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The Taft equation is a linear free energy relationship (LFER) used in physical organic chemistry in the study of reaction mechanisms and in the development of quantitative structure–activity relationships for organic compounds . It was developed by Robert W. Taft in 1952 [ 2 ] [ 3 ] [ 4 ] as a modification to the Hammett equation . [ 5 ] While the Hammett equation accounts for how field , inductive , and resonance effects influence reaction rates, the Taft equation also describes the steric effects of a substituent . The Taft equation is written as:
where log k s k CH 3 {\displaystyle \log {\frac {k_{s}}{k_{{\ce {CH3}}}}}} is the ratio of the rate of the substituted reaction compared to the reference reaction, ρ* is the sensitivity factor for the reaction to polar effects , σ* is the polar substituent constant that describes the field and inductive effects of the substituent, δ is the sensitivity factor for the reaction to steric effects, and E s is the steric substituent constant.
Polar substituent constants describe the way a substituent will influence a reaction through polar (inductive, field, and resonance) effects. To determine σ * Taft studied the hydrolysis of methyl esters (RCOOMe). The use of ester hydrolysis rates to study polar effects was first suggested by Ingold in 1930. [ 6 ] The hydrolysis of esters can occur through either acid and base catalyzed mechanisms , both of which proceed through a tetrahedral intermediate . In the base catalyzed mechanism the reactant goes from a neutral species to negatively charged intermediate in the rate determining (slow) step , while in the acid catalyzed mechanism a positively charged reactant goes to a positively charged intermediate.
Due to the similar tetrahedral intermediates, Taft proposed that under identical conditions any steric factors should be nearly the same for the two mechanisms and therefore would not influence the ratio of the rates. However, because of the difference in charge buildup in the rate determining steps it was proposed that polar effects would only influence the reaction rate of the base catalyzed reaction since a new charge was formed. He defined the polar substituent constant σ* as:
where log(k s /k CH 3 ) B is the ratio of the rate of the base catalyzed reaction compared to the reference reaction, log(k s /k CH 3 ) A is ratio of a rate of the acid catalyzed reaction compared to the reference reaction, and ρ* is a reaction constant that describes the sensitivity of the reaction series. For the definition reaction series, ρ* was set to 1 and R = methyl was defined as the reference reaction (σ* = zero). The factor of 1/2.48 is included to make σ* similar in magnitude to the Hammett σ values .
Although the acid catalyzed and base catalyzed hydrolysis of esters gives transition states for the rate determining steps that have differing charge densities , their structures differ only by two hydrogen atoms. Taft thus assumed that steric effects would influence both reaction mechanisms equally. Due to this, the steric substituent constant E s was determined from solely the acid catalyzed reaction, as this would not include polar effects. E s was defined as:
where k s is the rate of the studied reaction and k CH 3 {\displaystyle {\ce {{\mathit {k}}_{CH3}}}} is the rate of the reference reaction (R = methyl). δ is a reaction constant that describes the susceptibility of a reaction series to steric effects. For the definition reaction series δ was set to 1 and E s for the reference reaction was set to zero. This equation is combined with the equation for σ* to give the full Taft equation.
From comparing the E s values for methyl, ethyl , isopropyl , and tert-butyl , it is seen that the value increases with increasing steric bulk. However, because context will have an effect on steric interactions [ 7 ] some E s values can be larger or smaller than expected. For example, the value for phenyl is much larger than that for tert -butyl. When comparing these groups using another measure of steric bulk, axial strain values , the tert -butyl group is larger. [ 8 ]
In addition to Taft's steric parameter E s , other steric parameters that are independent of kinetic data have been defined. Charton has defined values v that are derived from van der Waals radii . [ 9 ] [ 10 ] Using molecular mechanics , Meyers has defined V a values that are derived from the volume of the portion of the substituent that is within 0.3 nm of the reaction center. [ 11 ]
Similar to ρ values for Hammett plots, the polar sensitivity factor ρ* for Taft plots will describe the susceptibility of a reaction series to polar effects. When the steric effects of substituents do not significantly influence the reaction rate the Taft equation simplifies to a form of the Hammett equation:
The polar sensitivity factor ρ* can be obtained by plotting the ratio of the measured reaction rates ( k s ) compared to the reference reaction ( k CH 3 {\displaystyle {\ce {{\mathit {k}}_{CH3}}}} ) versus the σ* values for the substituents. This plot will give a straight line with a slope equal to ρ*. Similar to the Hammett ρ value:
Similar to the polar sensitivity factor, the steric sensitivity factor δ for a new reaction series will describe to what magnitude the reaction rate is influenced by steric effects. When a reaction series is not significantly influenced by polar effects, the Taft equation reduces to:
A plot of the ratio of the rates versus the E s value for the substituent will give a straight line with a slope equal to δ. Similarly to the Hammett ρ value, the magnitude of δ will reflect to what extent a reaction is influenced by steric effects:
Since E s values are large and negative for bulkier substituents, it follows that:
When both steric and polar effects influence the reaction rate the Taft equation can be solved for both ρ* and δ through the use of standard least squares methods for determining a bivariant regression plane . Taft outlined the application of this method to solving the Taft equation in a 1957 paper. [ 12 ]
The Taft equation is often employed in biological chemistry and medicinal chemistry for the development of quantitative structure–activity relationships (QSARs). In a recent example, Sandri and co-workers [ 13 ] have used Taft plots in studies of polar effects in the aminolysis of β-lactams . They have looked at the binding of β-lactams to a poly(ethyleneimine) polymer , which functions as a simple mimic for human serum albumin (HSA). The formation of a covalent bond between penicillins and HSA as a result of aminolysis with lysine residues is believed to be involved in penicillin allergies . As a part of their mechanistic studies Sandri and co-workers plotted the rate of aminolysis versus calculated σ* values for 6 penicillins and found no correlation, suggesting that the rate is influenced by other effects in addition to polar and steric effects. | https://en.wikipedia.org/wiki/Taft_equation |
The Tage Erlander Prize ( Tage Erlanders pris ) is a prize awarded by the Royal Swedish Academy of Science "for research in Natural Sciences and Technology"
in four fields (Physics, Chemistry, Technology and Biology). The prize is awarded on a rolling schedule: every year the prize is awarded for research in one of these fields.
The prize commemorates Tage Erlander , who was the prime minister of Sweden from 1946 to 1969.
Its 2024 recipient is mathematician Lilian Matthiesen . [ 1 ]
This Sweden -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tage_Erlander_Prize |
Tagged queuing is a method for allowing a hardware device or controller to process commands received from a device driver out of order. It requires that the device driver attaches a tag to each command which the controller or device can later use to identify the response to the command.
Tagged queueing can speed up processing considerably if a controller serves devices of very different speeds, such as an SCSI controller serving a mix of CD-ROM drives and high-speed disks . In such cases if a request to fetch data from the CD-ROM is shortly followed by a request to read from the disk, the controller doesn't have to wait for the CD-ROM to fetch the data, it can instead instruct the disk to fetch the data and return the value to the device driver, while the CD-ROM is probably still seeking. [ 1 ]
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tagged_queuing |
Tagging of postage stamps means that the stamps are printed on luminescent paper or with luminescent ink to facilitate automated mail processing . Both fluorescence and phosphorescence are used. The same stamp may have been printed with and without these luminescent features. The two varieties are referred to as tagged and untagged , respectively.
Letters and postcards fed into an automated mail processing plant are illuminated with ultraviolet light . The reaction of the luminescent features of the stamps on this illumination is used to position the mail items such that the stamps can be cancelled , and that the significant parts of the address such as postcodes may be read and the mail be sorted accordingly.
The luminescent features of the stamps are generally invisible or barely visible to the human eye in normal illumination. However, they can be identified under ultraviolet light similar to how it is done in the postal machinery. In general, fluorescent features can be identified with UV light of a longer wavelength than needed for phosphorescent features (see below).
The luminescent substance ("taggant") can be printed over the whole surface of the stamp, the main design, the margins only, single bands or bars or other patterns, or can be added to the paper itself.
The tagging pattern can also be varied to enable the sorting of mail according to the service class.
Upon absorption of light, fluorescent materials emit light upon of a longer wavelength (lower energy) than the absorbed radiation, but cease to do so once immediately, when the illumination is stopped. The tagging of stamps uses substances that absorb ultraviolet light of wavelengths between 300 nm and 450 nm (" Black light ", UVA, long-wave UV) and emit light in the visible spectrum. Under UV illumination they usually glow a greenish or yellowish colour.
It must not be confused with the "whitening" of paper. [ 1 ] It is achieved by adding optical brighteners that usually re-emit light in the blue region of the spectrum, making the paper appear whiter by compensating a perceived deficit in reflected colours of these wavelengths.
Phosphorescent materials release the absorbed energy only slowly, so that they exhibit an "afterglow". Materials for stamp tagging absorb ultraviolet light of wavelengths between 180 nm and 300 nm (UVC, short-wave UV) and emit light of a greenish or reddish colour depending on the substances used.
Fluorescent stamps can be detected with a black light fluorescent tube . Phosphorescent stamps can be detected using a shortwave UV lamp. The effects of both processes can be recorded photographically. Lamps for both ranges of wavelengths as well as combinations of both are available. Care must be taken when using UV lamps, since their light can damage the eyes. [ 2 ]
The first tagged stamps of Canada were issued in 1962 with vertical phosphorescent bands. In 1972, fluorescent general tagging was introduced, initially as vertical bars, now normally on all four sides of the stamp. [ 3 ]
Deutsche Bundespost started issuing stamps on fluorescent Lumogen paper in 1960 in connection with trials for automated mail processing in the Darmstadt area. Fluorescent paper was generally used for stamps of Deutsche Bundespost and Deutsche Bundespost Berlin from 1961 on. [ 4 ] Deutsche Post AG continues to use this technology. Deutsche Post of the GDR did not use luminescent tagging on stamps.
Luminescent tagging has been added to postage stamps of the United Kingdom since the Wilding issues of 1959 in the shape of vertical bands. Stamps of the current Machin series have been printed with one or two such "phosphor bands"; [ 5 ] those for second-class mail bear only one such band, those for first-class mail bear two. The positions of the bands may vary: stamps from booklets may have shortened, notched, or inset bands that do not extend onto neighbouring gutters to avoid the use of the latter instead of stamps for franking. [ 6 ] Due to the presence of optical brighteners in many printing papers, phosphorescent materials were chosen for stamp tagging in the UK. [ 7 ]
The US Post Office Department started experiments with fluorescent compounds in the early 1960s. An 8¢ air mail stamp issued in 1963 was the first stamp printed for trials with new cancelling machines. [ 8 ] The 5¢ City Delivery issue of 1963 was the first commemorative issue produced with tagging. [ 9 ]
Precancelled stamps and service-inscribed stamps are not usually tagged because they need not be routed through the cancelling equipment.
Since luminescent ink or luminescent paper are only delivered to specialist printers, tagging also serves as an anti-counterfeiting measure, similar to the practice on banknotes .
When Deutsche Post of the GDR expanded automated mail processing in the 1980s, they did not use luminescent tagging, but used sideways illumination to identify the shadows of the stamp perforation in order to position mail items in cancelling and sorting machinery. Red light was used for this purpose, giving a good contrast to ordinary writing ink colours and enabling machine reading of postcodes. Some issues of Postal cards were printed entirely in orange to facilitate the latter process. However, the colours of the imprinted stamps was later changed to those of the usual definitives of the corresponding value, and simulated perforations were added around the stamp design to help locate the stamp position. | https://en.wikipedia.org/wiki/Tagging_(stamp) |
Tagging of Pacific Predators ( TOPP ) began in 2000 as one of many projects formed by Census of Marine Life , [ 1 ] [ 2 ] an organization whose goal is to help understand and explain the diversity and abundances of the ocean in the past, present, and future. [ 2 ] After they were formed, TOPP began by building a coalition of researchers from all over the world to find and study predators of the Pacific Ocean. [ 3 ] Since then, they have satellite-tagged 22 different species and more than 2,000 animals. [ 4 ] These animals include elephant seals , great white sharks , leatherback turtles , squid , albatrosses , and more. [ 5 ] [ 6 ]
Through the efforts of TOPP, information never before accessed by humans was now available, such as migration routes and ecosystems, [ 1 ] [ 7 ] but from the animals', rather than human, aspects. [ 8 ] It also became possible to learn about the ocean itself through use of the animals, because they can go where humans cannot. We learn through their everyday actions, and through these data, researchers have been able to determine better ways of protecting endangered species, such as the leatherback turtle. [ 1 ] [ 9 ] [ 10 ]
The TOPP program ended in 2010, but the research team ceased work in 2016.
In addition to the NOAA's Pacific Fisheries Ecosystems Lab , Stanford's Hopkins Marine Lab , and University of California, Santa Cruz 's Long Marine Laboratory, [ 1 ] [ 11 ] a wide range of people contribute to the success of the program, including marine biologists , oceanographers , engineers, computer programmers, journalists, graphic designers, educators and the public who provide support. [ 1 ]
Many different types of tags are used in the TOPP tagging program, each designed for different marine animals and different types of data. [ 12 ] Archival tags, though small, are very powerful, and can last up to 10 years. [ 1 ] Researchers surgically implant them into the bellies of tuna, where the tags record, as often as every few seconds, pressure (for depth of dives), ambient light (to estimate location), internal and external body temperature, and, in some cases, speed of travel. [ 1 ] The tags are small and light enough to be attached to the outside of an animal, such as the tail feathers of red-footed boobies. However, they do have a drawback, they have to be retrieved. So, they are useful for fish likely to be caught as seafood, such as bluefin or yellowfin tuna, [ 13 ] or animals that return to rookeries or nesting beaches, such as boobies and leatherback turtles. [ 1 ] [ 14 ]
Pop-up satellite archival tags (PSATs), also known simply as pop-up archival tags (PATs), are just like archival tags, except they are designed to release at a preset time, like 30 days. They then float to the surface and send their data via an Argos satellite back to the laboratory for two weeks, which is the life of its battery. [ 1 ] Even when the battery dies, the data are saved on the tag, so if it is ever recovered, the whole data set can be downloaded. This tag is useful for animals that do not spend a lot of time at the surface, and are not caught often. [ 1 ] Numerous white sharks have been tagged with this tag. [ 15 ] [ 16 ] [ 17 ] [ 18 ] The tags are attached to white sharks by inserting a small surgical titanium anchor into it. [ 19 ] On elephant seals, the tag is glued to the fur. [ 20 ] Connecting the tag to the anchor is a thin line that loops around a metal pin at the base of the tag. [ 1 ] This metal pin is connected to a battery. A clock in the tag turns the battery on at a preprogrammed time. When the battery turns on, the attachment pin dissolves. The tag floats to the surface and starts transmitting data to one of the Argos satellites. [ 1 ]
Spot tags, or smart positioning, or temperature transmitting tags, are ideal tags for air-breathing marine animals (seals, whales and sea turtles) and animals that often swim close to the surface (salmon sharks, blue sharks and makos). [ 1 ] When the antenna breaks the surface, it sends data to a satellite. The data include pressure, speed, and water temperature. Location is estimated by calculating the Doppler shift in the transmission signal in successive transmissions. [ 1 ] When the animal goes beneath the surface, a saltwater switch turns off the tag. The tag, made by Wildlife Computers, lasts about two years. [ 21 ]
Satellite relay data logger (SRDL) tags compress data so more information can be transmitted through the Argos satellite. [ 1 ] These can be outfitted with CTD tags to record the salinity, temperature and depth data oceanographers need to identify ocean currents and water. [ 1 ] Elephant seals, sea lions and leatherback sea turtles wear these tags.
TOPP has also set up programs to engage the public in learning about marine science and marine conservation. One of its programs is called Elephant Seal Homecoming Days, which was started by TOPP in 2008 for the months the northern elephant seals return to Año Nuevo State Reserve during the breeding season. TOPP picks around 10 of the many they tagged to become "spokes-seals" for the public, allowing them to see what it is like being one of the biggest seals in the world. [ 22 ] Elephant seals are prime candidates for tagging because they tend to return to the same beach every year to breed, yielding high tag recovery. [ 23 ] They can gather immense amounts of information because they can swim for thousands of miles. [ 24 ] [ 25 ] The seals are issued their own Facebook profile that is kept up for them by undergraduate volunteer keepers. [ 26 ] Facebook profiles keep the public updated on their progress while they give birth and proceed to take care of their pups. Two seals are the stars of TOPP Elephant Seal Homecoming Days. The first is Penelope Seal, who has been part of the program since 2008. The second one is new as of 2009, and his name is Stelephant Colbert (in reference to Stephen Colbert of The Colbert Report ). [ 27 ]
Stelephant started out as a pup in a small harem of northern elephant seals located at Año Nuevo State Reserve. [ 28 ] He grew into one of the most famous elephant seals ever, even appearing on The Colbert Report . At around eight years old, Stelephant weighed 4,500-pound (2,000 kg). He gets by on his aggressiveness and determination to get as close to females as possible. Almost any male is turned away unless they wish to fight. [ 29 ] These fights rarely last long, however, because Stelephant is a seasoned fighter.
After Stelephant Colbert was weaned, he spent the next few weeks on the beach waiting to join adult seals out in the northern Pacific Ocean. He returned the next year to the same beach on the California coast as a yearling, where he was exiled to the outskirts of the harems with the other yearlings. The next few years were spent building muscle and growing large to be able to compete with the larger males. During his third to fifth years, Stelephant practiced fighting with other young males to build strength; while he did compete with the adults, he spent most of his days far away from females. [ 30 ] When he was around seven years old with his proboscis finally grown in, he was ready to compete with the adult males for the alpha position.
Stelephant has been involved with the University of California, Santa Cruz Long Marine Lab researchers, and was tagged last spring to track his foraging habits while swimming and diving off the Aleutian Islands in the northern Pacific Ocean.
Stelephant was issued his own Facebook page, where people can keep track of his whereabouts and his status updates. [ 31 ] [ 32 ] He is the most talked-about seal in the program in that he was featured in the Santa Cruz Sentinel, the Associated Press, and many additional media organizations. [ 33 ] [ 34 ] [ 35 ] The biggest media event was when Stephen Colbert of The Colbert Report , after whom Stelephant is named, spoke of his seal on his show. [ 36 ] [ 37 ] [ 38 ] Stelephant was featured in one of the show's episodes, creating a media buzz. [ 39 ] Since then, Stelephant's fan base has skyrocketed bringing attention to TOPP and the UCSC Long Marine Lab Stelephant is associated with. [ 40 ] [ 41 ] Stelephant allowed TOPP to engage the public in learning about northern elephant seals, marine science, and marine conservation.
Stelephant is now an alpha male at Año Nuevo State Reserve, and will soon return to the ocean to forage once again. Stelephant was featured on Oceans Google Earth, which allows the user to explore the oceans. [ 42 ] [ 43 ] His tags were recovered in January and he was not tagged again.
Penelope Seal is an elephant seal and one of the many marine animals tagged as a part of the TOPP program.
Penelope was born as a "little" 90 lb (41 kg) pup in early January 1998 at the Año Nuevo State Reserve on the California coast. [ 44 ] Since then, she has become a 1,500 lb (680 kg) full grown female northern elephant seal with six pups of her own, and many fans who have contributed to her fame. [ 45 ] From five years old, she has had one pup a year, all at Año Nuevo State Reserve. Penelope Seal is known as gnarly seal and a survivor, considering 50% of her species will die before they reach maturity. [ 46 ] She has dark-brown fur that she molts (or sheds) once a year. The beach is a known hangout spot of hers and one can usually spot her basking in the sun, or if its raining, cuddling up to another elephant seal. She has had multiple relationships, never known to be a monogamous seal, and in fact living in a polygynous society. The group, or harem, is made up of many other females and an alpha male with which Penelope has been known to hang out. [ 45 ] However, Penelope has also been seen socializing with some other beta males around the harem, much to the dismay of the alpha.
Penelope spent her early life as a weaner , or weaned pup, at Año Nuevo State Reserve. During this time she fasted on the beach waiting until she was brave enough to go out into the open ocean for the first time. The next year she became a yearling and spent her days being shunned from her harem, since she was still small compared to the others. Years two and three were spent much the same way. During this period she learned a lot about how to become a successful predator, like being able to travel all the way out into the northern Pacific Ocean and back again without any maps, and making it to the same place every time, which can be around an 8,000 mi (13,000 km) journey. However, when she was four years old, she became pregnant with her first pup and gave birth to him 10 months later at Año Nuevo State Reserve.
Penelope is tagged with a small tag and collects data for the TOPP team's researchers. Penelope's tag has a time depth recorder, [ 47 ] which documents the dive depth, dive duration, and light levels. [ 48 ] Elephant seals are incredible divers and allow scientists to learn about their unique behavior and the environment in which they live. Elephant seals are prime candidates for tagging because they tend to return to the same beach every year to breed, yielding high tag recovery. [ 23 ] They can gather immense amounts of information because they can swim for thousands of miles. [ 24 ] [ 25 ] Penelope herself has been involved with the University of California, Santa Cruz Long Marine Lab researchers and was tagged last year.
Penelope has since had her seventh pup, and is living on the beaches of Año Nuevo State Reserve. She was not tagged this year. Stelephant Colbert has since taken on the challenge of promoting elephant seals, and has been quite successful. [ 49 ]
Penelope is featured on the new version of Ocean Google Earth , which allows the user to explore the oceans. Penelope can be located at Año Nuevo State Reserve. [ 50 ] [ 51 ] [ 52 ] [ 53 ]
Penelope Seal has been the star of Elephant Seal Homecoming Days for the past two years. She enabled TOPP to engage the public in learning about northern elephant seals, marine science, and marine conservation.
The Great Turtle Race is an international sea turtle conservation event that brings together corporate sponsors and conservation organizations. The race tracks sea turtles as they move toward feeding areas south of the Galapagos Islands after nesting at Playa Grande in Costa Rica's Las Baulas National Park, the primary nesting area for leatherbacks in the Pacific.
