oxidizing power also makes it a useful catalyst for organic chlorination reactions . [ 3 ]
Iodine trichloride reacts with concentrated hydrochloric acid , forming tetrachloroiodic acid : [ 4 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Iodine_trichloride |
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