TOPP is a cosponsor of The Great Turtle Race, and tracking technology created and used by TOPP is how the turtles are tracked. The race occurred in 2007 and 2008 in April. [ 54 ] | https://en.wikipedia.org/wiki/Tagging_of_Pacific_Predators |
The Taiga of North America is a Level I ecoregion of North America designated by the Commission for Environmental Cooperation (CEC) in its North American Environmental Atlas .
The taiga ecoregion includes much of interior Alaska as well as the Yukon forested area, and extends on the west from the Bering Sea to the Richardson Mountains in on the east, with the Brooks Range on the north and the Alaska Range on the south end. It is a region with a vast mosaic of habitats and a fragile yet extensive patchwork of ecological characteristics. All aspects of the region such as soils and plant species, hydrology, and climate interaction, and are affected by climate change, new emerging natural resources, and other environmental threats such as deforestation. These threats alter the biotic and abiotic components of the region, which lead to further degradation and to various endangered species.
The main type of soil in the taiga is Spodosol. These soils contain a Spodic horizon, a sandy layer of soil that has high accumulations of iron and aluminum oxides, which lays underneath a leached A horizon. The color contrast between the Spodic horizon and the overlying horizon is very easy to identify. The color change is the result of the migration of iron and aluminum oxides from small, but consistent amounts of rainfall from the top horizon to the lower horizon of the soil.
The decomposition of organic matter is very slow in the taiga because of the cold climate and low moisture. With the slow decomposition of organic matter, nutrient cycling is very slow and the nutrient level of the soil is also very low. The soils in the taiga are quite acidic as well. A relatively small amount of rainfall coupled with the slow decomposition of organic material allows the acidic plant debris to sit and saturate the top horizons of the soil profile.
As a result of the infertile soil, only a few plant species can really thrive in the taiga. The common plant species in the taiga are coniferous trees. Not only do conifer trees thrive in acidic soils, they actually make the soil more acidic. Acidic leaflitter (or needles) from conifers falls to the forest floor and the precipitation leaches the acids down into the soil. Other species that can tolerate the acidic soils of the taiga are lichens and mosses, yellow nutsedge, and water horsetail. The depth to bedrock has an effect on the plants that grow well in the taiga as well. A shallow depth to bedrock forces the plants to have shallow roots, limiting overall stability and water uptake.
Beaver , Canadian lynx , bobcat , wolverine , and snowshoe hare are all keystone species in the taiga area. These species are keystone because they have learned to adapt to the cold climate of the area and are able to survive year-round.
These species survive year-round in taiga by changing fur color and growing extra fur. They have adapted to use each other to survive too. All of the predators depend on the snowshoe hare at some point during the year. All of the species also depend on forests in the area for shelter.
The taiga is inhabited by many species, some of which are endangered, and include the Canadian lynx, gray wolf, and grizzly bear. The Canadian lynx is one well-known animal to inhabit the North American taiga region and is listed as threatened in the U.S. The mother lynx will have a litter of about 4 kittens in the spring. Following the birth, the female is the sole caretaker, nursing them for about 5 months and teaching them to hunt. They will stay with her until the next breeding season. According to the USDS Forest Service, protection for the lynx has increased since 2000, which marks the date it became protected under the Endangered Species Act. Since much of the lynx's habitat is land managed by the agency, efforts to maintain and increase the habitat for the Canadian lynx using forest management plans are underway.
The taiga region is also interspersed with various plant species. The endangered or threatened species include Labrador tea, lady's slipper orchid, helleborine orchid, longleaf pine, lingonberry plant, Newfoundland pine marten, Methuselahs beard, lodgepole pine, and Scots pine. The life history of longleaf pine is a tree species that has been around for quite some time and can reach more than 250 years in age. To begin the tree's life, a seed falls from the parent in October to late November awaiting water to begin germination in a few weeks. Those individuals that make it, will enter what is known as the grass stage. During this stage, the roots are established, and the bud of the tree is protected from fire. Years later, the longleaf will reach about 6–10 feet (1.8–3.0 m) in height and the diameter will increase with time. Somewhere around 30 years after the trees will begin to produce cones with fertile seeds and average about 110 feet (34 m) at maturity. One recent study discusses the effects of logging in the 1950s on pine species. Since then, conservation efforts have increased the number of pine (and other) tree species. The Nature Conservancy is prioritizing its protection efforts to rebuild long-leaf pine forests through land purchases, conservation easements, and management of land sites. Restoration is also a large part of efforts to ensure the long-leaf pine remains extant. By planting seedlings, controlling competitive vegetation, and controlling burning methods, scientists and volunteers are working to increase the number of long-leaf pine.
Watersheds characterize much of the taiga ecoregion as interconnecting rivers, streams, lakes, and coastlines. Due to a cool climate, low evaporation levels keep moisture levels high and enable water to have serious influences on ecosystems. The vast majority of water in the taiga is freshwater, occupying lakes and rivers.
Many watersheds are dominated by large rivers that dump huge amounts of freshwater into the ocean such as the Lena river in Central Siberia. This exportation of freshwater helps control the thermohaline circulation and the global climate. Flow rates of taiga rivers are variable and "flashy" due to the presence of permafrost that keeps water from percolating deep into the soil. Due to global warming, flow rates have increased as more of the permafrost melts every year. In addition to "flashy" flow levels, the permafrost in the taiga allows dissolved inorganic nitrogen and organic carbon levels in the water to be higher while calcium, magnesium, sulfate, and hydrogen bicarbonate levels are shown to be much lower. As a dominant characteristic in the soil, permafrost also influences the degree to which water percolates into the soil. Where there is year-long permafrost, the water table is located much deeper in the soil and is less available to organisms, while discontinuous permafrost provides much shallower access.
Lakes that cover the taiga are characteristically formed by receding glaciers and therefore have many unique features. The vast majority of lakes and ponds in the taiga ecoregion are oligotrophic and have much higher levels of allochthonous versus autochthonous matter. This is due to glacier formation and has implications for how trophic levels interact with limiting nutrients. These oligotrophic lakes show organic nitrogen and carbon as more limiting nutrients for trophic growth over phosphorus. This contrasts sharply with mesotrophic or eutrophic lakes from similar climates.
When we [ who? ] look at the climate of the taiga, we [ who? ] are looking at average temperatures, abiotic factors such as precipitation, and circulatory patterns. According to the study in Global Change Biology, the average yearly temperatures across the Alaskan and Canadian taiga ranged from −26.6 °C to 4.8 °C. This indicates the extremely cold weather the taiga has for the majority of the year. As for precipitation, the majority of it is snow, but rain is also an important factor. According to The International Journal of Climatology , precipitation in the form of rain ranged from 40 mm average in August, to 15 mm average in April over a multi-year study. Rain is not the only kind of precipitation that affects the taiga; the main factor in precipitation is usually snow. According to CEC Ecological Regions of North America, snow and freshwater ice can occupy the taiga for half to three-quarters of the year. A CEC Ecological Regions of North America document states that the lowest average precipitation is on the western side of taiga; can be as little as 200 mm and on the east coast it can be as high as exceeding 1,000 mm. As for circulatory patterns, we're [ who? ] finding that the temperature increases have led to a season shift. Global Change Biology also has noted with the change in temperature over time, as well as the overall climate change, the growing season has lengthened. Their findings illustrate that the growing season has grown 2.66 days per ten years. This growing season change as a result of global warming is having an extreme effect on the taiga.
Climate change has played its role in threatening the taiga ecoregion. Equally as harmful are the human effects like deforestation, however, many associations and regulations are working to protect the taiga and reverse the damage. Climate change is resulting in rising temperatures and decreases in moisture, which causes parasites and other insects to be more active thus causing tree stress and death. Thawing permafrost has led to many forests experiencing less stability and they become "drunken forests" (the decrease in soil stability causes the trees to lean or fall over). Increased tree death then leads to a carbon dioxide outflux, thus further propagating the increases in global warming. It is essential for climate change to be combated with global action, which is what the Kyoto Protocol in 1997 was created to do. Other measures to protect the taiga would be to prohibit unsustainable deforestation, switch to renewable energy, and protect old-growth forests, (they sequester the most carbon dioxide). The taiga also suffers from more direct human effects such as logging and mining sites. Logging has been a very profitable business in the region, however, fragmentation of forests leads to loss of habitats, relocation of keystone species, increases in erosion, increases in magnitude and frequency of flooding, and altered soil composition. Regions in which permafrost has thawed and trees have fallen take centuries to recover. Canadian and Russian governments enacted a Protection Belt, which covers 21.1 million ha, and initiatives like the Far East Association for the use of non-timber forest products, give economic significance to the forests while avoiding logging. In addition to logging, studies have measured over 99,300 tones of airborne pollutants from just one metal-extracting plant over a 50-year span. These pollutants are 90% sulfur dioxide, which is a precursor to acid rain. Other emissions include nitrogen oxides, sulfurous anhydrides, and inorganic dust. Forests in a 50 kilometres (31 mi) radius of these sites can serve little to no biological services once affected, and there has been the little appearance of protection measures to regulate mining plants.
Over the next 100 years, global annual mean temperatures are expected to rise by 1.4−5.8 °C, but changes in high latitudes where the boreal biome exists will be much more extreme (perhaps as much as a 10 °C rise). The warming observed at high latitudes over the past 50 years exceeds the global average by as much as a factor of 5 (2–3 °C in Alaska versus the 0.53° global mean).
The effects of increased temperature on boreal forest growth have varied, often depending on tree species, site type, and region, as well as whether or not the warming is accompanied by increases or decreases in precipitation. However, studies of tree rings from all parts of the boreal zone have indicated an inverse growth response to temperature, likely as a result of direct temperature and drought stress. As global warming increases, negative effects on growth are likely to become more widespread as ecosystems and species will be unable to adapt to increasingly extreme environmental conditions.
Perhaps the most significant effect of climate change on the boreal region is the increase in the severity of disturbance regimes, particularly fire and insect outbreaks. Fire is the dominant type of disturbance in boreal North America, but the past 30-plus years have seen a gradual increase in fire frequency and severity as a result of warmer and drier conditions. From the 1960s to the 1990s, the annual area burned increased from an average of 1.4 to 3.1 million hectares per year. Insect outbreaks also represent an increasingly significant threat. Historically, temperatures have been low enough in the wintertime to control insect populations, but under global warming, many insects are surviving and reproducing during the winter months, causing severe damage to forests across the North American boreal. The main culprits are the mountain pine beetle in the western provinces of British Columbia and Alberta, and the spruce bark beetle in Alaska.
Taiga (boreal forests) has amazing natural resources that are being exploited by humans. Human activities have a huge effect on the taiga ecoregions mainly through extensive logging, natural gas extraction, and mine-fracking. This results in the loss of habitat and increases the rate of deforestation. It is important to use natural resources but its key to use natural resources sustainably and not over-exploit them. In recent years rules and regulations have been set in place to conserve the forests to reduce the number of trees that are cut. There has been an increase in oil extraction and mining throughout the United States and Canada. Exploitation of tar sands oil reserves has increased mining. This is a large operation that started in Alberta Canada. Oil extraction has a direct effect on the taiga forests because the most valuable and abundant oil resources come from taiga forests. Tar sands have affected over 75% of the habitat in the Alberta taiga forest due to the clearing of the forests and the oil ponds that come from the extraction. These tar sands also create awful toxic oil ponds that affect wildlife and surrounding vegetation. Oil extraction also affects the forest soil, which harms tree and plant growth.
Today, the world population has an increasingly high ecological footprint and a large part of that has to do with the population's carbon footprint. As a result of that, oil supplies have increased, which has spread across the U.S. and into other countries. This is detrimental to natural ecosystems. Taiga is the largest region and is seeing major consequences of our actions on extracting oil and natural gas. This is also causing climate change temperatures to increase at a rapid rate, which is affecting wildlife and forests. However, even though Human activities are responsible for the exploitation of these natural resources humans are the solution and have the tools to fix this issue. It is crucial that humans reduce the consumption rate of these natural resources to increase environmental conditions. | https://en.wikipedia.org/wiki/Taiga_of_North_America |
The Taiji program is a proposed Chinese satellite -based gravitational-wave observatory . [ 1 ] [ 2 ] It is scheduled for launch in 2033 [ 3 ] to study ripples in spacetime caused by gravitational waves . The program consists of a triangle of three spacecraft orbiting the Sun linked by laser interferometers .
There are two alternative plans for Taiji. One is to take a 20 percent share of the European Space Agency's LISA project; the other is to launch China's own satellites by 2033 to authenticate the ASE project. [ 4 ] Like LISA, the Taiji spacecraft would be 3 million kilometers apart, making them sensitive to as similar range of frequencies, [ 5 ] [ 6 ] although Taiji is proposed to perform better in some of that range. [ 7 ]
'Taiji Program' is the ELISA Program proposed by ESA , and the predecessor of the ELISA Program is the LISA Program cooperated by ESA and NASA . Similar to the configuration of the three networking satellites in the LISA Program, the three satellites in the Taiji Program also rotate around their centroid. The centroid also revolves in orbit around the Sun. The difference is that the phases of the LISA system, Earth system and Taiji system are different. With the Earth as the reference, the phase of the LISA system is 20 degrees behind that of the Programet, and the phase of the Taiji system is 20 degrees ahead of that of the Earth. [ 8 ] In addition, the Tai Chi Program is part of the proposed space-based gravitational wave observatories Program, the other parts of which are the Chinese Academy of Sciences ( CAS ) Tianqin Program and the European Space Agency (ESA) Laser Interferometer Space Antenna (LISA) and the Decimal Hertz Interferometer Gravitational-Wave Observatory ( DECIGO ) led by the Japan Aerospace Exploration Agency ( JAXA ). [ 9 ] In December 2021, a study pointed out that the gravitational wave detection network combined with Taiji and LISA will accurately measure the Hubble constant greater than 95.5% within ten years. [ 10 ] Moreover, The LISA-Taiji network has the potential to detect more than twenty stellar binary black holes (sBBHs), for which the error in luminous distance measurement is in the range of 0.05−0.2, and the relative error in sky positioning is in the range of 1−100deg2 In the range. [ 11 ]
The main scientific goal of the Taiji Program is to measure the mass, spin and distribution of black holes through the precise measurement of gravitational waves, to explore how intermediate-mass seed black holes develop if dark matter can produce black seed holes, and how enormous and supermassive black holes grow from black seed holes; Look for traces of the earliest generation of stars' genesis, development, and death, give direct restrictions on the intensity of primordial gravitational waves , and detect the polarization of gravitational waves, providing direct observational data for revealing the nature of gravity. [ 12 ] Gravitational waves can provide a clear picture of the universe because they are weakly linked to matter, and the information provided can be used in conjunction with information from telescopes and particle detectors. [ 13 ] The precise measurement of gravitational waves allows for in-depth and thorough investigation of the universe's large-scale structure, the birth and development of galaxies , and other topics; Better develop and establish a quantum theory of gravity beyond Einstein's general theory of relativity, reveal the nature of gravity, and help understand dark matter, the nature of energy, the formation of black holes , and cosmic inflation, [ 14 ] Gravitational waves can transmit information that electromagnetic waves cannot. [ 15 ] At the same time, the forward-looking technology developed from this is of great significance for improving the technical level of space science and deep space exploration; It will also play a positive role in applications such as inertial navigation, Earth science, global environmental change, and high-precision satellite platform construction. [ 16 ]
In 2008, the Chinese Academy of Sciences began demonstrating the feasibility of space gravitational wave detection, proposing the "Taiji Program" for China's space gravitational wave detection, and establishing the "single satellite, dual satellite, three satellites" and "three steps" development strategy and road map； and in August 2018, the "Taiji Program" single-satellite program was implemented in the Space Science (Phase II) Strategic Pilot Science and Technology Special Neutral Program and the first step in the three-step process was launched, that is, the Taiji-1 satellite. [ 17 ]
On August 31, 2019, Taiji-1 satellite was launched from the Jiuquan Satellite Launch Center. [ 18 ] In July 2021, "Taiji-1" has completed all the preset experimental tasks and achieved the highest precision space laser interferometry in China. It has achieved the first full performance verification of the two types of micro-push technology of Microbull-level radiofrequency ion and Hall, and took the lead in realizing the breakthrough of two non-drug control technologies in China. [ 19 ]
The optical metrology system and the non-resistance control system, both of which are part of Taiji-2 satellites, were confirmed by the Taiji-1 satellite mission; The mission's success also gave sufficient backing for the creation of Taiji-2 satellite; However, because Taiji-1 satellite only has one satellite, there is no way to test the inter-satellite laser link; The relevant unit expects to launch two satellites (Taiji-2) in 2023-2025 to clear obstacles for Taiji-3 satellites. [ 20 ] And it is expected to launch an equilateral triangle gravitational wave detection star group composed of three satellites around 2030. [ 21 ]
The scientific application unit and user of Taiji-1 in this Program is UCAS . The Taiji Program and the ground support system are managed by China's National Space Science Center, while the satellite system is developed by the Chinese Academy of Sciences' Institute of Microsatellite Innovation; the Institute of Precision Measurement Science and Technology Innovation, Chinese Academy of Sciences, Institute of Mechanics, Chinese Academy of Sciences, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Changchun Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Singapore University of Science and Technology, Singapore Nanyang Technological University, and the Institute of Precision Measurement Science and Technology Innovation, Chinese Academy of Sciences are among the cooperative units involved in payload development. [ 22 ] In addition, the Chinese Academy of Sciences established the gravitational wave cosmic polar laboratory in Hangzhou in April 2021. [ 23 ] | https://en.wikipedia.org/wiki/Taiji_program |
In Chinese philosophy , a taijitu ( Chinese : 太極圖 ; pinyin : tàijítú ; Wade–Giles : tʻai⁴chi²tʻu² ) is a symbol or diagram ( 圖 ; tú ) representing taiji ( 太極 ; tàijí ; 'utmost extreme') in both its monist ( wuji ) and its dualist ( yin and yang ) forms in application is a deductive and inductive theoretical model. Such a diagram was first introduced by Neo-Confucian philosopher Zhou Dunyi of the Song Dynasty in his Taijitu shuo ( 太極圖說 ).
The Daozang , a Taoist canon compiled during the Ming dynasty , has at least half a dozen variants of the taijitu . The two most similar are the Taiji Xiantiandao and wujitu ( 無極圖 ; wújítú ) diagrams, both of which have been extensively studied since the Qing period for their possible connection with Zhou Dunyi's taijitu . [ 2 ]
Ming period author Lai Zhide simplified the taijitu to a design of two interlocking spirals with two black-and-white dots superimposed on them, became synonymous with the Yellow River Map . [ 3 ] [ further explanation needed ] This version was represented in Western literature and popular culture in the late 19th century as the "Great Monad", [ 4 ] this depiction became known in English as the "yin-yang symbol" since the 1960s. [ 5 ] The contemporary Chinese term for the modern symbol is referred to as "the two-part Taiji diagram" ( 太極兩儀圖 ).
Ornamental patterns with visual similarity to the "yin yang symbol" are found in archaeological artefacts of European prehistory ; such designs are sometimes descriptively dubbed "yin yang symbols" in archaeological literature by modern scholars. [ 6 ] [ 7 ] [ 8 ]
The taijitu consists of five parts. Strictly speaking, the "yin and yang symbol", itself popularly called taijitu , represents the second of these five parts of the diagram.
The term taijitu in modern Chinese is commonly used to mean the simple "divided circle" form ( ), but it may refer to any of several schematic diagrams that contain at least one circle with an inner pattern of symmetry representing yin and yang .
While the concept of yin and yang dates to Chinese antiquity, [ 10 ] the interest in "diagrams" ( 圖 tú ) is an intellectual fashion of Neo-Confucianism during the Song period (11th century), and it declined again in the Ming period, by the 16th century. During the Mongol Empire and Yuan dynasty , Taoist traditions and diagrams were compiled and published in the encyclopedia Shilin Guangji by Chen Yuanjing . [ 11 ]
The original description of a taijitu is due to Song era philosopher Zhou Dunyi (1017–1073), author of the Taijitu shuo ( 太極圖說 ; "Explanation of the Diagram of the Supreme Ultimate"), which became the cornerstone of Neo-Confucianist cosmology. His brief text synthesized aspects of Chinese Buddhism and Taoism with metaphysical discussions in the Yijing .
Zhou's key terms Wuji and Taiji appear in the opening line 無極而太極 , which Adler notes could also be translated "The Supreme Polarity that is Non-Polar".
Non-polar ( wuji ) and yet Supreme Polarity ( taiji )! The Supreme Polarity in activity generates yang ; yet at the limit of activity it is still. In stillness it generates yin ; yet at the limit of stillness it is also active. Activity and stillness alternate; each is the basis of the other. In distinguishing yin and yang , the Two Modes are thereby established. The alternation and combination of yang and yin generate water, fire, wood, metal, and earth. With these five [phases of] qi harmoniously arranged, the Four Seasons proceed through them. The Five Phases are simply yin and yang ; yin and yang are simply the Supreme Polarity; the Supreme Polarity is fundamentally Non-polar. [Yet] in the generation of the Five Phases, each one has its nature. [ 12 ]
Instead of usual Taiji translations "Supreme Ultimate" or "Supreme Pole", Adler uses "Supreme Polarity" (see Robinet 1990) because Zhu Xi describes it as the alternating principle of yin and yang , and:
insists that taiji is not a thing (hence "Supreme Pole" will not do). Thus, for both Zhou and Zhu, taiji is the yin-yang principle of bipolarity, which is the most fundamental ordering principle, the cosmic "first principle." Wuji as "non-polar" follows from this.
Since the 12th century, there has been a vigorous discussion in Chinese philosophy regarding the ultimate origin of Zhou Dunyi's diagram. Zhu Xi (12th century) insists that Zhou Dunyi had composed the diagram himself, against the prevailing view that he had received it from Daoist sources. Zhu Xi could not accept a Daoist origin of the design, because it would have undermined the claim of uniqueness attached to the Neo-Confucian concept of dao . [ 11 ]
While Zhou Dunyi (1017–1073) popularized the circular diagram, [ 13 ] the introduction of "swirling" patterns first appears in the Ming period and representative of transformation.
Zhao Huiqian ( 趙撝謙 , 1351–1395) was the first to introduce the "swirling" variant of the taijitu in his Liushu benyi ( 六書本義 , 1370s). The diagram is combined with the eight trigrams ( bagua ) and called the "River Chart spontaneously generated by Heaven and Earth".
By the end of the Ming period, this diagram had become a widespread representation of Chinese cosmology . [ 14 ]
The dots were introduced in the later Ming period (replacing the droplet-shapes used earlier, in the 16th century) and are encountered more frequently in the Qing period . [ 15 ] The dots represent the seed of yin within yang and the seed of yang within yin; the idea that neither can exist without the other and are never absolute.
Lai Zhide's design is similar to the gakyil ( dga' 'khyil or "wheel of joy") symbols of Tibetan Buddhism ; but while the Tibetan designs have three or four swirls (representing the Three Jewels or the Four Noble Truths , i.e. as a triskele and a tetraskelion design), Lai Zhide's taijitu has two swirls, terminating in a central circle. [ 16 ]
The Ming-era design of the taijitu of two interlocking spirals was a common yin-yang symbol in the first half of the 20th century. The flag of South Korea , originally introduced as the flag of Joseon era Korea in 1882, shows this symbol in red and blue. This was a modernisation of the older (early 19th century) form of the Bat Quai Do used as the Joseon royal standard.
The symbol is referred to as taijitu , simply taiji [ 19 ] [ 20 ] [ 21 ] (or the Supreme Ultimate in English), [ 22 ] hetu [ 1 ] or "river diagram", "the yin-yang circle", [ 23 ] or wuji , as wuji was viewed synonymously with the artistic and philosophical concept of taiji by some Taoists, [ 9 ] [ 24 ] including Zhou . [ 9 ] [ 25 ] Zhou viewed the dualistic and paradoxical relationship between the concepts of taiji and wuji , which were and are often thought to be opposite concepts, as a cosmic riddle important for the "beginning...and ending" of a life. [ 26 ]
The names of the taijitu are highly subjective and some interpretations of the texts they appear in would only call the principle of taiji those names rather than the symbol. [ further explanation needed ]
Since the 1960s, the He tu symbol, which combines the two interlocking spirals with two dots, has more commonly been used as a yin-yang symbol. [ citation needed ] compare with
In the standard form of the contemporary symbol, one draws on the diameter of a circle two non-overlapping circles each of which has a diameter equal to the radius of the outer circle. One keeps the line that forms an "S", and one erases or obscures the other line. [ 27 ] In 2008 the design was also described by Isabelle Robinet as a "pair of fishes nestling head to tail against each other". [ 28 ]
The Soyombo symbol of Mongolia may be prior to 1686.
It combines several abstract shapes, including a Taiji symbol illustrating the mutual complement of man and woman. In socialist times, it was alternatively interpreted as two fish symbolizing vigilance, because fish never close their eyes. [ 29 ]
The modern symbol has also been widely used in martial arts, particularly tai chi , [ 30 ] and Jeet Kune Do , since the 1970s. [ 31 ] In this context, it is generally used to represent the interplay between hard and soft techniques .
The dots in the modern " yin-yang symbol " have been given the additional interpretation of "intense interaction" between the complementary principles, i.e. a flux or flow to achieve harmony and balance. [ 32 ]
Similarities can be seen in Neolithic – Eneolithic era Cucuteni–Trypillia culture on the territory of current Ukraine and Romania. Patterns containing ornament looking like Taijitu from archeological artifacts of that culture were displayed in the Ukraine pavilion at the Expo 2010 in Shanghai, China. [ 36 ]
The interlocking design is found in artifacts of the European Iron Age . [ 6 ] [ 37 ] Similar interlocking designs are found in the Americas: Xicalcoliuhqui .
While this design appears to become a standard ornamental motif in Iron-Age Celtic culture by the 3rd century BC, found on a wide variety of artifacts, it is not clear what symbolic value was attached to it. [ 38 ] [ 39 ] Unlike the Chinese symbol, the Celtic yin-yang lack the element of mutual penetration, and the two halves are not always portrayed in different colors. [ 40 ] Comparable designs are also found in Etruscan art . [ 7 ]
Unicode features the he tu symbol in the Miscellaneous Symbols block, at code point U+262F ☯ YIN YANG . The related "double body symbol" is included at U+0FCA ࿊ TIBETAN SYMBOL NOR BU NYIS -KHYIL , in the Tibetan block.
The Soyombo symbol , which includes a taijitu, is available in Unicode as the sequence U+11A9E 𑪞 + U+11A9F 𑪟 + U+11AA0 𑪠. | https://en.wikipedia.org/wiki/Taijitu |
The tail flick test is a test of the pain response in animals, similar to the hot plate test . It is used in basic pain research and to measure the effectiveness of analgesics , by observing the reaction to heat. It was first described by D'Amour and Smith in 1941. [ 1 ]
Most commonly, an intense light beam is focused on the animal's tail and a timer starts. When the animal flicks its tail, the timer stops and the recorded time (latency) is a measure of the pain threshold . [ 2 ] Alternate methods can be used to apply heat , such as immersion in hot water. [ 3 ]
Alternately, a dolorimeter with a resistance wire with a constant heat flow may be used. For the tail flick test, the wire is attached to the tail of the organism, and the wire applies heat to the tail. The researcher then records the latency to tail flick. [ 4 ]
Researchers testing the effectiveness of drugs on the pain threshold often use the tail flick test to measure the extent to which the drug being tested has reduced the amount of pain felt by the model organism. [ 5 ]
Both laboratory mice and rats are a common model organism for these tests. These rodents are usually given analgesics , which are responsible for weakening the response to pain. Under these weakened responses to pain, with effectiveness often peaking about 30 minutes after ingestion, researchers test the effectiveness of the drugs by exposing the tail to constant heat and measuring how long it takes to flick, signaling its response to the pain. [ 6 ] [ 7 ] Naloxone and naltrexone , two opioid antagonists, have been used to study pain sensitivity in relation to exercise in mice. [ 8 ]
Experimental tests of the tail flick testing method showed that the temperature of the skin of the tail plays a major role in the critical temperature, i.e., the temperature at which the tail flicks in response to pain. Researchers found that if the tail has been exposed to a cooler temperatures before the test, then the critical temperature decreases. [ 9 ]
Through use of the tail flick test, researchers have found that genetics play a role in pain sensation and the effectiveness of analgesics. A mouse of one genetic line may be more or less tolerant of pain than a mouse of another genetic line. Also, a mouse of one genetic line may experience a higher or lower effectiveness of an analgesic than a mouse of another genetic line. Using this test, researchers can also begin to identify genes that play a role in pain sensation. For example, the Calca gene (see WikiGenes CALCA ) is primarily responsible for the variability in thermal (heat) nociception . [ 10 ] The Sprawling mutation (see WikiGenes Swl ) resulted in a moderate sensory neuropathy but the mutation did not affect nociceptive modality or motor function in the mice. The mice with the Sprawling mutation were unable to sense the pain, but their other sensory functions were unaffected. [ 11 ]
The tail flick test is one test to measure heat-induced pain in animals. This reflexive response is an indicator of pain sensitivity in an organism and reduction of pain sensitivity produced by analgesics. Limitations of this test include: the need for more research with murine subjects, and determining the validity of applying observed pain responses from animals to humans. [ 12 ] Also, researchers have found that skin temperature can significantly affect the results of the tail flick test and it is important to consider this effect when performing the test. [ 13 ] Lastly, many thermal tests do not distinguish between opioid agonists and mixed agonist-antagonists, and consequently a tail flick test for mice using cold water in place of heat has been developed to allow that distinction. [ 14 ] | https://en.wikipedia.org/wiki/Tail_flick_test |
A tail lift (term used in the UK, also called a "liftgate" in North America) is a mechanical device permanently installed on the rear of a work truck, van , or lorry , and is designed to facilitate the handling of goods from ground level or a loading dock to the level of the vehicle bed, or vice versa.
The majority of tail lifts are hydraulic or pneumatic in operation, although they can be mechanical, and are controlled by an operator using an electric relay switch .
Using a tail lift can make it unnecessary to use machinery such as a forklift truck to load heavy items on to a vehicle. A tail lift can also bridge the difference in height between a loading dock and the vehicle load bed.
Tail lifts are available for many sizes of vehicle, from standard vans to articulated lorries, and standard models can lift anywhere up to 2500kg. Some heavy-duty models can even exceed this limit, making them suitable for industrial applications where extreme loads need to be transported. [ 1 ]
Tail lifts are most often categorized by design type. Tail lift design types include Parallel Arm, Railgate, Column, Cantilever, Tuckunder, and Slider. [ 1 ]
Parallel Arm lifts support lower lifting capacities and are commonly installed on pickup trucks and service truck bodies. The parallel "arms" attach to both sides of the lifting platform and guide the platform out and away from the liftgate mainframe. Parallel Arm designs can either feature two hydraulic cylinders applying force directly to the lifting platform or a single hydraulic cylinder using some version of a cable-pulley system.
This type is particularly popular among small businesses and service industries due to its cost-effectiveness and simplicity in design.
Railgate lifts are very similar in design to Column Lifts but (generally) support lower lifting capacities. Railgate lifts get their name from the "outrails" which install directly to the vehicle body and serve as the guides for the liftgate platform. Platforms on railgates are larger than those of parallel arm lifts and, like column lifts, fix at a 90° angle from the outrails and lift completely vertically. [ 2 ]
Railgate lifts are often preferred for deliveries that require stable, consistent vertical motion, such as fragile or sensitive goods.
Column lifts are "beefier" versions of railgates, supporting some of the highest lifting capacities of any type of hydraulic lift. Like Railgate lifts, Column Lifts feature "tracks" that install directly onto the vehicle body. From the tracks a folding platform extends and lifts completely vertically.
Column lifts have the advantage of being able to lift to a higher level than the load bed, also known as "above bed travel," and are therefore preferable for vehicles with bed heights lower than standard dock height.
The disadvantages of column lifts include that the platform is only usually able to operate at a 90° angle from the track, meaning that on uneven surfaces, the lift will not meet the ground properly.
Their robust design makes them a common choice for logistics companies handling large-scale operations.
Cantilever lifts work by a set of rams attached to the chassis of the vehicle. These rams are on hinges, allowing them to change angle as they expand or contract. By using the rams in sequence, the working platform can either be tilted, or raised and lowered.
Cantilever lifts have the advantage of being able to tilt, which means they can often form a ramp arrangement, which may be more appropriate for some applications. It also means that it can be easier to load or unload on uneven ground.
On Tuckunder lifts, the lifting platform may be folded and stored underneath the load bed of the vehicle, leaving the option of it not being used when at a loading dock, and giving access and egress for operators without the need to operate the lift. Common tuckunder designs are either single- or dual-cylinder, with dual-cylinder designs supporting higher lifting capacities.
The Maxon company claims to have invented the first tuckunder lift in 1957 under the brand name Tuk-A-Way. [ 3 ]
Slider Lift designs, like tuckunders, are characterized by folding and storing directly underneath the vehicle bed. However, slider designs feature lifting platforms that "slide" out from underneath the vehicle bed (instead of lowering and unfolding). Slider lift designs support some of the highest lifting capacities of any type of hydraulic lift.
In North America, "liftgate" is the commonly used term for a hydraulic lift installed at the rear of a vehicle that can be used to mechanically load or unload cargo.
In the automobile industry, "liftgate" is also used to refer to the automatic rear door of a van, minivan, or crossover SUV type vehicle. This opening system is also sometimes called a "rear hatch."
Modern liftgates often come with advanced features such as remote control operation, automatic height adjustment, and integrated safety mechanisms, making them more user-friendly and efficient. | https://en.wikipedia.org/wiki/Tail_lift |
The tail vein or caudal vein is the largest vein in the tail of a vertebrate animal. It leads directly into the posterior cardinal vein of the posterior trunk in fishes. The mammal caudal vein (the middle caudal vein ) leads to the inferior vena cava .
The caudal vein is one of the many places from which a laboratory worker can withdraw blood from a mouse specimen . The process does not require the death of the mouse, assuming that it does not exceed the established standard that "no more than two blood samples are taken per session and in any one 24-hour period...The lateral tail vein is usually used and 50 μl to 0.2 ml of blood can be obtained per sample depending on the size of the animal." [ 1 ]
Warming an animal to the ideal 39°C may be necessary to cause dilation of the veins and allow for easier processing. | https://en.wikipedia.org/wiki/Tail_vein |
In mining , tailings or tails are the materials left over after the process of separating the valuable fraction from the uneconomic fraction ( gangue ) of an ore . Tailings are different from overburden , which is the waste rock or other material that overlies an ore or mineral body and is displaced during mining without being processed. Waste valorization is the evaluation of waste and residues from an economic process in order to determine their value in reuse or recycling , as what was gangue at the time of separation may increase with time or more sophisticated recovery processes.
The extraction of minerals from ore can be done two ways: placer mining , which uses water and gravity to concentrate the valuable minerals, or hard rock mining , which pulverizes the rock containing the ore and then relies on chemical reactions to concentrate the sought-after material. In the latter, the extraction of minerals from ore requires comminution , i.e., grinding the ore into fine particles to facilitate extraction of the target element(s). Because of this comminution, tailings consist of a slurry of fine particles, ranging from the size of a grain of sand to a few micrometres. [ 1 ] Mine tailings are usually produced from the mill in slurry form, which is a mixture of fine mineral particles and water. [ 2 ]
Tailings are likely to be dangerous sources of toxic chemicals such as heavy metals , sulfides , and radioactive content. These chemicals are especially dangerous when stored in water in ponds behind tailings dams . These ponds are also vulnerable to major breaches or leaks from the dams, causing environmental disasters , such as the Mount Polley disaster in British Columbia . Because of these and other environmental concerns such as groundwater leakage , toxic emissions and bird death, tailing piles and ponds have received more scrutiny, especially in developed countries, but the first UN-level standard for tailing management was only established 2020. [ 3 ]
There are a wide range of methods for recovering economic value, containing, or otherwise mitigating the impacts of tailings. However, internationally, these practices are poor, sometimes violating human rights.
Tailings are also called mine dumps, culm dumps, slimes, refuse, leach residue, slickens, or terra-cone (terrikon). [ citation needed ]
The effluent from the tailings from the mining of sulfidic minerals has been described as "the largest environmental liability of the mining industry". [ 4 ] These tailings contain large amounts of pyrite (FeS 2 ) and Iron(II) sulfide (FeS), which are rejected from the sought-after ores of copper and nickel, as well as coal. Although harmless underground, these minerals are reactive toward air in the presence of microorganisms, which if not properly managed lead to acid mine drainage .
Between 100 million and 280 million tons of phosphogypsum waste are estimated to be produced annually as a consequence of the processing of phosphate rock for the production of phosphate fertilizers. [ 5 ] In addition to being useless and abundant, phosphogypsum is radioactive due to the presence of naturally occurring uranium , thorium , and their daughter isotopes . Depending on the price achievable on the uranium market , extraction of the uranium content may be economically lucrative even absent other incentives, such as reducing the harm the radioactive heavy metals do to the environment.
Bauxite tailings is a waste product generated in the industrial production of aluminium . Making provision for the approximately 70 million tonnes (150 billion pounds) that is produced annually is one of the most significant problems in aluminium manufacturing. [ 6 ]
Red mud , now more frequently termed bauxite residue, is an industrial waste generated during the processing of bauxite into alumina using the Bayer process . It is composed of various oxide compounds, including the iron oxides which give its red colour. Over 97% of the alumina produced globally is through the Bayer process; for every tonne (2,200 lb) of alumina produced, approximately 1 to 1.5 tonnes (2,200 to 3,300 lb) of red mud are also produced; the global average is 1.23. Annual production of alumina in 2023 was over 142 million tonnes (310 billion pounds) resulting in the generation of approximately 170 million tonnes (370 billion pounds) of red mud. [ 7 ]
Due to this high level of production and the material's high alkalinity , if not stored properly, it can pose a significant environmental hazard. As a result, significant effort is being invested in finding better methods for safe storage and dealing with it such as waste valorization in order to create useful materials for cement and concrete . [ 8 ]
Coal refuse , also known as coal waste, rock, slag, coal tailings, waste material, rock bank, culm, boney, or gob (garbage of bituminous), is the material left over from coal mining , usually as tailings piles or spoil tips . For every tonne of hard coal generated by mining, 400 kg (880 lb) of waste material remains, which includes some lost coal that is partially economically recoverable. [ 9 ] Coal refuse is distinct from the byproducts of burning coal, such as fly ash .
Piles of coal refuse can have significant negative environmental consequences, including the leaching of iron , manganese , and aluminum residues into waterways and acid mine drainage . [ 10 ] The runoff can create both surface and groundwater contamination. [ 11 ] The piles also create a fire hazard, with the potential to spontaneously ignite . Because most coal refuse harbors toxic components, it is not easily reclaimed by replanting with plants like beach grasses. [ 12 ] [ 13 ]
Early mining operations often did not take adequate steps to make tailings areas environmentally safe after closure. [ 14 ] [ 15 ] Modern mines, particularly those in jurisdictions with well-developed mining regulations and those operated by responsible mining companies, often include the rehabilitation and proper closure of tailings areas in their costs and activities. For example, the Province of Quebec , Canada, requires not only the submission of a closure plan before the start of mining activity, but also the deposit of a financial guarantee equal to 100% of the estimated rehabilitation costs. [ 16 ] Tailings dams are often the most significant environmental liability for a mining project. [ 17 ]
Mine tailings may have economic value in carbon sequestration due to the large exposed surface area of the minerals. [ 18 ]
The fraction of tailings to ore can range from 90 to 98% for some copper ores to 20–50% of the other (less valuable) minerals. [ 19 ] The rejected minerals and rocks liberated through mining and processing have the potential to damage the environment by releasing toxic metals (arsenic and mercury being two major culprits), by acid drainage (usually by microbial action on sulfide ores), or by damaging aquatic wildlife that rely on clear water (vs suspensions). [ 20 ]
Tailings ponds can also be a source of acid drainage , leading to the need for permanent monitoring and treatment of water passing through the tailings dam; the cost of mine cleanup has typically been 10 times that of mining industry estimates when acid drainage was involved. [ 21 ]
The greatest danger of tailings ponds is dam failure, with the most publicized failure in the U.S. being the failure of a coal slurry dam in the West Virginia Buffalo Creek Flood of 1972, which killed 125 people; other collapses include the Ok Tedi environmental disaster in New Guinea , which destroyed the fishery of the Ok Tedi River . On average, worldwide, there is one big accident involving a tailings dam each year. [ 21 ]
Other disasters caused by tailings dam failures are, the 2000 Baia Mare cyanide spill and the Ajka alumina plant accident . In 2015, the iron ore tailings dam failure at the Germano mine complex in Minas Gerais, Brazil, was the country's biggest environmental disaster. The dam breach caused the death of 19 people due to flooding of tailings slime downstream and affected some 400 km of the Doce river system with toxic effluence and out into the Atlantic Ocean.
Tailings deposits tend to be located in rural areas or near marginalized communities, such as indigenous communities . The Global Industry Standard on Tailings Management (GISTM) recommends that "a human rights due diligence process is required to identify and address those that are most at risk from a tailings facility or its potential failure." [ 22 ]
Historically, tailings were disposed of in the most convenient manner, such as in downstream running water or down drains . Because of concerns about these sediments in the water and other issues, tailings ponds came into use. The sustainability challenge in the management of tailings and waste rock is to dispose of material, such that it is inert or, if not, stable and contained, to minimise water and energy inputs and the surface footprint of wastes and to move toward finding alternate uses. [ 20 ]
Bounded by impoundments (an impoundment is a dam), these dams typically use "local materials" including the tailings themselves, and may be considered embankment dams . [ 1 ] Traditionally, the only option for tailings storage was to contain the tailings slurry with locally available earthen materials. [ 23 ] This slurry is a dilute stream of the tailings solids within water that was sent to the tailings storage area. The modern tailings designer has a range of tailings products to choose from depending upon how much water is removed from the slurry prior to discharge. It is increasingly common for tailings storage facilities to require special barriers like bituminous geomembranes (BGMs) to contain liquid tailings slurries and prevent impact to the surrounding environment. [ 24 ] The removal of water not only can create a better storage system in some cases (e.g. dry stacking, see below) but can also assist in water recovery which is a major issue as many mines are in arid regions. In a 1994 description of tailings impoundments, however, the U.S. EPA stated that dewatering methods may be prohibitively expensive except in special circumstances. [ 1 ] Subaqueous storage of tailings has also been used. [ 1 ]
Tailing ponds are areas of refused mining tailings where the waterborne refuse material is pumped into a pond to allow the sedimentation (meaning separation) of solids from the water. The pond is generally impounded with a dam, and known as tailings impoundments or tailings dams. [ 1 ] It was estimated in 2000 that there were about 3,500 active tailings impoundments in the world. [ 17 ] The ponded water is of some benefit as it minimizes fine tailings from being transported by wind into populated areas where the toxic chemicals could be potentially hazardous to human health; however, it is also harmful to the environment. Tailing ponds are often somewhat dangerous because they attract wildlife such as waterfowl or caribou as they appear to be a natural pond, but they can be highly toxic and harmful to the health of these animals. Tailings ponds are used to store the waste made from separating minerals from rocks, or the slurry produced from tar sands mining. Tailings are sometimes mixed with other materials such as bentonite to form a thicker slurry that slows the release of impacted water to the environment.
There are many different subsets of this method, including valley impoundments, ring dikes, in-pit impoundments, and specially dug pits. [ 1 ] The most common is the valley pond, which takes advantage of the natural topographical depression in the ground. [ 1 ] Large earthen dams may be constructed and then filled with the tailings. Exhausted open pit mines may be refilled with tailings. In all instances, due consideration must be made to contamination of the underlying water table, among other issues. Dewatering is an important part of pond storage, as the tailings are added to the storage facility the water is removed – usually by draining into decant tower structures. The water removed can thus be reused in the processing cycle. Once a storage facility is filled and completed, the surface can be covered with topsoil and revegetation commenced. However, unless a non-permeable capping method is used, water that infiltrates into the storage facility will have to be continually pumped out into the future.
Paste tailings is a modification to the conventional methods of disposal of tailings (pond storage). Conventional tailings slurries are composed of a low percent of solids and relatively high water content (normally ranging from 20% to 60% solids for most hard rock mining) and when deposited into the tailings pond the solids and liquids separate. In paste tailings the percent of solids in the tailings slurry is increased through the use of paste thickeners to produce a product where the minimal separation of water and solids occurs and the material is deposited into a storage area as a paste (with a consistency somewhat like toothpaste). Paste tailings has the advantage that more water is recycled in the processing plant and therefore the process is more water efficient than conventional tailings and there is a lower potential for seepage. However the cost of the thickening is generally higher than for conventional tailings and the pumping costs for the paste are also normally higher than for conventional tailings as positive displacement pumps are normally required to transport the tailings from the processing plant to the storage area. Paste tailings are used in several locations around the world including Sunrise Dam in Western Australia and Bulyanhulu Gold Mine in Tanzania. [ 25 ]
Tailings do not have to be stored in ponds or sent as slurries into oceans, rivers, or streams. There is a growing use of the practice of dewatering tailings using vacuum or pressure filters so the tailings can then be stacked. [ 26 ] This saves water which potentially reduces the impacts on the environment in terms of a reduction in the potential seepage rates, space used, leaves the tailings in a dense and stable arrangement and eliminates the long-term liability that ponds leave after mining is finished. However although there are potential merits to dry stacked tailings these systems are often cost prohibitive due to increased capital cost to purchase and install the filter systems and the increase in operating costs (generally associated electricity consumption and consumables such as filter cloth) of such systems. [ citation needed ]
While disposal into exhausted open pits is generally a straightforward operation, disposal into underground voids is more complex. A common modern approach is to mix a certain quantity of tailings with waste aggregate and cement, creating a product that can be used to backfill underground voids and stopes . A common term for this is high-density paste fill (HDPF). HDPF is a more expensive method of tailings disposal than pond storage, however it has many other benefits as it can significantly increase the stability of underground excavations by providing a means for ground stress to be transmitted across voids – rather than having to pass around them – which can cause mining induced seismic events like that suffered previously at the Beaconsfield Mine Disaster .
Usually called riverine tailings disposal (RTD). In most environments, not a particularly environmentally sound practice, it has seen significant utilisation in the past, leading to such spectacular environmental damage as done by the Mount Lyell Mining & Railway Company in Tasmania to the King River , or the poisoning from the Panguna mine on Bougainville Island , which led to large-scale civil unrest on the island, and the eventual permanent closing of the mine. [ 21 ]
As of 2005, only three mines operated by international companies continued to use river disposal: The Ok Tedi mine , the Grasberg mine [ 21 ] and the Porgera mine , all on New Guinea . This method is used in these cases due to seismic activity and landslide dangers which make other disposal methods impractical and dangerous.
Commonly referred to as STD (Submarine Tailings Disposal) or DSTD (Deep Sea Tailings Disposal). Tailings can be conveyed using a pipeline then discharged so as to eventually descend into the depths. Practically, it is not an ideal method, as the close proximity to off-shelf depths is rare. When STD is used, the depth of discharge is often comparatively shallow, and extensive damage to the seafloor can result due to covering by the tailings product. [ 27 ] If the density and temperature of the tailings product is not controlled, it may travel long distances, or even float to the surface.
This method is used by the gold mine on Lihir Island ; its waste disposal has been viewed by environmentalists [ who? ] as highly damaging, while the owners claim that it is not harmful. [ 21 ]
Phytostabilisation is a form of phytoremediation that uses hyperaccumulator plants for long-term stabilisation and containment of tailings, by sequestering pollutants in soil near the roots. The plant's presence can reduce wind erosion, or the plant's roots can prevent water erosion, immobilise metals by adsorption or accumulation, and provide a zone around the roots where the metals can precipitate and stabilise. Pollutants become less bioavailable and livestock, wildlife, and human exposure is reduced. This approach can be especially useful in dry environments, which are subject to wind and water dispersion. [ 28 ]
Considerable effort and research continues to be made into discovering and refining better methods of tailings disposal. Research at the Porgera Gold Mine is focusing on developing a method of combining tailings products with coarse waste rock and waste muds to create a product that can be stored on the surface in generic-looking waste dumps or stockpiles. This would allow the current use of riverine disposal to cease. Considerable work remains to be done. However, co-disposal has been successfully implemented by several designers including AMEC at, for example, the Elkview Mine in British Columbia.
During extraction of the oil from oil sand, tailings consisting of water, silt, clays, and other solvents are also created. This solid will become mature fine tailings by gravity. Foght et al (1985) estimated that there are 10 3 anaerobic heterotrophs and 10 4 sulfate-reducing prokaryotes per milliliter in the tailings pond, based on conventional most probable number methods. Foght set up an experiment with two tailings ponds and an analysis of the archaea , bacteria , and the gas released from tailings ponds showed that those were methanogens . As the depth increased, the moles of CH 4 released actually decreased. [ 29 ]
Siddique (2006, 2007) states that methanogens in the tailings pond live and reproduce by anaerobic degradation, which will lower the molecular weight from naphtha to aliphatic , aromatic hydrocarbons, carbon dioxide and methane. Those archaea and bacteria can degrade the naphtha, which was considered as waste during the procedure of refining oil. Both of those degraded products are useful. Aliphatic, aromatic hydrocarbons and methane can be used as fuel in the humans' daily lives. In other words, these methanogens improve the coefficient of utilization. Moreover, these methanogens change the structure of the tailings pond and help the pore water efflux to be reused for processing oil sands. Because the archaea and bacteria metabolize and release bubbles within the tailings, the pore water can go through the soil easily. Since they accelerate the densification of mature fine tailings, the tailings ponds are enabled to settle the solids more quickly so that the tailings can be reclaimed earlier. Moreover, the water released from the tailings can be used in the procedure of refining oil. Reducing the demand of water can also protect the environment from drought. [ 30 ]
As mining techniques and the price of minerals improve, it is not unusual for tailings to be reprocessed using new methods, or more thoroughly with old methods, to recover additional minerals. Extensive tailings dumps of Kalgoorlie / Boulder in Western Australia were re-processed profitably in the 1990s by KalTails Mining. [ 31 ]
A machine called the PET4K Processing Plant has been used in a variety of countries for the past 20 years to remediate contaminated tailings. [ 32 ]
The UN and business communities developed an international standard for tailings management in 2020 after the critical failure of the Brumadinho dam disaster . [ 3 ] The program was convened by United Nations Environment Programme (UNEP), International Council on Mining and Metals (ICMM) and the Principles for Responsible Investment . [ 3 ] | https://en.wikipedia.org/wiki/Tailings |
The Tainter gate is a type of radial arm floodgate used in dams and canal locks to control water flow. It is named for its inventor, the Wisconsin structural engineer Jeremiah Burnham Tainter . [ 1 ]
Tainter, an employee of the lumber firm Knapp, Stout and Co. , invented the gate in 1886 for use on the company's dam that forms Lake Menomin in the United States. [ 1 ]
A side view of a Tainter gate resembles a slice of pie with the curved part of the piece facing the source or upper pool of water and the tip pointing toward the destination or lower pool. The curved face or skinplate of the gate takes the form of a wedge section of cylinder . The straight sides of the pie shape, the trunnion arms, extend back from each end of the cylinder section and meet at a trunnion which serves as a pivot point when the gate rotates. [ 2 ]
Pressure forces on a submerged body act perpendicular to the body's surface. The design of the Tainter gate results in every pressure force acting through the centre of the imaginary circle of which the gate is a section, so that all resulting pressure force acts through the pivot point of the gate, making construction and design easier.
When a Tainter gate is closed, water bears on the convex (upstream) side. When the gate is rotated, the rush of water passing under the gate helps to open and close the gate. The rounded face, long radial arms and bearings allow it to close with less effort than a flat gate. Tainter gates are usually controlled from above with a chain/gearbox/ electric motor assembly.
A critical factor in Tainter gate design is the amount of stress transferred from the skinplate through the radial arms and to the trunnion, with calculations pertaining to the resulting friction encountered when raising or lowering the gate. Some older systems have had to be modified to allow for frictional forces which the original design did not anticipate. [ 3 ] In 1995, too much stress during an opening resulted in a gate failure at Folsom Dam in northern California.
The Tainter gate is used in water control dams and locks worldwide. The Upper Mississippi River basin alone has 321 Tainter gates, and the Columbia River basin has 195. A Tainter gate is also used to divert the flow of water to San Fernando Power Plant on the Los Angeles Aqueduct . [ 1 ] | https://en.wikipedia.org/wiki/Tainter_gate |
Tairus ( Russian : Тайрус , a portmanteau of Тай ско ( Thai ) and Рус ский ( Russian )) is a synthetic gemstone manufacturer. It was formed in 1989 as part of Mikhail Gorbachev 's perestroika initiative to establish a joint venture between the Russian Academy of Sciences and Tairus Created Gems Co Ltd. of Bangkok, Thailand. Today Tairus is a major supplier of hydrothermally grown gemstones to the jewellery industry. Later, Tairus became a privately held enterprise, operating out of its Bangkok distribution hub under the trade name Tairus, owned by Tairus Created Gems Co Ltd. of Bangkok, Thailand.
In the beginning, the team was led by the scientist and developer of the hydrothermal process, the late Alexander Lebedev, whose name was kept secret by the Soviet regime for many years, and Walter Barshai , who was appointed to be the Chairman of the Board of the Joint Venture Tairus. Their objective was to grow and to supply emeralds , rubies , sapphires , alexandrite and other gems to the jewelry industry. The driving force was late Academician Nikolai Dobretsov, former President of the Siberian Branch of the Russian Academy of Sciences . Tairus has achieved many scientific breakthroughs. For example, the development of the hydrothermally grown corundum , aquamarine and the development of a revolutionary process of horizontal crystallization for growing corundum (ruby), chrysoberyl and alexandrite.
After many years of development, scientists at Tairus had succeeded to commercially grow emeralds in a laboratory environment that resemble in color and have gemological properties that “overlap natural emeralds from various localities, especially those of low alkali-bearing stones from Colombia” ( The Journal of Gemmology, 2006, Vol. 30, Nos 1/2, 59-74 ).
The following are synthetic gemstones that were developed by Tairus scientists; they are alternately referred to as Tairus stones (e.g. "Tairus Ruby"). | https://en.wikipedia.org/wiki/Tairus |
The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots . [ 1 ] The Tait conjectures involve concepts in knot theory such as alternating knots , chirality , and writhe . All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.
Tait came up with his conjectures after his attempt to tabulate all knots in the late 19th century. As a founder of the field of knot theory, his work lacks a mathematically rigorous framework, and it is unclear whether he intended the conjectures to apply to all knots, or just to alternating knots . It turns out that most of them are only true for alternating knots. [ 2 ] In the Tait conjectures, a knot diagram is called "reduced" if all the "isthmi", or "nugatory crossings" have been removed.
Tait conjectured that in certain circumstances, crossing number was a knot invariant , specifically:
Any reduced diagram of an alternating link has the fewest possible crossings.
In other words, the crossing number of a reduced, alternating link is an invariant of the knot. This conjecture was proved by Louis Kauffman , Kunio Murasugi (村杉 邦男), and Morwen Thistlethwaite in 1987, using the Jones polynomial . [ 3 ] [ 4 ] [ 5 ] A geometric proof, not using knot polynomials, was given in 2017 by Joshua Greene . [ 6 ]
A second conjecture of Tait:
An amphicheiral (or acheiral) alternating link has zero writhe.
This conjecture was also proved by Kauffman and Thistlethwaite . [ 3 ] [ 7 ]
The Tait flyping conjecture can be stated:
Given any two reduced alternating diagrams D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} of an oriented, prime alternating link: D 1 {\displaystyle D_{1}} may be transformed to D 2 {\displaystyle D_{2}} by means of a sequence of certain simple moves called flypes . [ 8 ]
The Tait flyping conjecture was proved by Thistlethwaite and William Menasco in 1991. [ 9 ] The Tait flyping conjecture implies some more of Tait's conjectures:
Any two reduced diagrams of the same alternating knot have the same writhe.
This follows because flyping preserves writhe. This was proved earlier by Murasugi and Thistlethwaite . [ 10 ] [ 7 ] It also follows from Greene's work. [ 6 ] For non-alternating knots this conjecture is not true; the Perko pair is a counterexample. [ 2 ] This result also implies the following conjecture:
Alternating amphicheiral knots have even crossing number. [ 2 ]
This follows because a knot's mirror image has opposite writhe. This conjecture is again only true for alternating knots: non-alternating amphichiral knot with crossing number 15 exist. [ 11 ] | https://en.wikipedia.org/wiki/Tait_conjectures |
The Taiwan Typhoon and Flood Research Institute (TTFRI) was a research institute which is part of the National Applied Research Laboratories of Taiwan . It was merged into the National Science and Technology Center for Disaster Reduction in 2018.
The Taiwan Typhoon and Flood Research Institute was inaugurated in 2011 in the city of Taichung . Lee Cheng-shang was the inaugural Director. [ 1 ]
TTFRI is a coordinator of research into quantitative precipitation forecasting. [ 2 ]
TTFRI has worked with the Central Weather Bureau to develop a radar assimilation system which has increased the accuracy of the six hour rainfall forecast by twenty percent. [ 3 ]
In 2018 TTFRI began a project to improve the flood management of Cayo District in Belize in partnership with the Belizean Government which is one of Taiwan's few remaining official diplomatic allies. [ 4 ]
In 2015 TTFRI acquired a set of UAVs from Australia for use their typhoon research program. Early attempts to acquire UAVs in 2005 were scrapped due to stricter air traffic controls imposed as a result of global terrorism. [ 5 ] | https://en.wikipedia.org/wiki/Taiwan_Typhoon_and_Flood_Research_Institute |
A takadai ( 高台 ) , also called kōdai , is a frame used for making kumihimo , a type of Japanese braid . The braids created on the takadai are flat (3D effects can be achieved) as opposed to the braids created on the marudai [ 1 ] which have a round or polygonal section. The threads are attached to weighted bobbins called tamas and lay on wood pieces with pegs that are called koma . A wooden sword is used to lightly beat the braid once the braiding has been done. The braiding progresses on a 'V' front, as opposed to weaving on a regular loom that progresses on a straight front.
The art that is worked on the takadai is a braid, not a weave. Although many of the patterns used on this braiding stand resemble the up and down motion of a weave, since each thread takes a turn at being both the weft and the warp , it is a braid.
On the takadai it is possible to make intricate patterns using a technique called "pick-up braids". The braid has two sides of two contrasting colors and is usually linked on the edges. Then a pattern is formed by interchanging strands from the bottom braid to the upper braid, and by changing the braiding sequence. [ 2 ]
This textile arts article is a stub . You can help Wikipedia by expanding it .
This article related to art or architecture in Japan is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Takadai |
The Takahashi Taxol total synthesis published by Takashi Takahashi in 2006 is one of several methods in taxol total synthesis . [ 1 ] The method starts from geraniol and differs from the other 6 published methods that it is a formal synthesis (the final product is baccatin III which lacks the amide tail found in taxol itself) and that it is racemic (the product baccatin III is optically inactive). A key feature of the published procedure is that several synthetic steps (construction of rings A, B and C) were performed in an automated synthesizer on a scale up to 300 gram and that purification steps were also automated.
Ring A was synthesised starting from geraniol 1 and involved acylation ( acetic anhydride , DMAP , Et 3 N ) to 2 , epoxidation ( N-bromosuccinimide , tBuOH / H 2 O then triethylamine ) to 3 , radical cyclisation ( titanocene dichloride , manganese , triethylborane , 2,6-lutidine ) to 4 , alcohol protection ( ethyl vinyl ether , camphorsulfonic acid ) to 5 , alcohol deprotection (NaOH, MeOH/THF/H 2 O) to alcohol 6 , Parikh-Doering oxidation to aldehyde 7 , isomerization ( DBU ) to enone 8 , organic reduction ( sodium borohydride ) to alcohol 9 , alcohol protection ( TBSCl , Et 3 N) to TBS ether 10 , hydrazone formation ( H 2 NNHTs ) to 11 and finally vinyl bromide formation ( tBuLi , 1,2-Dibromoethane ) in 12 .
The synthesis of ring C also required hydroxygeranyl acetate 2 . Subsequent steps were allylic oxidation ( SeO 2 , tBuO 2 H , salicylic acid ) to aldehyde 13 , then carbonyl reduction ( NaBH 4 ) to alcohol 14 , then epoxidation ( VO(acac) 2 , tBuO 2 H ) to 15 , then alcohol protection ( MPM trichloroacetimidate ) to MPM ether 16 , then radical cyclisation ( titanocene dichloride , manganese , triethylborane , TMSCl , K 2 CO 3 ) to alcohol 17 , alcohol protection ( BOMCl , DIPEA ) to benzyloxymethyl ether 18 , acetate hydrolysis ( NaOH ) and Ley oxidation to aldehyde 19 .
Ring A ( 12 ) and ring C ( 19 ) reacted together to alcohol 20 in a Shapiro reaction ( tBuLi , CeCl 3 ) in a similar way as in the Nicolaou Taxol total synthesis . Subsequent steps were epoxidation ( VO(acac) 2 , tBuO 2 H ) to 21 , reduction ( LiAlH 4 ) to the diol and alcohol protection (aqueous KOH , BnBr , Bu 4 NHSO 4 ) to benzyl ether 22 , alcohol protection ( Me 2 SiHCl , imidazole ) and oxidation ( DDQ ) to DMS ether 23 , tosylation ( TsCl , DMAP ) to 24 , deprotection to diol ( TBAF ) and reprotection ( TMSOTf , 2,6-lutidine , DIPEA ) as TMS ether 25 , Ley oxidation to aldehyde 26 , cyanohydrin formation ( TMSCN , 18-crown-6 , KCN ) and alcohol protection ( ethyl vinyl ether , camphorsulfonic acid ) to EE ether 27 .
Cyclisation of 27 took place by alkylation ( LiN(TMS) 2 , dioxane, microwave irradiation ) to tricycle 28 . Subsequent steps were cyanohydrin hydrolysis ( camphorsulfonic acid ), TMS deprotection ( KOH ) and allylic oxidation ( SeO 2 , tBuO 2 H , salicylic acid ) to ketone 29 , then Upjohn dihydroxylation to triol 30 , then acylation ( AcCl , DMAP ) and mesitylation ( MsCl , DMAP) to 31 , then benzyl group and benzyloxy group removal ( hydrogenation / Palladium on carbon ) followed by carbonate protection ( triphosgene , pyridine ) to 32 , then secondary alcohol protection ( TESCl , pyridine ) and primary alcohol deprotection ( potassium carbonate ) to diol 33 , then oxetane formation ( DIPEA , HMPA ) to 34 , then acylation ( Ac 2 O , DMAP), then benzoylation ( phenyllithium ) to 35 , then oxidation ( tBuOK , (PhSeO) 2 O , THF ) to the acyloin 36 , then isomerisation ( tBuOK ) and acylation ( Ac 2 O , DMAP, pyr ) to 37 , then oxidation at the allylic position ( PCC , celite , NaOAc , benzene ), ketone group oxidation ( NaBH 4 ) and TES protecting group removal ( HF · pyr ) to baccatin III ( 38 ). | https://en.wikipedia.org/wiki/Takahashi_Taxol_total_synthesis |
Takai olefination in organic chemistry describes the organic reaction of an aldehyde with a diorganochromium compound to form an alkene . It is a name reaction , named for Kazuhiko Takai , who first reported it in 1986. [ 1 ] In the original reaction, the organochromium species is generated from iodoform or bromoform and an excess of chromium(II) chloride and the product is a vinyl halide . One main advantage of this reaction is the E -configuration of the double bond that is formed. According to the original report, existing alternatives such as the Wittig reaction only gave mixtures.
In the reaction mechanism proposed by Takai, chromium(II) is oxidized to chromium(III) eliminating two equivalents of a halide. The geminal carbodianion complex thus formed (determined as [Cr 2 Cl 4 (CHI)(THF) 4 ]) [ 2 ] [ 3 ] reacts with the aldehyde in a 1,2-addition along one of the carbon to chromium bonds and in the next step both chromium bearing groups engage in an elimination reaction . In Newman projection it can be seen how the steric bulks of chromium groups and the steric bulks of the alkyl and halogen groups drive this reaction towards anti elimination. [ 4 ]
Prior to the introduction of this chromium-based protocol, olefination reactions generally gave Z alkenes or mixtures of isomers. [ 1 ] Similar olefination reactions had been performed using a variety of reagents such as zinc and lead chloride; [ 5 ] however, these olefination reactions often lead to the formation of diols—the McMurry reaction —rather than the methylenation or alkylidenation of aldehydes. [ 6 ] To circumvent this issue, the Takai group examined the synthetic potential of chromium(II) salts.
The reaction primarily employs the use of aldehydes, but ketones may be used. However, ketones do not react as well as aldehydes; thus, for a compound with both aldehyde and ketone groups, the reaction can target just the aldehyde group and leave the ketone group intact. [ 1 ]
The drawbacks to the reaction include the fact that stoichiometrically, four equivalents of chromium chloride must be used, since there is a reduction of two halogen atoms. [ 3 ] Ways to limit the amount of chromium chloride exist, namely by utilization of zinc equivalent, [ 7 ] but this method remains unpopular.
In a second publication the scope of the reaction was extended to diorganochromium intermediates bearing alkyl groups instead of halogens: [ 8 ] | https://en.wikipedia.org/wiki/Takai_olefination |
A take-off warning system or TOWS is a set of warning signals required on most commercial aircraft, designed to alert the pilots of potentially-dangerous errors in an aircraft's take-off configuration.
There are numerous systems on board an aircraft that must be set in the proper configuration to allow it to take off safely. Prior to every flight, the flight officers use checklists to verify that each of the many systems is operating and has been configured correctly. Due to the inevitability of human error , even the checklist procedure can lead to failures to properly configure the aircraft.
Several improper configurations can leave an aircraft completely unable to become airborne—these conditions can easily result in fatal hull loss accidents. In order to reduce this, all major nations now mandate something similar to the US requirement that on (nearly) "all airplanes with a maximum weight more than 6,000 pounds and all jets [...] a takeoff warning system must be installed". This system must meet the following requirements:
(a) The system must provide to the pilots an aural warning that is automatically activated during the initial portion of the takeoff roll if the airplane is in a configuration that would not allow a safe takeoff. The warning must continue until—
(b) The means used to activate the system must function properly for all authorized takeoff power settings and procedures and throughout the ranges of takeoff weights, altitudes, and temperatures for which certification is requested. [ 1 ]
TOWS is designed to sound a warning for numerous other dangerous errors in the take-off configuration, such as the flaps and slats not being extended when the throttles are opened while the aircraft is on the ground. The alert is typically in the form of an audible warning horn accompanied by a voice message that indicates the nature of the configuration error.
A number of aircraft disasters due to improper configuration have happened in spite of the presence of a functional take-off warning system: | https://en.wikipedia.org/wiki/Take-off_warning_system |
In psychology , the take-the-best heuristic [ 1 ] is a heuristic (a simple strategy for decision-making ) which decides between two alternatives by choosing based on the first cue that discriminates them, where cues are ordered by cue validity (highest to lowest). In the original formulation, the cues were assumed to have binary values (yes or no) or have an unknown value. The logic of the heuristic is that it bases its choice on the best cue (reason) only and ignores the rest.
Psychologists Gerd Gigerenzer and Daniel Goldstein discovered that the heuristic did surprisingly well at making accurate inferences in real-world environments, such as inferring which of two cities is larger. The heuristic has since been modified and applied to domains from medicine , artificial intelligence , and political forecasting . [ 2 ] [ 3 ] The heuristic has been used to accurately model how experts, such as airport customs officers [ 4 ] and professional burglars, make decisions; [ 5 ] the model often makes better predictions of human behavior than more complex models that assume experts integrate all available cues. [ 6 ] [ 7 ]
Theories of decision making typically assume that all relevant reasons (features or cues) are searched and integrated into a final decision. Yet under uncertainty (as opposed to risk), the relevant cues are typically not all known, nor are their precise weights and the correlations between cues. In these situations, relying only on the best cue available may be a reasonable alternative that allows for fast, frugal, and accurate decisions. This is the logic of a class of heuristics known as "one-reason decision making," which includes take-the-best. [ 8 ] Consider cues with binary values (0, 1), where 1 indicates the cue value that is associated with a higher criterion value. The task is to infer which of two alternatives has the higher criterion value. An example is which of two NBA teams will win the game, based on cues such as home match and who won the last match. The take-the-best heuristic entails three steps to make such an inference: [ 9 ]
Search rule : Look through cues in the order of their validity.
Stopping rule : Stop search when the first cue is found where the values of the two alternatives differ.
Decision rule : Predict that the alternative with the higher cue value has the higher value on the outcome variable.
The validity v of a cue is given by v = C/(C+W), where C is the number of correct inferences when a cue discriminates, and W is the number of wrong inferences, all estimated from samples.
Consider the task to infer which object, A or B, has a higher value on a numerical criterion. As an example imagine someone having to judge whether the German city of Cologne has a larger population than the other German city of Stuttgart. This judgment or inference has to be based on information provided by binary cues, like "Is the city a state capital?". From a formal point of view, the task is a categorization: A pair (A, B) is to be categorized as X A > X B or X B > X A (where X denotes the criterion), based on cue information.
Cues are binary; this means they assume two values and can be modeled, for instance, as having the values 0 and 1 (for "yes" and "no"). They are ranked according to their cue validity , defined as the proportion of correct comparisons among the pairs A and B, for which it has different values, i.e., for which it discriminates between A and B. Take-the-best analyses each cue, one after the other, according to the ranking by validity and stopping the first time a cue discriminates between the items and concluding that the item with the larger value has also a larger value on the criterion.
The matrix of all objects of the reference class, from which A and B have been taken, and of the cue values which describe these objects constitutes a so-called environment. Gigerenzer and Goldstein, who introduced take-the-best (see Gerd Gigerenzer and Daniel Goldstein , D. G. (1996) [ 10 ] ) considered, as a walk-through example, precisely pairs of German cities. yet only those with more than 100,000 inhabitants. The comparison task for a given pair (A,B) of German cities in the reference class, consisted in establishing which one has a larger population, based on nine cues. Cues were binary-valued, such as whether the city is a state capital or whether it has a soccer team in the national league.
The cue values could be modeled by 1s (for "yes") and 0s (for "no") so that each city could be identified with its "cue profile", i.e., a vector of 1s and 0s, ordered according to the ranking of cues.
The question was: How can one infer which of two objects, for example, city A with cue profile (100101010) and
city B with cue profile (100010101) ,
scores higher on the established criterion, i.e., population size?
The take-the-best heuristic simply compares the profiles lexicographically, just as numbers written in base two are compared: the first cue value is 1 for both, which means that the first cue does not discriminate between A and B. The second cue value is 0 for both, again with no discrimination. The same happens for the third cue value, while the fourth cue value is 1 for A and 0 for B, implying that A is judged as having a higher value on the criterion.
In other words, X A > X B if and only if (100101010) > (100010101) .
Mathematically this means that the cues found for the comparison allow a quasi- order isomorphism between the objects compared on the criterion, in this case cities with their populations, and their corresponding binary vectors. Here "quasi" means that the isomorphism is, in general, not perfect, because the set of cues is not perfect.
What is surprising is that this simple heuristic has a great performance compared with other strategies. One obvious measure for establishing the performance of an inference mechanism is determined by the percentage of correct judgements. Furthermore, what matters most is not just the performance of the heuristic when fitting known data, but when generalizing from a known training set to new items.
Czerlinski, Goldstein and Gigerenzer compared several strategies with take-the-best: a simple tallying, or unit weight model (also called "Dawes' rule" in that literature), a weighted linear model on the cues weighted by their validities (also called "Franklin's rule" in that literature), linear regression, and Minimalist. Their results show the robustness of take-the-best in generalization.
For example, consider the task of selecting the bigger city of two cities when
The percent correct was roughly 74% for regression, take-the-best, unit weight linear. More specifically, the scores were 74.3%, 74.2%, and 74.1%, so regression won by a small margin.
However, the paper also considered generalization (also known as out-of-sample prediction).
In this case, when 10,000 different random splits were used, regression had on average 71.9% correct, Take-the-best had 72.2% correct, and unit with linear had 71.4% correct. The take-the-best heuristic was more accurate than regression in this case. [ 13 ] | https://en.wikipedia.org/wiki/Take-the-best_heuristic |
Take Back The Tech is a collaborative global campaign that connects the issue of violence against women and information and communications technology (ICT). It aims to raise awareness on the way violence against women is occurring on ICT platforms such as the Internet and mobile phones , and to call for people to use ICT in activism to end violence against women. [ 1 ]
It was initiated by the Association for Progressive Communications , Women's Networking Support Programme, in 2006. Since then, the campaign has been taken up and organised by individuals, collectives and non-governmental organizations in at least 24 countries. [ 2 ]
The name Take Back The Tech ! was inspired by the Take Back the Night (or Reclaim the Night ) movement, an international feminist march and rally to take direct action against rape and other forms of violence against women. [ 1 ]
The campaign highlights the way that violence against women is taking new forms through the use of ICT. This includes:
Research indicates that the majority of technology-enabled forms of violence victims are women. [ 3 ] [ 4 ]
The campaign also recognises that the gender digital divide contributes to unequal power relations that create enabling contexts for violence against women to occur. [ 5 ] To address this disparity, campaigners are encouraged to:
Throughout the year, Take Back the Tech! runs smaller campaigns, publishes articles and zines and organizes events. [ 6 ]
In 2013, Bytes for All, Pakistan , the human rights organization that runs the local Take Back The Tech! campaign in Pakistan , was awarded the Avon Communication Award under the 'Innovative Campaign Award' category for leading an exemplary national campaign in Pakistan. The award was presented by Salma Hayek at the United Nations Headquarters . [ 7 ] | https://en.wikipedia.org/wiki/Take_Back_The_Tech! |
Takeda Oncology (originally Millennium Pharmaceuticals ) is a biopharmaceutical company based in Cambridge , Massachusetts . It is a fully owned subsidiary of Takeda Pharmaceutical .
Takeda Oncology's research , development and commercialization activities focused in two therapeutic areas: oncology and inflammation to develop a line of new product candidates. It was one of the first companies to systematically search for genes linked to disease, [ 1 ] although none of the drugs which it is marketing or has in clinical trial, with one partial exception, have been the results of that research. [ 1 ]
It is particularly known for bringing bortezomib (marketed as Velcade ) through clinical trials to approval for treatment of patients with multiple myeloma by the U.S. FDA, but has a growing clinical development pipeline of other product candidates.
On May 14, 2008, Japanese company Takeda Pharmaceutical announced the completion of its acquisition of Millennium for US$25.00 per share in cash—a deal worth $8.8 billion. Takeda completed the acquisition through a tender offer and subsequent merger as a wholly owned subsidiary Millennium: The Takeda Oncology Company - with the name being simplified to Takeda Oncology in 2013.
Millennium was founded by Mark J. Levin, CEO, in Cambridge, Massachusetts in 1993. [ citation needed ] In its early years, Millennium focused on building science and business teams.
Beginning in 1994, Millennium created more than 20 strategic alliances with pharmaceutical and biotechnology companies. These alliances provided Millennium with close to $2 billion of committed funding that was used to develop and enhance its pipeline.
A merger with Leukosite in 1999 brought the company its first drug close-to-market, Campath (alemtuzumab) Injection, and additional investigational drugs in clinical trials. [ 2 ] In 2000, a merger with Cambridge Discovery Chemistry gave Millennium a strong presence in the United Kingdom and added to the organization more than 100 scientists with expertise in chemistry. [ 3 ] In a strategic business decision, Campath was later sold to the Millennium partner for the drug, ILEX Oncology, which in turn was acquired by Genzyme .
In February 2002, there was a further merger with COR Therapeutics [ 4 ] [ 5 ] —among the largest such mergers in the history of the biotech industry at that time. In addition to creating a strong pipeline of novel therapeutics, the merger added cardiovascular research and drug development to the company's other key therapeutic areas: oncology and inflammation. The merger also brought Integrilin ( eptifibatide ) Injection, an intravenous anti- platelet drug for patients with severe cardiovascular diseases , into the Millennium fold. Millennium partnered with Schering-Plough on the development and marketing of Integrilin until 2005 when Millennium licensed the exclusive U.S. commercialization and development rights of Integrilin to Schering-Plough. [ 6 ]
In May 2003 Velcade was launched for the treatment of relapsed and refractory multiple myeloma – a cancer of the blood. At the time, the U.S. Food and Drug Administration (FDA) granted approval for the treatment of multiple myeloma for patients who had not responded to at least two other therapies for the disease. Velcade — the first FDA-approved proteasome inhibitor — reached the market in record time and represented the first treatment in more than a decade to be approved for patients with multiple myeloma.
In late December 2007, Millennium successfully submitted a supplemental new drug application ( sNDA ) to the FDA for Velcade for previously untreated multiple myeloma. [ 7 ] The sNDA submitted to the FDA for this indication included data from the Phase III VISTA [ 8 ] study, a large, well-controlled international clinical trial, comparing a Velcade-based regimen to a traditional standard of care. Priority review was granted by the FDA in January 2008. On June 20, 2008, the FDA approved VELCADE in combination for patients with previously untreated multiple myeloma. This means that Millennium can market Velcade to patients who have not had any prior therapies for multiple myeloma (a first-line therapy).
In May 2008, Takeda Pharmaceutical Company Limited purchased Millennium for $8.8 billion. [ 9 ] Millennium was operating as an independent subsidiary, serving as the global center of excellence in oncology under its new name: "Millennium: The Takeda Oncology Company". This global footprint includes oncology research and marketing strategy and oversight. In addition to Cambridge, MA, oncology resources include facilities in San Diego, San Francisco, Tsukuba and Osaka with Millennium as the global hub for this critical and rapidly expanding therapeutic area.
In 2014, Takeda dropped the "Millennium" brand; effectively renaming the subsidiary into Takeda Oncology. [ 10 ] [ 11 ] However, Millennium Pharmaceuticals Inc is still a legal holding of Takeda and its name is still used for on-going legal processes. [ citation needed ]
In October 2023, Takeda began the process of withdrawing its drug, Exkivity in the United States after ending a trial for the drug early due to failure in the late-stage study. [ 12 ] Exkivity is a drug for adults with non-small cell lung cancer. [ 13 ] The FDA granted approval for the trial in 2021. [ 12 ]
Velcade is the first oncology drug marketed and promoted by Millennium. Velcade was granted FDA approval little more than four and a half years after initiation of the first clinical trial. To discover and develop such treatments, the Company focuses on key molecular pathways that play crucial roles in underlying disease processes, and on identifying therapeutically significant differences that may exist among people . Takeda Oncology applies this approach broadly throughout its R&D program to develop novel treatments not just for cancer but also for a number of other important diseases.
Velcade was co-developed by Millennium Pharmaceuticals. and Johnson & Johnson Pharmaceutical Research & Development. Takeda is responsible for commercialization of Velcade in the U.S. and Janssen-Cilag is responsible for commercialization in Europe and the rest of the world. Janssen Pharmaceutical K.K. is responsible for commercialization in Japan. For a limited period of time, Takeda and Ortho Biotech. are co- promoting Velcade in the U.S. Approved in 85 countries, at least 85,000 patients have been treated with Velcade globally. | https://en.wikipedia.org/wiki/Takeda_Oncology |
In the study of dynamical systems , a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of that system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms ), but it does not preserve the geometric shape of structures in phase space .
Takens' theorem is the 1981 delay embedding theorem of Floris Takens . It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions.
It is the most commonly used method for attractor reconstruction . [ 1 ]
Delay embedding theorems are simpler to state for discrete-time dynamical systems .
The state space of the dynamical system is a ν -dimensional manifold M . The dynamics is given by a smooth map
Assume that the dynamics f has a strange attractor A ⊂ M {\displaystyle A\subset M} with box counting dimension d A . Using ideas from Whitney's embedding theorem , A can be embedded in k -dimensional Euclidean space with
That is, there is a diffeomorphism φ that maps A into R k {\displaystyle \mathbb {R} ^{k}} such that the derivative of φ has full rank .
A delay embedding theorem uses an observation function to construct the embedding function. An observation function α : M → R {\displaystyle \alpha :M\to \mathbb {R} } must be twice-differentiable and associate a real number to any point of the attractor A . It must also be typical , so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function
is an embedding of the strange attractor A in R k . {\displaystyle \mathbb {R} ^{k}.}
Suppose the d {\displaystyle d} -dimensional
state vector x t {\displaystyle x_{t}} evolves according to an unknown but continuous
and (crucially) deterministic dynamic. Suppose, too, that the
one-dimensional observable y {\displaystyle y} is a smooth function of x {\displaystyle x} , and “coupled”
to all the components of x {\displaystyle x} . Now at any time we can look not just at
the present measurement y ( t ) {\displaystyle y(t)} , but also at observations made at times
removed from us by multiples of some lag τ : y t + τ , y t + 2 τ {\displaystyle \tau :y_{t+\tau },y_{t+2\tau }} , etc. If we use k {\displaystyle k} lags, we have a k {\displaystyle k} -dimensional vector. One might expect that, as the
number of lags is increased, the motion in the lagged space will become
more and more predictable, and perhaps in the limit k → ∞ {\displaystyle k\to \infty } would become
deterministic. In fact, the dynamics of the lagged vectors become
deterministic at a finite dimension; not only that, but the deterministic
dynamics are completely equivalent to those of the original state space (precisely, they are related by a smooth, invertible change of coordinates,
or diffeomorphism). In fact, the theorem says that determinism appears once you reach dimension 2 d + 1 {\displaystyle 2d+1} , and the minimal embedding dimension is often less. [ 2 ] [ 3 ]
Takens' theorem is usually used to reconstruct strange attractors out of experimental data, for which there is contamination by noise. As such, the choice of delay time becomes important. Whereas for data without noise, any choice of delay is valid, for noisy data, the attractor would be destroyed by noise for delays chosen badly.
The optimal delay is typically around one-tenth to one-half the mean orbital period around the attractor. [ 4 ] [ 5 ] | https://en.wikipedia.org/wiki/Takens's_theorem |
A Takeoff Acceleration Monitoring System automates the pilot monitoring of Distance to Go (DTG), "to sense, in a timely fashion the development of insufficient acceleration, which would extend the takeoff roll, perhaps precipitously". [ 1 ]
Over the years, recommendations have been made to develop a Take Off Performance Management System. The NLR and NASA developed TOPMS prototypes. However, these systems were never operationally introduced. [ 2 ]
EASA established two working groups (WGs) to address this issue. WG-88 focussed on the specification and standardization of On-Board Weight and Balance Systems
(OBWBS), an ongoing effort for what is considered to be a feasible option. WG-94 focussed on standards and operational conditions for a TOPMS; it WG-94 was concluded early 2017, considering that TOPMS was not feasible, in particular due to limitations in technology and data availability. [ 2 ]
A version suitable for detecting gross errors, which can be integrated in existing avionics, has been proposed by National Aerospace Laboratory (NLR), KLM , and Martinair . [ 2 ]
A 2019 research paper explores the cause of a July 2017 serious incident , caused by erroneous data entry, where such system could have been useful. It "summarises a basic takeoff acceleration monitoring system and the effect this would have had on the July 2017 event". [ 3 ]
This aviation -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Takeoff_Acceleration_Monitoring_System |
In mathematics , Takeuti's conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-order logic has cut-elimination (Takeuti 1953). It was settled positively:
Takeuti's conjecture is equivalent to the 1-consistency of second-order arithmetic in the sense that each of the statements can be derived from each other in the weak system of primitive recursive arithmetic (PRA) . It is also equivalent to the strong normalization of the Girard/Reynold's System F .
This mathematical logic -related article is a stub . You can help Wikipedia by expanding it .
This logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Takeuti's_conjecture |
In the mathematical fields of set theory and proof theory , the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal , which acts as the limit of the range of Buchholz's psi function and Feferman's theta function. [ 1 ] [ 2 ] It was named by David Madore, [ 2 ] after Gaisi Takeuti , Solomon Feferman and Wilfried Buchholz. It is written as ψ 0 ( ε Ω ω + 1 ) {\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})} using Buchholz's psi function, [ 3 ] an ordinal collapsing function invented by Wilfried Buchholz, [ 4 ] [ 5 ] [ 6 ] and θ ε Ω ω + 1 ( 0 ) {\displaystyle \theta _{\varepsilon _{\Omega _{\omega }+1}}(0)} in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. [ 7 ] [ 8 ] It is the proof-theoretic ordinal of several formal theories:
This set theory -related article is a stub . You can help Wikipedia by expanding it .
This article about a number is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Takeuti–Feferman–Buchholz_ordinal |
Takt time , or simply takt , is a manufacturing term to describe the required product assembly duration that is needed to match the demand. Often confused with cycle time , takt time is a tool used to design work and it measures the average time interval between the start of production of one unit and the start of production of the next unit when items are produced sequentially. For calculations, it is the time to produce parts divided by the number of parts demanded in that time interval. [ 1 ] The takt time is based on customer demand; if a process or a production line are unable to produce at takt time, either demand leveling, additional resources, or process re-engineering is needed to ensure on-time delivery.
For example, if the customer demand is 10 units per week, then, given a 40-hour workweek and steady flow through the production line, the average duration between production starts should be 4 hours, ideally. This interval is further reduced to account for things like machine downtime and scheduled employee breaks.
Takt time is a borrowing of the Japanese word takuto taimu ( タクトタイム ) , which in turn was borrowed from the German word Taktzeit , meaning 'cycle time'. The word was likely introduced to Japan by German engineers in the 1930s. [ 2 ]
The word originates from the Latin word "tactus" meaning "touch, sense of touch, feeling". [ 3 ] Some earlier meanings include: (16th century) "beat triggered by regular contact, clock beat", then in music "beat indicating the rhythm" and (18th century) "regular unit of note values". [ 4 ]
Takt time has played an important role in production systems even before the industrial revolution . From 16th-century shipbuilding in Venice, mass-production of Model T by Henry Ford , synchronizing airframe movement in the German aviation industry and many more. Cooperation between the German aviation industry and Mitsubishi brought takt to Japan, where Toyota incorporated it in the Toyota Production System (TPS). [ 3 ]
James P. Womack and Daniel T. Jones in The Machine That Changed the World (1990) [ 5 ] and Lean Thinking (1996) [ 6 ] introduced the world to the concept of " lean ". Through this, Takt was connected to lean systems. In the Toyota Production System (TPS), takt time is a central element of the just-in-time pillar (JIT) of this production system.
Assuming a product is made one unit at a time at a constant rate during the net available work time, the takt time is the amount of time that must elapse between two consecutive unit completions in order to meet the demand.
Takt time can be first determined with the formula: [ 7 ]
Where T = Takt time (or takt), e.g. [work time between two consecutive units] T a = Net time available to work during the period, e.g. [work time per period] D = Demand (customer demand) during the period, e.g. [units required per period]
Net available time is the amount of time available for work to be done. This excludes break times and any expected stoppage time (for example scheduled maintenance , team briefings, etc.).
Example : If there are a total of 8 hours (or 480 minutes) in a shift (gross time) less 30 minutes lunch, 30 minutes for breaks (2 × 15 mins), 10 minutes for a team briefing and 10 minutes for basic maintenance checks, then the net Available Time to Work = 480 - 30 - 30 - 10 - 10 = 400 minutes.
If customer demand were 400 units a day and one shift was being run, then the line would be required to output at a minimum rate of one part per minute in order to be able to keep up with customer demand.
Takt time may be adjusted according to requirements within a company. For example, if one department delivers parts to several manufacturing lines, it often makes sense to use similar takt times on all lines to smooth outflow from the preceding station. Customer demand can still be met by adjusting daily working time, reducing down times on machines, and so on.
Takt time is common in production lines that move a product along a line of stations that each performs a set of predefined tasks.
With the adoption of lean thinking in the construction industry , takt time has found its way into the project-based production systems of the industry. Starting with construction methods that have highly repetitive products like bridge construction, tunnel construction, and repetitive buildings like hotels and residential high-rises, implementation of takt is increasing. [ 3 ]
According to Koskela (1992), an ideal production system has continuous flow and creates value for the customer while transforming raw materials into products. [ 8 ] Construction projects use critical path method (CPM) or program evaluation and review technique (PERT) for planning and scheduling. These methods do not generate flow in the production and tend to be vulnerable to variation in the system. Due to common cost and schedule overruns, industry professionals and academia have started to regard CPM and PERT as outdated methods that often fail to anticipate uncertainties and allocate resources accurately and optimally in a dynamic construction environment. [ 9 ] This has led to increasing developments and implementation of takt.
Takt, as used in takt planning or takt-time planning (TTP) for construction, is considered one of the several ways of planning and scheduling construction projects based on their utilization of space rather than just time, as done traditionally in the critical path method. Also, to visualize and create flow of work on a construction site, utilization of space becomes essential. [ 10 ] Some other space scheduling methods include:
In manufacturing, the product being built keeps moving on the assembly line, while the workstations are stationary. On contrary, construction product, i.e. the building or infrastructure facilities being constructed, is stationary and the workers move from one location to another. [ 14 ]
Takt planning needs an accurate definition of work at each workstation, which in construction is done through defining spaces, called "zones". Due to the non-repetitive distribution of work in construction, achieving work completion within the defined takt for each zone, becomes difficult. Capacity buffer is used to deal with this variability in the system. [ 15 ]
The rationale behind defining these zones and setting the takt is not standardized and varies as per the style of the planner. Work density method (WDM) is one of the methods being used to assist in this process. Work density is expressed as a unit of time per unit of area. For a certain work area, work density describes how much time a trade will require to do their work in that area (zone), based on: [ 16 ]
Once a takt system is implemented there are a number of benefits:
Once a takt system is implemented there are a number of problems: | https://en.wikipedia.org/wiki/Takt_time |
Talairach coordinates , also known as Talairach space , is a 3-dimensional coordinate system (known as an 'atlas') of the human brain, which is used to map the location of brain structures independent from individual differences in the size and overall shape of the brain. It is still common to use Talairach coordinates in functional brain imaging studies and to target transcranial stimulation of brain regions. [ 1 ] However, alternative methods such as the MNI Coordinate System (originated at the Montreal Neurological Institute and Hospital ) have largely replaced Talairach for stereotaxy and other procedures. [ 2 ]
The coordinate system was first created by neurosurgeons Jean Talairach and Gabor Szikla in their work on the Talairach Atlas in 1967, creating a standardized grid for neurosurgery. [ 3 ] [ 4 ] The grid was based on the idea that distances to lesions in the brain are proportional to overall brain size (i.e., the distance between two structures is larger in a larger brain). In 1988 a second edition of the Talairach Atlas came out that was coauthored by Tournoux, and it is sometimes known as the Talairach-Tournoux system. This atlas was based on single post-mortem dissection of a human brain. [ 5 ]
The Talairach Atlas uses Brodmann areas as the labels for brain regions. [ 6 ]
The Talairach coordinate system is defined by making two anchors, the anterior commissure and posterior commissure , lie on a straight horizontal line. [ 7 ] Since these two points lie on the midsagittal plane, the coordinate system is completely defined by requiring this plane to be vertical. Distances in Talairach coordinates are measured from the anterior commissure as the origin (as defined in the 1998 edition). The y-axis points posterior and anterior to the commissures, the left and right is the x-axis, and the z-axis is in the ventral-dorsal (down and up) directions. [ 8 ] Once the brain is reoriented to these axes, the researchers must also outline the six cortical outlines of the brain: anterior, posterior, left, right, inferior, and superior. [ 9 ] In the 1967 atlas the left is with positive coordinates while in the 1988 atlas the left has negative coordinates.
By defining standard anatomical landmarks that could be identified on different subjects (the anterior and posterior commissures), it became easier to spatially warp an individual brain image obtained through Magnetic Resonance Imaging (MRI), positron emission tomography (PET) and other imaging methods to this standard Talairach space. One can then make inferences about tissue identity at a specific location by referring to the atlas.
The Brodmann areas is an illustration of a cytoarchitectonic map of the human brain that was published by Korbinan Brodmann in his 1909 monogram. Brodmann's map splits the cerebral cortex into 43 differing parts, which become visible in cell-body stained histological sections. Years later, a large group of neuroscientists still utilize Brodmann's map for the localization of neuroimaging data that is obtained in living human brains. [ 10 ]
Some neuroimaging techniques, in fact, purported the use of Brodmann's area as a guideline for Talairach coordinates. Further, these technologies showcase that experimental tasks in a common reference space become possible through imaging the living human brain by registering function and having architectonic data performed and defined.
Brodmann's map proved useful in varying neuroimaging software packages and stereotaxic atlases, such as the Talairach atlas. This atlas also serves as a demonstration of the inherent problems (i.e., impressions of matches between areal borders and sulcal landmarks may lead to wrong conclusions in terms of localization of cytoarchitectonic borders or the usage of Brodmann's map without knowledge of the text that accompanies the drawing misleading researchers to false conclusions). [ 10 ]
The MNI coordinate system, also referred to as MNI space, are multiple stereotaxic brain coordinate systems created by the Montreal Neurological Institute and Hospital . [ 11 ] [ 12 ] [ 13 ] Similar to the Talairach coordinates, MNI coordinates can be used to describe the location of particular brain structures, without having to take into account any individual brain differences. However, as the Talairach system is "unrepresentative of the population at large", MNI coordinates were developed from MRI data garnered from many individuals, in an effort to create a neural coordinate system that could be more generalizable. [ 14 ] MNI coordinates have been matched to Tailarach coordinates to allow landmarks to correspond. [ 14 ]
The original MNI space was MNI 305, which was created from 305 Tailarach aligned images, from which a mean brain image was taken. [ 11 ] [ 15 ] MNI 152 (also known as ICBM 152) was created later with higher resolution MRI images that were registered to MNI 305, and from which a mean was taken. [ 12 ] [ 13 ] [ 15 ] MNI 152 is itself further made up of different atlases, with different constraints, such as linearly/non-linearly aligned and symmetric or non-symmetric brain hemispheres. [ 12 ]
Most neuroimaging software packages are able to convert from Talairach to MNI coordinates. However, disparities between MNI and Talairach coordinates can impede the comparison of results across different studies. This problem is most prevalent in situations where coordinate disparities should be corrected to reduce error, such as coordinate-based meta-analyses. There is a possibility that these disparities can be assuaged through the Lancaster transform , which can be adopted in order to minimize the variability in the literature regarding spatial normalization strategies. [ 16 ]
Non-linear registration is the process of mapping Talairach coordinates to subject-specific coordinates and generating a non-linear map in an attempt to compensate for actual shape differences between the two. Registration is usually nonlinear because the Talairach atlas is not a simple rigid transform (or even affine transform ) of a subject brain. [ 17 ]
The Talairach atlas is still commonly used in terms of the neuroimaging techniques that are available, but the lack of a three-dimensional model of the original brain makes it difficult for researchers to map locations from three-dimensional anatomical MRI images to the atlas automatically. Previous methods such as MNI have tried to assuage this issue through linear and piecewise mapping between the Talairach and the MNI template, but can only account for differences in overall brain orientation and size and thus cannot correctly account for actual shape differences. [ 18 ]
Current target brains are not suitable for current research (i.e., are average, can only be used in low-resolution MRI target brain mapping studies or are single brain). Optimized high-resolution brain template (HRBT), a high-resolution MRI target brain, is a technique that can aid in the issues named above. This optimization can be performed to help reduce individual anatomical biases of the original ICBM HBRT. The optimized HRBT is more adept at anatomically matching groups of brains. [ 19 ] | https://en.wikipedia.org/wiki/Talairach_coordinates |
Talimogene laherparepvec , sold under the brand name Imlygic among others, is a biopharmaceutical medication used to treat melanoma that cannot be operated on; it is injected directly into a subset of lesions which generates a systemic immune response against the recipient's cancer. [ 5 ] The final four year analysis from the pivotal phase 3 study upon which TVEC was approved by the FDA showed a 31.5% response rate with a 16.9% complete response (CR) rate. There was also a substantial and statistically significant survival benefit in patients with earlier metastatic disease (stages IIIb-IVM1a) and in patients who hadn't received prior systemic treatment for melanoma. The earlier stage group had a reduction in the risk of death of approximately 50% with one in four patients appearing to have met, or be close to be reaching, the medical definition of cure. [ 6 ] Real world use of talimogene laherparepvec have shown response rates of up to 88.5% with CR rates of up to 61.5%. [ 7 ]
Around half of people treated with talimogene laherparepvec in clinical trials experienced fatigue and chills; around 40% had fever, around 35% had nausea, and around 30% had flu-like symptoms as well as pain at the injection site. The reactions were mild to moderate in severity; 2% of people had severe reactions and these were generally cellulitis . [ 8 ]
Talimogene laherparepvec is a genetically engineered herpes virus (an oncolytic herpes virus ). Two genes were removed – one that shuts down an individual cell's defenses, and another that helps the virus evade the immune system – and a gene for human GM-CSF was added. The drug works by replicating in cancer cells, causing them to burst; it was also designed to stimulate an immune response against the patient's cancer, which has been demonstrated by multiple pieces of data, including regression of tumors which have not been injected with talimogene laherparepvec. [ 9 ] [ 5 ]
The drug was created and initially developed by BioVex, Inc. and was continued by Amgen , which acquired BioVex in 2011. [ 10 ] It was one of the first oncolytic immunotherapy approved globally; it was approved in the US in October 2015 and approved in Europe in December 2015. [ 11 ] [ 9 ] [ 12 ]
Talimogene laherparepvec is delivered by injecting it directly into tumors, thereby creating a systemic anti-tumor immune response. [ 2 ]
In the US, talimogene laherparepvec is FDA approved to treat Stage IIIb-IVM1c melanoma patients for whom surgical intervention is not appropriate and with tumors which can be directly injected; the EMA approved population in Europe is for Stage IIIb-IVM1a. [ 2 ] [ 8 ]
Talimogene laherparepvec has been shown to extend survival in patients with Stage IIIb-IVM1a melanoma and patients who have not received prior systemic therapy for melanoma. [ 13 ]
Around half of people treated with talimogene laherparepvec in clinical trials experienced fatigue and chills; around 40% had fever, around 35% had nausea, and around 30% had flu-like symptoms as well as pain at the injection site. The reactions were mild to moderate in severity; 2% of people had severe reactions and these were generally cellulitis . [ 8 ]
More than 10% of people had edema , headache, cough, vomiting, diarrhea, constipation, muscle pain, or joint pain. Between 1% and 10% developed cold sores, pain or infection in the lesion, anemia , immune mediated events (like vasculitis, pneumonia, worsening psoriasis, glomerulonephritis and vitiligo [ 14 ] ), dehydration, confusion, anxiety, depression, dizziness, insomnia, ear pain, fast heart beating , deep vein thrombosis , high blood pressure, flushing, shortness of breath when exercising, sore throat, symptoms of the common cold , stomach pain, back pain, groin pain, weight loss, or oozing from the injection site. [ 8 ]
Talimogene laherparepvec is taken up by normal cells and cancer cells like the wild type herpes simplex virus , it is cleared in the same way. [ 8 ]
Talimogene laherparepvec directly destroys the cancer cells it infects, inducing a systemic immune response against the patient's cancer. [ 9 ] [ 5 ]
The virus invades both cancerous and healthy cells, but it cannot productively replicate in healthy tissue because it lacks Infected cell protein 34.5 (ICP34.5). When cells are infected with a virus they shut down and die, but ICP34.5 blocks this stress response , allowing the virus to hijack the cell's translation machinery to replicate itself. A herpesvirus lacking the gene coding for ICP34.5 cannot replicate in normal tissue. However, in many cancer cells the stress response is already disrupted, so a virus lacking ICP34.5 can still replicate in tumors. After the virus has replicated many times, the cell swells and finally bursts, killing the cell and releasing the copies of the virus, which can then infect nearby cells. [ 5 ] [ 15 ]
While talimogene laherparepvec is using the cell's translation machinery to replicate, it also uses it to make the cell create GM-CSF . GM-CSF is secreted or released when the cancer cell bursts, attracting dendritic cells to the site, which pick up the tumor antigens, process them, and then present them on their surface to cytotoxic (killer) T cells which in turn sets off an immune response. [ 9 ] [ 5 ]
Talimogene laherparepvec is a biopharmaceutical drug; it is an oncolytic herpes virus that was created by genetically engineering a strain of herpes simplex virus 1 (HSV-1) taken from a person infected with the virus, rather than a laboratory strain. [ 9 ] Both copies of the viral gene coding for ICP34.5 were deleted and replaced with the gene coding for human GM-CSF , and the gene coding for ICP47 was removed. [ 9 ] [ 5 ] [ 16 ] In wild herpes virus, ICP47 suppresses the immune response to the virus; it was removed because the drug was designed with the intention of activating the immune system. [ 5 ]
The first oncolytic virus to be approved by a regulatory agency was a genetically modified adenovirus named H101 by Shanghai Sunway Biotech. It gained regulatory approval in 2005 from China's State Food and Drug Administration (SFDA) for the treatment of head and neck cancer. [ 17 ] Talimogene laherparepvec is the world's first approved oncolytic immunotherapy, i.e. it was also designed to provide systemic anti-tumor effects through the induction of an anti-tumor immune response. [ 18 ]
Talimogene laherparepvec was created and initially developed by BioVex, Inc. under the brand OncoVEX GM-CSF . Development was continued by Amgen , which acquired BioVex in 2011. [ 10 ] [ 9 ] BioVex was founded in 1999, based on research by Robert Coffin at University College London , [ 19 ] and moved its headquarters to Woburn, Massachusetts in 2005, leaving about half its employees in the UK. [ 20 ]
The phase II clinical trial in melanoma was published in 2009 [ 21 ] and the phase III trial was published in 2013. [ 22 ]
Talimogene laherparepvec was approved by the US Food and Drug Administration to treat melanoma in October 2015. It was the first approval of an oncolytic virus and the first approval of a gene therapy in the West. [ 11 ] It was approved by the European Medicines Agency in December of that year. [ 8 ] [ 9 ]
Amgen estimated that talimogene laherparepvec would be priced at US$65,000 per patient at the time it was approved. [ 23 ]
As of 2016, talimogene laherparepvec has been studied in early stage clinical trials in pancreatic cancer , soft-tissue sarcoma , and head and neck squamous-cell carcinoma ; it had also been tested in combination with checkpoint inhibitors ipilimumab and pembrolizumab . [ 9 ] | https://en.wikipedia.org/wiki/Talimogene_laherparepvec |
talk.origins (often capitalised to Talk.Origins or abbreviated as t.o. ) is a Usenet discussion forum concerning the origins of life , and evolution . Its official purpose is to draw such debates out of the science newsgroups , such as sci.bio.evolution and sci.bio.paleontology. With the general decline of evolution/creationism debate and discontinuation of Google Groups support for Usenet in February 2024, Talk.Origins is now largely defunct.
The first post to talk.origins was a starter post by Mark Horton , dated 5 September 1986. [ 1 ]
In the early 1990s, a number of FAQs on various topics were being periodically posted to the newsgroup . In 1994, Brett J. Vickers established an anonymous FTP site to host the collected FAQs of the newsgroup. In 1995, Vickers started the TalkOrigins Archive web site as another means of hosting the talk.origins FAQs. It maintains an extensive FAQ on topics in evolutionary biology , geology and astronomy , with the aim of representing the views of mainstream science . It has spawned other websites, notably TalkDesign "a response to the intelligent design movement", Evowiki, and the Panda's Thumb weblog .
The group was originally created as the unmoderated newsgroup net.origins as a 'dumping ground' for all the various flame threads 'polluting' other newsgroups, then renamed to talk.origins as part of the Great Renaming . Subsequently, after discussion on the newsgroup, the group was voted to be moderated in 1997 by the normal USENET RFD/CFV process, in which only spam and excessive crossposting are censored. The moderator for the newsgroup is David Iain Greig [ 2 ] (and technically Jim Lippard as alternate/backup).
The group is characterized by a long list of in-crowd jokes like the fictitious University of Ediacara, [ 3 ] the equally fictitious Evil Atheist Conspiracy [ 4 ] which allegedly hides all the evidence supporting Creationism , a monthly election of the Chez Watt -award for "statements that make you go 'say what', or some such", [ 5 ] pun cascades, a strong predisposition to quoting Monty Python , and a habit of calling penguins "the best birds". [ citation needed ]
Apart from the humor, the group includes rebuttals to creationist claims. There is an expectation that any claim is to be backed up by actual evidence, preferably in the form of a peer-reviewed publication in a reputable journal . [ citation needed ] The group as a whole votes for a PoTM-award (Post of The Month), which makes it into the annals of TalkOrigins Archive . [ 6 ] | https://en.wikipedia.org/wiki/Talk.origins |
Talk pages are where people discuss how to make content on Wikipedia the best that it can be. You can use this page to start a discussion with others about how to improve the " Dicalcium ruthenate " page. | https://en.wikipedia.org/wiki/Talk:Dicalcium_ruthenate |
Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles . [ 1 ] The series of eclipses separated by a repeat of one of these intervals is called an eclipse series .
Eclipses may occur when Earth and the Moon are aligned with the Sun , and the shadow of one body projected by the Sun falls on the other. So at new moon , when the Moon is in conjunction with the Sun, the Moon may pass in front of the Sun as viewed from a narrow region on the surface of Earth and cause a solar eclipse . At full moon , when the Moon is in opposition to the Sun, the Moon may pass through the shadow of Earth, and a lunar eclipse is visible from the night half of Earth. The conjunction and opposition of the Moon together have a special name: syzygy ( Greek for "junction"), because of the importance of these lunar phases .
An eclipse does not occur at every new or full moon, because the plane of the Moon's orbit around Earth is tilted with respect to the plane of Earth's orbit around the Sun (the ecliptic ): so as viewed from Earth, when the Moon appears nearest the Sun (at new moon) or furthest from it (at full moon), the three bodies are usually not exactly on the same line.
This inclination is on average about 5° 9′, much larger than the apparent mean diameter of the Sun (32′ 2″), the Moon as viewed from Earth's surface directly below the Moon (31′ 37″), and Earth's shadow at the mean lunar distance (1° 23′).
Therefore, at most new moons, Earth passes too far north or south of the lunar shadow, and at most full moons, the Moon misses Earth's shadow. Also, at most solar eclipses, the apparent angular diameter of the Moon is insufficient to fully occlude the solar disc, unless the Moon is around its perigee , i.e. nearer Earth and apparently larger than average. In any case, the alignment must be almost perfect to cause an eclipse.
An eclipse can occur only when the Moon is on or near the plane of Earth's orbit, i.e. when its ecliptic latitude is low. This happens when the Moon is around either of the two orbital nodes on the ecliptic at the time of the syzygy . Of course, to produce an eclipse, the Sun must also be around a node at that time – the same node for a solar eclipse or the opposite node for a lunar eclipse.
Up to three eclipses may occur during an eclipse season , a one- or two-month period that happens twice a year, around the time when the Sun is near the nodes of the Moon's orbit.
An eclipse does not occur every month, because one month after an eclipse the relative geometry of the Sun, Moon, and Earth has changed.
As seen from the Earth, the time it takes for the Moon to return to a node, the draconic month of 27.21 days, is less than the time it takes for the Moon to return to the same ecliptic longitude as the Sun: the synodic month of 29.53 days. The main reason is that during the time that the Moon has completed an orbit around the Earth, the Earth (and Moon) have completed about 1 ⁄ 13 of their orbit around the Sun: the Moon has to make up for this in order to come again into conjunction or opposition with the Sun. Secondly, the orbital nodes of the Moon precess westward in ecliptic longitude, completing a full circle in about 18.60 years, so a draconic month is shorter than a sidereal month of 27.32 days. In all, the difference in period between synodic and draconic month is nearly 2 + 1 ⁄ 3 days. Likewise, as seen from the Earth, the Sun passes both nodes as it moves along its ecliptic path. The period for the Sun to return to a node is called the eclipse or draconic year : about 346.6201 days, which is about 1 ⁄ 20 year shorter than a sidereal year .
If a solar eclipse occurs at one new moon, which must be close to a node, then at the next full moon the Moon is already more than a day past its opposite node, and may or may not miss the Earth's shadow. By the next new moon it is even further ahead of the node, so it is less likely that there will be a solar eclipse somewhere on Earth. By the next month, there will certainly be no event.
However, about 5 or 6 lunations later the new moon will fall close to the opposite node. In that time (half an eclipse year) the Sun will also have moved to the opposite node, so the circumstances will again be suitable for one or more eclipses.
The periodicity of solar eclipses is the interval between any two solar eclipses in succession, which will be either 1, 5, or 6 synodic months . [ 2 ] It is calculated that the earth will experience a total number of 11,898 solar eclipses between 2000 BCE and 3000 CE. A particular solar eclipse will be repeated approximately after every 18 years 11 days and 8 hours (6,585.32 days) of period, but not in the same geographical region. [ 3 ] A particular geographical region will experience a particular solar eclipse in every 54 years 34 days period. [ 2 ] Total solar eclipses are rare events, although they occur somewhere on Earth every 18 months on average, [ 4 ]
For the repetition of a solar eclipse, the geometric alignment of the Earth, Moon and Sun, as well as some parameters of the lunar orbit should be repeated. The following parameters and criteria must be repeated for the repetition of a solar eclipse:
These conditions are related with the three periods of the Moon's orbital motion, viz. the synodic month , anomalistic month and draconic month . In other words, a particular eclipse will be repeated only if the Moon will complete roughly an integer number of synodic, draconic, and anomalistic periods (223, 242, and 239) and the Earth-Sun-Moon geometry will be nearly identical to that eclipse. The Moon will be at the same node and the same distance from the Earth. Gamma (how far the moon is north or south of the ecliptic during an eclipse) changes monotonically throughout any single Saros series. The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January). When the Earth is near its average distance (March to April or September to October), the change in gamma is average.
For the repetition of a lunar eclipse, the geometric alignment of the Moon, Earth and Sun, as well as some parameters of the lunar orbit should be repeated. The following parameters and criteria must be repeated for the repetition of a lunar eclipse:
These conditions are related with the three periods of the Moon's orbital motion, viz. the synodic month , anomalistic month and draconic month . In other words, a particular eclipse will be repeated only if the Moon will complete roughly an integer number of synodic, draconic, and anomalistic periods (223, 242, and 239) and the Earth-Sun-Moon geometry will be nearly identical to that eclipse. The Moon will be at the same node and the same distance from the Earth. Gamma changes monotonically throughout any single Saros series. The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January). When the Earth is near its average distance (March to April or September to October), the change in gamma is average.
Another thing to consider is that the motion of the Moon is not a perfect circle. Its orbit is distinctly elliptic, so the lunar distance from Earth varies throughout the lunar cycle. This varying distance changes the apparent diameter of the Moon, and therefore influences the chances, duration, and type (partial, annular, total, mixed) of an eclipse. This orbital period is called the anomalistic month , and together with the synodic month causes the so-called " full moon cycle " of about 14 lunations in the timings and appearances of full (and new) Moons. The Moon moves faster when it is closer to the Earth (near perigee) and slower when it is near apogee (furthest distance), thus periodically changing the timing of syzygies by up to 14 hours either side (relative to their mean timing), and causing the apparent lunar angular diameter to increase or decrease by about 6%. An eclipse cycle must comprise close to an integer number of anomalistic months in order to perform well in predicting eclipses.
If the Earth had a perfectly circular orbit centered around the Sun, and the Moon's orbit was also perfectly circular and centered around the Earth, and both orbits were coplanar (on the same plane) with each other, then two eclipses would happen every lunar month (29.53 days). A lunar eclipse would occur at every full moon, a solar eclipse every new moon, and all solar eclipses would be the same type. In fact the distances between the Earth and Moon and that of the Earth and the Sun vary because both the Earth and the Moon have elliptic orbits. Also, both the orbits are not on the same plane. The Moon's orbit is inclined about 5.14° to Earth's orbit around the Sun. So the Moon's orbit crosses the ecliptic at two points or nodes. If a New Moon takes place within about 17° of a node, then a solar eclipse will be visible from some location on Earth. [ 5 ] [ 6 ] [ 7 ]
At an average angular velocity of 0.99° per day, the Sun takes 34.5 days to cross the 34° wide eclipse zone centered on each node. Because the Moon's orbit with respect to the Sun has a mean duration of 29.53 days, there will always be one and possibly two solar eclipses during each 34.5-day interval when the Sun passes through the nodal eclipse zones. These time periods are called eclipse seasons. [ 2 ] Either two or three eclipses happen each eclipse season. During the eclipse season, the inclination of the Moon's orbit is low, hence the Sun , Moon, and Earth become aligned straight enough (in syzygy ) for an eclipse to occur.
These are the lengths of the various types of months as discussed above (according to the lunar ephemeris ELP2000-85, valid for the epoch J2000.0; taken from ( e.g. ) Meeus (1991) ):
Note that there are three main moving points: the Sun, the Moon, and the (ascending) node; and that there are three main periods, when each of the three possible pairs of moving points meet one another: the synodic month when the Moon returns to the Sun, the draconic month when the Moon returns to the node, and the eclipse year when the Sun returns to the node. These three 2-way relations are not independent (i.e. both the synodic month and eclipse year are dependent on the apparent motion of the Sun, both the draconic month and eclipse year are dependent on the motion of the nodes), and indeed the eclipse year can be described as the beat period of the synodic and draconic months (i.e. the period of the difference between the synodic and draconic months); in formula:
as can be checked by filling in the numerical values listed above.
Eclipse cycles have a period in which a certain number of synodic months closely equals an integer or half-integer number of draconic months: one such period after an eclipse, a syzygy ( new moon or full moon ) takes place again near a node of the Moon's orbit on the ecliptic , and an eclipse can occur again. However, the synodic and draconic months are incommensurate: their ratio is not an integer number. We need to approximate this ratio by common fractions : the numerators and denominators then give the multiples of the two periods – draconic and synodic months – that (approximately) span the same amount of time, representing an eclipse cycle.
These fractions can be found by the method of continued fractions : this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions.
Since there may be an eclipse every half draconic month, we need to find approximations for the number of half draconic months per synodic month: so the target ratio to approximate is: SM / (DM/2) = 29.530588853 / (27.212220817/2) = 2.170391682
The continued fractions expansion for this ratio is:
The ratio of synodic months per half eclipse year yields the same series:
Each of these is an eclipse cycle. Less accurate cycles may be constructed by combinations of these.
This table summarizes the characteristics of various eclipse cycles, and can be computed from the numerical results of the preceding paragraphs; cf. Meeus (1997) Ch.9. More details are given in the comments below, and several notable cycles have their own pages. Many other cycles have been noted, some of which have been named. [ 12 ]
Any eclipse cycle, and indeed the interval between any two eclipses, can be expressed as a combination of saros ( s ) and inex ( i ) intervals. These are listed in the column "formula".
Any eclipse can be assigned to a given saros series and inex series. The year of a solar eclipse (in the Gregorian calendar ) is then given approximately by: [ 21 ]
When this is greater than 1, the integer part gives the year AD, but when it is negative the year BC is obtained by taking the integer part and adding 2. For instance, the eclipse in saros series 0 and inex series 0 was in the middle of 2884 BC.
A "panorama" of solar eclipses arranged by saros and inex has been produced by Luca Quaglia and John Tilley showing 61775 solar eclipses from 11001 BC to AD 15000 (see below). [ 22 ] Each column of the graph is a complete Saros series which progresses smoothly from partial eclipses into total or annular eclipses and back into partials. Each graph row represents an inex series. Since a saros, of 223 synodic months, is slightly less than a whole number of draconic months, the early eclipses in a saros series (in the upper part of the diagram) occur after the moon goes through its node (the beginning and end of a draconic month), while the later eclipses (in the lower part) occur before the moon goes through its node. Every 18 years, the eclipse occurs on average about half a degree further west with respect to the node, but the progression is not uniform.
Saros and inex number can be calculated for an eclipse near a given date.
Saros and inex numbers are also defined for lunar eclipses. A solar eclipse of given saros and inex series will be preceded a fortnight earlier by a lunar eclipse whose saros number is 26 lower and whose inex number is 18 higher, or it will be followed a fortnight later by a lunar eclipse whose saros number is 12 higher and whose inex number is 43 lower. As with solar eclipses, the Gregorian year of a lunar eclipse can be calculated as:
Lunar eclipses can also be plotted in a similar diagram, this diagram covering 1000 AD to 2500 AD. The yellow diagonal band represents all the eclipses from 1900 to 2100. This graph immediately illuminates that this 1900–2100 period contains an above average number of total lunar eclipses compared to other adjacent centuries.
This is related to the fact that tetrads (see above) are more common at present than at other periods. Tetrads occur when four lunar eclipses occur at four lunar inex numbers, decreseing by 8 (that is, a semester apart), which are in the range giving fairly central eclipses (small gamma ), and furthermore the eclipses take place around halfway between the earth's perihelion and aphelion. For example, in the tetrad of 2014-2015 (the so-called Four Blood Moons ), the inex numbers were 52, 44, 36, and 28, and the eclipses occurred in April and late September-early October. Normally the absolute value of gamma decreases and then increases, but because in April the sun is further east than its mean longitude , and in September/October further west than its mean longitude, the absolute values of gamma in the first and fourth eclipse are decreased, while the absolute values in the second and third are increased. The result is that all four gamma values are small enough to lead to total lunar eclipses. The phenomenon of the moon "catching up" with the sun (or the point opposite the sun), which is usually not at its mean longitude, has been called a "stern chase". [ 23 ]
Inex series move slowly through the year, each eclipse occurring about 20 days earlier in the year, 29 years later. This means that over a period of 18.2 inex cycles (526 years) the date moves around the whole year. But because the perihelion of Earth's orbit is slowly moving as well, the inex series that are now producing tetrads will again be halfway between Earth's perihelion and aphelion in about 586 years. [ 24 ]
One can skew the graph of inex versus saros for solar or lunar eclipses so that the x axis shows the time of year. (An eclipse which is two saros series and one inex series later than another will be only 1.8 days later in the year.) This shows the 586-year oscillations as oscillations that go up around perihelion and down around aphelion. | https://en.wikipedia.org/wiki/Talk:Eclipse_cycle/Temp |
Talk pages are where people discuss how to make content on Wikipedia the best that it can be. You can use this page to start a discussion with others about how to improve the " Geoffrey Wilson Greenwood " page. | https://en.wikipedia.org/wiki/Talk:Geoffrey_Wilson_Greenwood |
Please add Provenance Blockchain (provenance.io). The blockchain is the 2nd largest blockchain in terms of Total Locked Value (value that is locked in smart contracts on a blockchain) at $9.4B; the first largest is https://ethereum.org/en/ . References for this data include https://explorer.provenance.io/dashboard and https://www.linkedin.com/posts/rich-falk-wallace_without-quality-comps-youre-not-running-activity-7019283096264929280-j46-?utm_source=share&utm_medium=member_desktop .
The chain was found in 2018 by Mike Cagney ( https://en.wikipedia.org/wiki/Mike_Cagney ). The native utility token is HASH. It is *not* private or permissioned, it is programmable, and the Consensus is POS (Proof of Stake) leveraging https://tendermint.com . Swellbreak ( talk ) 17:24, 14 February 2023 (UTC) [ reply ]
Edits for submission can be previewed here.
Cleaned up dates for sorting and added info to existing entries.
Added entries with existing articles: Dash, Verge, & Firo.
Thanks, WikiMythril ( talk ) 11:13, 9 September 2023 (UTC) [ reply ]
This is a list of blockchains - decentralized, cryptographic databases - and other distributed ledgers .
BTC
( founded by Vitalik Buterin and others)
ETH
Please add PulseChain as another list of blockchains created by Richard Heart, ticker $PLS. Website PulseChain.com. Djkreditis ( talk ) 09:52, 26 February 2024 (UTC) [ reply ]
Several items are listed which are not blockchains (eg layer 2 protocols on Ethereum); also several have no sources and no existing Wikipedia pages. I suggest deleting these. GreyStar456 ( talk ) 17:11, 15 August 2024 (UTC) [ reply ]
the url connected to the "Base" chain network on this page is directing to the incorrect wikipedia site. the actual wiki page does not exist. 2601:47:4D00:592F:B410:606F:1908:62AA ( talk ) 04:33, 5 October 2024 (UTC) [ reply ]
Please add Gnosis Chain to this list. Native currency is GNO. Docs are here: https://docs.gnosischain.com/ Gnosisdao ( talk ) 15:14, 31 October 2024 (UTC) [ reply ]
Add to List of blockchains
Name: Sui
Date created: May 3, 2023
Created by: Mysten Labs
Native cryptocurrency: SUI
Consensus algorithm: PoS with Mysticeti
Programmable: Yes
Private: No
Permissioned: No
Finality: Sub-second
Notes: Move-based blockchain built by the originators of Meta 's Libra project. Waycun ( talk ) 16:44, 31 October 2024 (UTC) [ reply ]
A very good blockchain not seen in your list is ruby explorer, RUBY SCAN, rubyscain.io
with 2600000 users | https://en.wikipedia.org/wiki/Talk:List_of_blockchains |
I left the following feedback for the creator/future reviewers while reviewing this article: Thanks for the interesting article on a rather unusual topic!
Klbrain ( talk ) 20:05, 2 April 2024 (UTC) [ reply ] | https://en.wikipedia.org/wiki/Talk:The_Mixon |
TalkTalk TV Store (formerly blinkbox ) was a UK-based transactional (purchase and rental) video-on-demand (VoD) service available on Macintosh and Microsoft Windows computers, games consoles , tablet computers and Smart TVs . [ 1 ] Content was generally streamed, with downloading possible on Windows PC/laptops. The blinkbox brand had been extended to companion services offering digital music and books.
Tesco bought an 80% stake in the business in 2011 as part of a move into digital content. On 8 January 2015 the company sold blinkbox Movies to TalkTalk Group , who stated they intended to integrate the service into its own range of services. [ 2 ] Tesco sold blinkbox Music to Guvera on 26 January 2015, [ 3 ] and confirmed it would close its blinkbox Books division at the end of February 2015. [ 4 ] TalkTalk renamed the blinkbox Movies service in 2016. In May 2018 TalkTalk announced that they would close the service completely, with customers transferring to rival service Rakuten TV . [ 5 ] [ 6 ] [ 7 ]
The company was co-founded in 2006 by Michael Comish (he was blinkbox's chief executive officer until June 2013 when he became Tesco Group Digital Officer) [ 8 ] and Australian entrepreneur and current Managing Partner in Brookfield's Private Equity Group based in London, Adrian Letts. Letts was chief operating officer of the blinkbox group of services and MD of the Movies and TV service, both former senior executives of Channel 4 and Vodafone respectively. blinkbox was launched in October 2007 with the backing of a number of venture capital firms.
On 20 April 2011, Tesco acquired an 80% stake in blinkbox from Eden Ventures and Nordic Venture Partners, intending to use the company to boost its digital entertainment offering. [ 9 ] The Movies and TV service has around one million users per month.
In December 2014, Tesco was reported to be in negotiations with TalkTalk Group over the sale of blinkbox Movies, after talks with Vodafone fell through. [ 10 ] The sale to TalkTalk was completed on 8 January 2015. [ 11 ] TalkTalk eventually acquired the service along with a number of broadband customers, who transferred to the TalkTalk network.
In May 2018 TalkTalk announced that they would close the service completely for off-network customers who did not take TalkTalk's own broadband service where the more fully featured TalkTalk TV service is provided. Customers are being transferred to rival DTO service Rakuten TV. [ 5 ] [ 6 ] [ 7 ]
TalkTalk TV Store has content deals with over 50 of the world's leading content producers including HBO , BBC Worldwide , Disney , Warner Bros. , Universal Studios , Paramount Pictures , Sony Pictures Entertainment , FremantleMedia , All3Media , Revolver Entertainment and Aardman Animations .
In 2010 the company signed a deal with the Samsung Group allowing films to be streamed directly though any Internet@TV Samsung TVs. blinkbox also formed a content partnership with YouTube [ 12 ] allowing the site to carry blinkbox films on its new Movies section. The service was also the first in the UK to offer streaming film content through Sony 's PlayStation 3 , and Microsoft 's Xbox 360 consoles. [ 12 ]
Rental content can be streamed directly to a Windows PC, Mac, Xbox 360 , PlayStation 3 or Samsung Smart TV , and can be viewed any number of times within 48 hours. Buy-to-own content can be viewed without limit. The blinkbox app is built into all pre 2012 Samsung Smart TVs. Content is protected with Microsoft's Windows Media DRM digital rights management and requires Microsoft Silverlight to play. blinkbox cannot be used on Linux .
Historically blinkbox combined a free+pay model allowing users both to buy titles and also to watch free ad-supported titles. In December 2012 blinkbox stopped all ad-supported titles and shortly after launched www.clubcardtv.com, in which users with a Tesco Clubcard could watch titles free of charge, viewing advertisements targeted to them based on their buying habits.
On 28 October 2014, Tesco shut down the Clubcard TV service, stating that they were not getting the level of repeat usage that they had hoped for. [ 13 ]
In 2012, Tesco bought the online music streaming service WE7 and relaunched it in 2013 as blinkbox Music , aiming to compete with Spotify . [ 14 ] [ 15 ] blinkbox launched an ebook service, branded as blinkbox Books, in March 2014. [ 16 ] blinkbox Music was sold to Guvera on 26 January 2015. [ 3 ] After exclusive talks with Waterstones ended, Tesco also confirmed that blinkbox Books would close at the end of February 2015. [ 4 ] Customers' purchases were transferred to Kobo . [ 17 ] | https://en.wikipedia.org/wiki/TalkTalk_TV_Store |
Talkboy is a line of handheld voice recorder and sound novelty toys manufactured by Tiger Electronics in the 1990s. [ 1 ] The brand began as a result of a promotional tie-in with the 1992 film Home Alone 2: Lost in New York ; the most well-known product was the Deluxe Talkboy, a cassette recorder and player with a variable-speed voice changer that caused toy crazes over several holiday shopping seasons beginning in 1993.
The Talkboy was originally conceived as a prop for Home Alone 2 for the lead character Kevin MacCallister to use to outsmart adults. At the request of writer John Hughes and distributor 20th Century Fox , Tiger designed and built the prop. The company was given permission by the movie studio to sell a retail version of the toy, and it released two cassette recorders modeled after the film prop in 1992 and 1993, respectively. The original model did not have the voice changer of the film version and sold only moderately during the 1992 holiday shopping season. Tiger added the feature to the Deluxe model, which was released in April 1993. Following the release of Home Alone 2 on home video in July and a cross-promotion with Life cereal , interest in the toy spiked. Retailers had severely underestimated demand, and as a result the Deluxe Talkboy was one of the most highly sought-after toys during the 1993 holiday shopping season, selling out of stores across the United States. The product continued to be a best-seller in subsequent holiday shopping seasons. A pink version of the cassette recorder called Deluxe Talkgirl was released in 1995.
The success of the Talkboy cassette recorders spawned a product line of electronic sound novelty toys, including a phone, walkie talkies , and a radio . For subsequent recording devices, Tiger transitioned to digital technology, using solid-state storage and adding sound effects, beginning with Talkboy/Talkgirl F/X+ pens in 1995, which sold more than a million units in 45 days.
The original Talkboy model was a cassette player and recorder that was conceived as a prop for the 1992 film Home Alone 2: Lost in New York . [ 2 ] In the film, the main character Kevin McCallister (played by Macaulay Culkin ) uses the Talkboy to outsmart adults; [ 3 ] he successfully makes a reservation at the Plaza Hotel by slowing his voice down with the toy's variable-speed voice changer to pass himself off as an adult, and later he records incriminating statements by the burglars Marv and Harry. [ 2 ] [ 4 ] [ 5 ]
Originally, writer John Hughes specified in his script only that a futuristic recording device would be needed; [ 5 ] he and the film's distributor 20th Century Fox wanted something that was realistic yet appeared to be cutting edge . [ 4 ] Nancy Overfield-Delmar, the vice president of marketing for 20th Century Fox licensing and merchandising, said: "It was important to John that Kevin not use something already out in the marketplace. Kevin has to be one step ahead of other kids." [ 2 ]
Weeks before filming on Home Alone 2 was scheduled to begin, the toy-licensing deal for the film that Overfield was negotiating with Mattel fell through. Forced to find a last-minute replacement, she turned to toy licensee Tiger Electronics . [ 5 ] Overfield knew the company's co-founder and executive vice president Roger Shiffman from a previous licensing deal for the cartoon Bobby's World . [ 4 ] [ 5 ] Shiffman was persuaded to sign a deal to produce Home Alone 2 ' s toys, which also included "Monster Sap" spraying goo and a screaming backpack, with the promise of escalating royalty payments that would limit the risk to Tiger Electronics. [ 5 ]
To design the as-yet-undeveloped recorder prop, Hughes and Shiffman met at Shiffman's office several times after 20th Century Fox made an introduction between the two. [ 6 ] Hughes's original concept in the script was for Kevin "to have a gun", but Shiffman thought it was impractical since the character would need to travel with it through O'Hare International Airport in the movie. Shiffman told Hughes to let him work on the idea, [ 6 ] and his team at Tiger subsequently built a prototype in three weeks. [ 7 ] The ensuing design for the Talkboy featured a handle that would allow the device to slide onto a hand and a retracting microphone so it would look "more lifelike". [ 6 ]
Following the success of the original Home Alone (1990), which grossed $285 million in North America, the third-highest amount at the time, an extensive marketing and cross-promotion campaign was undertaken for Home Alone 2 . About 80 licensed products were released to tie-in with the movie, while corporate partners included Coca-Cola , Bloomingdale's , Kids "R" Us , and American Airlines . As part of the campaign, [ 8 ] Tiger was given permission by 20th Century Fox to sell a retail version of Talkboy in stores, [ 9 ] with Shiffman negotiating a "modest royalty" to build the brand. [ 6 ]
The original retail Talkboy model requires 4 AA batteries and uses a standard cassette tape. [ 2 ] However, it does not feature the voice changer of the film prop. [ 3 ] The toy was previewed at the American International Toy Fair in New York in February 1992, [ 10 ] and was released to market for that year's holiday shopping season , coinciding with the theatrical release of Home Alone 2 . Tiger spokeswoman Robin Plous said, "sales weren't very good because the product couldn't do everything it did in the film". [ 3 ] [ 11 ]
Complicating matters for Tiger, the company heard from retailer Toys "R" Us that many parents were complaining about their kids finding coarse language recorded on their newly purchased Talkboys. Although the cassette tape that was included with each unit was initially blank, the toy's packaging featured a "try me" element that allowed recording in stores but was being abused. To address the complaints, Tiger altered the packaging to prevent people from recording without first purchasing the toy. Even despite the changes, the issue persisted as some consumers purchased the toy, recorded profanities, and then returned it to stores where it was resold. [ 5 ]
Tiger created a second model of the cassette player and recorder, the Deluxe Talkboy, which added the variable-speed voice changer. [ 3 ] Audio can be sped up by recording on the "slow" setting and playing it back on the normal setting, or slowed down by recording on the normal setting and playing it back on the "slow" setting. [ 12 ] The toy was released in April 1993 for US$29.99, [ 4 ] [ 13 ] and was sold at 11 retailers: Caldor , FAO Schwarz , Fingerhut , Hills , Kay Bee Toys , Kmart , Musicland , Service Merchandise , Target , Toys "R" Us, and Wal-Mart . [ 4 ] [ 12 ] [ 14 ]
The Deluxe Talkboy was a sleeper hit , catching many retailers off-guard with its popularity during the 1993 holiday shopping season. [ 4 ] Tiger spokesman Marc J. Rosenberg said that retailers at the American International Toy Fair earlier that February had not predicted such high consumer demand, placing orders that were 300 to 400 percent below what demand actually turned out to be; only Toys "R" Us had ordered enough units, according to Shiffman. [ 4 ] [ 15 ] Tiger did not anticipate high demand for a second-season product, [ 3 ] and manufactured only as many units as retailers had ordered. [ 4 ] Interest in Talkboy grew after the July 27 release of Home Alone 2 on VHS , [ 5 ] which included an advertising insert that confirmed the toy was a real product; 10 million copies of the movie sold by December. [ 6 ] [ 12 ] Sales of Talkboy continued to increase as the result of a cross-promotion with Life cereal that advertised the toy and the Home Alone 2 video cassette on the side panel of cereal boxes. [ 16 ]
Tiger relied on the Home Alone 2 movie tie-in and the advertising insert in home copies to promote the toy. [ 12 ] Once demand surged, retailers tried to increase their orders, but Tiger found itself with insufficient time to produce enough units. [ 11 ] On Black Friday , one store sold out of 288 Talkboys within half a day. [ 12 ] Ultimately, Tiger was forced to pull all television commercials for the toy after Thanksgiving . [ 11 ] Rosenberg said this was done "because [they] didn't want to deceive anyone" about the product's availability. [ 13 ] Exact sales figures were not released, but Tiger spokespeople said in December that it had sold "hundreds of thousands" of Talkboys while facing demand for around 2 million units. [ 17 ]
By mid-December, the company's telephone switchboard was handling more than 500 calls per day regarding the toy. [ 15 ] Tiger employees recounted stories of the tactics that desperate callers resorted to. One person claimed to have a terminally ill child who needed a Talkboy, while another claimed to be from NBC News . [ 3 ] [ 15 ] One caller claimed to represent Rolling Stones guitarist Keith Richards and said he needed four Talkboys; after consulting Tiger's lawyers, Rosenberg agreed to accommodate him in exchange for autographed CDs. [ 5 ] Other callers tried to bribe Tiger employees or said they were willing to fly anywhere in the United States to purchase 10 Talkboys. [ 3 ] [ 15 ] Shiffman said, "There is not a story we haven't heard". [ 12 ]
Target reportedly ceased issuing rain checks for Talkboys after receiving more than 20,000 requests from customers. [ 15 ] By mid-December, the retailer had been out of stock of the toy for a "couple of weeks" and confirmed any additional shipments would not go onto store shelves due to their commitment to fulfill rain checks. [ 18 ] Eighteen days before Christmas , one Toys "R" Us store in Clinton Township, Michigan, had a waiting list of more than 500 people for the Talkboy. [ 3 ] Rosenberg said that special security agents were required to meet each of Tiger's air shipments arriving from Asia for protection. [ 15 ] The company's manufacturing plants in Hong Kong were running 24 hours a day to produce Talkboys in an attempt to keep up with demand, [ 12 ] and daily air shipments were being delivered overnight across the United States. [ 4 ] [ 15 ] Only three stores – Toys "R" Us, Kmart, and Wal-Mart – were scheduled to receive shipments the week before Christmas. [ 12 ] Tiger said it would continue shipping Talkboys past New Year's Day. [ 15 ] Rosenberg blamed the frenzied demand for Talkboys on retailers shifting away from the inventory practice of stockpiling, [ 17 ] while the St. Petersburg Times faulted the burgeoning "just-in-time" delivery model through which stores used computer-based delivery systems to handle advance ordering. [ 11 ]
In December 1993, Rosenberg said that Tiger anticipated "high demand for [the Deluxe Talkboy] going all the way through [1994]", and predicted that 20th Century Fox releasing Home Alone 2 for television broadcasts "would spur demand all over again". [ 12 ] The Deluxe Talkboy was popular once again during the 1994 holiday shopping season, with many media outlets reporting it as one of the hottest-selling toys. [ 19 ] [ 20 ] [ 21 ] Tiger said in November 1994 that it expected the Deluxe Talkboy to sell out for a second consecutive year. [ 22 ] Kmart and Target stores reported swift sales around Thanksgiving, [ 23 ] [ 24 ] with one Target location in Fort Myers, Florida , selling out of the toy on Black Friday within an hour of their 7 a.m. opening. [ 25 ] Wal-Mart indicated in mid-December that it was experiencing extended shortages of the toy. [ 26 ] Within the last two weeks of the shopping season, Michelle Healy of Gannett News Service said the Deluxe Talkboy was "almost impossible to find". Roger Goddu of Toys "R" Us called it "without question the single most surprising item in our entire inventory". [ 27 ]
In 1995, Tiger released the Deluxe Talkgirl, a pink Deluxe Talkboy that was marketed to girls. Shiffman said, "We think the [Talkboy] name may have prevented us from reaching the full market". [ 28 ]
The Talkboy cassette recorder was popular again in 1996, with Playthings listing it as a "standout" on its survey of the year's most popular toys. [ 29 ] It continued to sell well into the 1997 holiday shopping season, according to The Morning Call . [ 30 ] Speaking about the Talkboy's staying power, Chris Byrne, editor of Market Focus: Toys , said that it sold "phenomenally well because it's a good toy". [ 31 ]
The success of the Talkboy cassette recorders spawned a product line of electronic toys. [ 31 ] In 1995, [ 32 ] Tiger released the Talkboy F/X+ and Talkgirl F/X+, which are writing pens with a 12-second recorder, three-speed playback, and six buttons that play sound effects. [ 28 ] [ 33 ] Designed by Ralph Osterhout for Tiger, the toy combined solid-state storage and a voice-recording computer chip into the form factor of a pen. Shiffman called it a "breakthrough product in the industry" for making digital recording technology available in a low-cost toy. [ 32 ] The technology for altering voice pitch and modulation was licensed from its developer Janese Swanson . [ 34 ] The product retailed for $20 and sold a million units in 45 days. [ 32 ] A poll conducted by Arthur Andersen LLP found the Talkboy F/X+ pen to be one of the most popular toys among shoppers during Thanksgiving weekend in 1995. [ 35 ] In March 1997, Tiger began a year-long promotion with Nabisco to place offers for Talkboy F/X+ toys on 30 million packages of Oreos , Chips Ahoy! , Nutter Butters , and Ritz Bits , along with a million-dollar television advertisement campaign. [ 36 ]
The trademark for the Talkboy brand was allowed to lapse in 1999, nearly a year after Tiger Electronics was acquired by Hasbro . [ 5 ] [ 43 ]
Due to his involvement in the toy's creation, Rosenberg acquired the nickname "Talkboy" within the toy industry, [ 44 ] and had a personalized license plate created reading the same. [ 45 ]
Several media outlets acknowledged the Talkboy on lists of the most popular holiday toys and past toy crazes, including USA Today , [ 46 ] Metro , [ 47 ] Statesman Journal , [ 48 ] Livingly.com , [ 49 ] [ 50 ] Rakuten , [ 51 ] SILive.com , [ 52 ] and CNN.com . [ 53 ] Talkboy was ranked by Thrillist as the 38th-greatest movie prop of all time. [ 6 ] Complex ranked it 75th on its list of the 90 best gadgets of the 1990s, [ 54 ] while ABC News included it on a similar list of tech toys from the decade. [ 55 ] | https://en.wikipedia.org/wiki/Talkboy |
Tally marks , also called hash marks , are a form of numeral used for counting . They can be thought of as a unary numeral system .
They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks , were also historically used for this purpose.
Counting aids other than body parts appear in the Upper Paleolithic . The oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africa's Late Stone Age .
The so-called Wolf bone is a prehistoric artifact discovered in 1937 in Czechoslovakia during excavations at Dolní Věstonice , Moravia , led by Karl Absolon . Dated to the Aurignacian , approximately 30,000 years ago, the bone is marked with 55 marks which may be tally marks. The head of an ivory Venus figurine was excavated close to the bone. [ 1 ]
The Ishango bone , found in the Ishango region of the present-day Democratic Republic of Congo , is dated to over 20,000 years old. Upon discovery, it was thought to portray a series of prime numbers . In the book How Mathematics Happened: The First 50,000 Years , Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC , with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10." [ 2 ] Alexander Marshack examined the Ishango bone microscopically, and concluded that it may represent a six-month lunar calendar . [ 3 ]
Tally marks are typically clustered in groups of five for legibility. The cluster size 5 has the advantages of (a) easy conversion into decimal for higher arithmetic operations and (b) avoiding error, as humans can far more easily correctly identify a cluster of 5 than one of 10. [ citation needed ]
Roman numerals , the Brahmi and Chinese numerals for one through three (一 二 三), and rod numerals were derived from tally marks, as possibly was the ogham script. [ 7 ]
Base 1 arithmetic notation system is a unary positional system similar to tally marks. It is rarely used as a practical base for counting due to its difficult readability.
The numbers 1, 2, 3, 4, 5, 6 ... would be represented in this system as [ 8 ]
Base 1 notation is widely used in type numbers of flour; the higher number represents a higher grind.
In 2015, Ken Lunde and Daisuke Miura submitted a proposal to encode various systems of tally marks in the Unicode Standard . [ 9 ] However, the box tally and dot-and-dash tally characters were not accepted for encoding, and only the five ideographic tally marks (正 scheme) and two Western tally digits were added to the Unicode Standard in the Counting Rod Numerals block in Unicode version 11.0 (June 2018). Only the tally marks for the numbers 1 and 5 are encoded, and tally marks for the numbers 2, 3 and 4 are intended to be composed from sequences of tally mark 1 at the font level. | https://en.wikipedia.org/wiki/Tally_marks |
A tally stick (or simply a tally [ 1 ] ) was an ancient memory aid used to record and document numbers, quantities, and messages. Tally sticks first appear as animal bones carved with notches during the Upper Palaeolithic ; a notable example is the Ishango Bone . Historical reference is made by Pliny the Elder (AD 23–79) about the best wood to use for tallies, and by Marco Polo (1254–1324) who mentions the use of the tally in China. Tallies have been used for numerous purposes such as messaging and scheduling, and especially in financial and legal transactions, to the point of being currency .
There are two principal forms of tally sticks: the single tally and the split tally. A comparable example of this primitive counting device can be found in various types of prayer beads .
A number of anthropological artefacts have been conjectured to be tally sticks:
The single tally stick was an elongated piece of bone, ivory, wood, or stone which is marked with a system of notches (see: Tally marks ). The single tally stick serves predominantly mnemonic purposes. Related to the single tally concept are messenger sticks (used by, e.g., Inuit tribes), the knotted cords, khipus or quipus , as used by the Inca . Greek historian Herodotus ( c. 484 – c. 425 BC ) reported the use of a knotted cord by Darius I of Persia ( r. 522 – 486 BC ).
The split tally became a prevalent technique in medieval Europe, a time characterised by a scarcity of coinage and widespread illiteracy, to document bilateral exchanges and debts. Typically fashioned from squared hazelwood , the stick was inscribed with a series of notches before being split lengthwise. Each party in the transaction retained one half of the marked stick, both pieces bearing identical records. Over the years, this method was refined to the point of becoming virtually impervious to tampering. One such refinement was to make the two halves of the stick of different lengths. The longer part was called stock and was given to the stock holder , [ 5 ] which had advanced money (or other items) to the receiver. The shorter portion of the stick was called foil and was given to the party which had received the funds or goods. Using this technique each of the parties had an identifiable record of the transaction. The natural irregularities in the surfaces of the tallies where they were split would mean that only the original two halves would fit back together perfectly, and so would verify that they were matching halves of the same transaction. If one party tried to unilaterally change the value of his half of the tally stick by adding more notches, the absence of those notches would be apparent on the other party's tally stick. The split tally was accepted as legal proof in medieval courts and the Napoleonic Code (1804) still makes reference to the tally stick in Article 1333. [ 6 ] Along the Danube and in Switzerland the tally was still used in the 20th century in rural economies.
The most prominent and best recorded use of the split tally stick or "nick-stick" [ 7 ] [ 8 ] being used as a form of currency [ 9 ] was when Henry I introduced the tally stick system in medieval England in around 1100. The tally sticks recorded a payment of taxes, but soon began to circulate in a secondary discount market, being accepted as payment for goods and services at a discount since they could be later presented to the treasury as proof of taxes paid. Then tally sticks began to be issued in advance, in order to finance war and other royal spending, and circulated as "wooden money". [ 10 ] [ 11 ]
The system of tally marks of the Exchequer is described in The Dialogue Concerning the Exchequer as follows:
The manner of cutting is as follows. At the top of the tally a cut is made, the thickness of the palm of the hand, to represent a thousand pounds; then a hundred pounds by a cut the breadth of a thumb; twenty pounds, the breadth of the little finger; a single pound, the width of a swollen barleycorn; a shilling rather narrower; then a penny is marked by a single cut without removing any wood.
The cuts were made the full width of the stick so that, after splitting, the portion kept by the issuer (the stock ) exactly matched the piece (the foil ) given as a receipt. Each stick had to have the details of the transaction written on it, in ink, to make it a valid record. [ 12 ]
Royal tallies (debt of the Crown) also played a role in the formation of the Bank of England at the end of the 17th century. In 1697, the bank issued £1 million worth of stock in exchange for £800,000 worth of tallies at par and £200,000 in bank notes. This new stock was said to be "engrafted". The government promised not only to pay the Bank interest on the tallies subscribed but to redeem them over a period of years. The "engrafted" stock was then cancelled simultaneously with the redemption. [ 13 ]
The split tally of the Exchequer remained in continuous use in England until 1826, when the conditions required for activation of the Receipt of the Exchequer Act 1783 (23 Geo. 3. c. 82), the death of the last Exchequer Chamberlain, came about. [ 14 ] In 1834, following the passing of 4 Will. 4. c .15, tally sticks representing six centuries' worth of financial records were ordered to be burned in two furnaces in the Houses of Parliament . [ 15 ] The resulting fire set the chimney ablaze and then spread until most of the building was destroyed . [ 9 ] This event was described by Charles Dickens in an 1855 article on administrative reform. [ 16 ]
Tally sticks feature in the design of the entrance gates to The National Archives at Kew . | https://en.wikipedia.org/wiki/Tally_stick |
A tallyman is an individual who keeps a numerical record with tally marks , historically often on tally sticks .
In Ireland , it is common for political parties to provide private observers when ballot boxes are opened. These tallymen keep a tally of the preferences of visible voting papers and allow an early initial estimate of which candidates are likely to win in the drawn-out single transferable vote counting process. [ 1 ] Since the public voting process is by then complete, it is usual for tallymen from different parties to share information.
Another possible definition is a person who called to literally do a head count, presumably on behalf of either the town council or the house owners. This is rumoured to have occurred in Liverpool , in the years after the First World War . Mechanical tally counters can make such head counts easier, by removing the need to make any marks.
In poorer parts of England (including the north and the East End of London ), the tallyman was the hire purchase collector, who visited each week to collect the payments for goods purchased on the 'never never', or hire purchase. These people still had such employment up until the 1980s.
The title tallyman extended to the keeper of a village pound as animals were often held against debts, and tally sticks were used to prove they could be released. | https://en.wikipedia.org/wiki/Tallyman |
The Tama Iron Cobra is a bass drum pedal line made between 1992 and the present. They are a full-baseplate, double chain or Kevlar strap drive, professional quality bass drum pedal used by many leading drummers . The Iron Cobra comes in both single and double pedal configurations with 3 different drive choices and, recently, a longboard Speed Cobra option. Tama also produces an Iron Cobra Junior pedal for beginner to intermediate audiences. The pedal's initial popularity was due to its adjustability, but it has since become known for its durability over time.
Tama Drums introduced the first Iron Cobra pedal prototype in 1992. This pedal had some of the eventual features of the production Iron Cobras but, with quickly machined, blocky parts, it looked quite a bit less refined than the eventual consumer version.
In 1993, the first-generation Iron Cobra went on the market. The pedals looked similar to previous Tama models, having the typical silver, squared off footboard of the Tama pedals of the 1980s. The main difference with the new Cobra was the added adjustability of many of the components. The pedal was offered in a slightly less adjustable standard version and a full-featured professional version. The pedal was mostly the native silver color of the metal, with a few black painted details.
In 1998, the second-generation Iron Cobra was released. This version had a more refined, curved footboard, updated branding and logo, and a black powder-coated finish. The bearings were upgraded and the hoop clamp was upgraded. All of the adjustment mechanisms were refined for the second generation, as the pedal had built its name on adjustability the past 5 years. It was during this time when the standard non-professional version became known as the Iron Cobra Junior. The Junior version had only a single chain drive, as opposed to the double chain of the full professional version. [ 1 ]
In 2011, the third generation Iron Cobra came out with a new set of features. The pedal retained the black styling of the second generation with a few changes in features. This Cobra had a footboard surface that was smoother, the bearing housing was redesigned, the cam shapes were altered slightly, and the baseplate came with a feature called the Cobra Coil. The Cobra coil was a spring mounted beneath the footboard that was supposed to increase speed and responsiveness. Unlike the other upgrades, this Cobra Coil feature is unique to the Iron Cobra line and has come to be seen as a gimmick, rather than a true upgrade, adding very little to the playability of the pedals. [ 2 ]
In 2013, the third-generation pedals were joined by two new variants, the duo-glide, which allowed the cam shape to be altered, [ 3 ] and the Speed Cobra, a longboard version with a unique light sprocket cam, new fastball bearing type, and a different beater shape. [ 4 ]
Since the beginning the Iron Cobra was known for adjustability. Through all the generations, the pedal has had the ability to adjust the footboard angle independently of the beater shaft angle, adjust the beater head angle, adjust the hoop clamp size, adjust the beater shaft length, adjust the feel of the beater with a sliding weight, and adjust the spring tension. Not all of these adjustments are universally possible on all other pedals from other manufacturers. [ 5 ] The cam shape is not adjustable on the Iron Cobra, except on the third gen. duo-glide model, but the pedal has historically been offered with two different optional cams. The rolling-glide cam is round for a smooth action, while the power glide cam changes radius mid-stroke for more acceleration. While most of the Iron Cobra pedals are double chain drive, the flexi-glide version has a Kevlar strap drive. Since at least the second generation, the pedals have come with a hard plastic carrying case. [ 6 ]
Prominent drummers to use the pedal include Zach Hill , Dave Lombardo , Scott Travis , Joey Jordison , Mike Portnoy , Gavin Harrison , Stewart Copeland , Lars Ulrich , and Derrick Plourde . | https://en.wikipedia.org/wiki/Tama_Iron_Cobra |
Tamarin Prover is a computer software program for formal verification of cryptographic protocols . [ 1 ] It has been used to verify Transport Layer Security 1.3, [ 2 ] ISO/IEC 9798, [ 3 ] DNP3 Secure Authentication v5, [ 4 ] WireGuard , [ 5 ] [ 6 ] [ 7 ] [ 8 ] and the PQ3 Messaging Protocol of Apple iMessage . [ 9 ]
Tamarin is an open source tool, written in Haskell , [ 10 ] built as a successor to an older verification tool called Scyther. [ 11 ] Tamarin has automatic proof features, but can also be self-guided. [ 11 ] In Tamarin lemmas that representing security properties are defined. [ 12 ] After changes are made to a protocol, Tamarin can verify if the security properties are maintained. [ 12 ] The results of a Tamarin execution will either be a proof that the security property holds within the protocol, an example protocol run where the security property does not hold, or Tamarin could potentially fail to halt . [ 12 ] [ 10 ] | https://en.wikipedia.org/wiki/Tamarin_Prover |
In model theory , a discipline within the field of mathematical logic , a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah , tameness as a property of AEC was first isolated by Grossberg and VanDieren, [ 1 ] who observed that tame AECs were much easier to handle than general AECs.
Let K be an AEC with joint embedding, amalgamation, and no maximal models. Just like in first-order model theory, this implies K has a universal model-homogeneous monster model C {\displaystyle {\mathfrak {C}}} . Working inside C {\displaystyle {\mathfrak {C}}} , we can define a semantic notion of types by specifying that two elements a and b have the same type over some base model M {\displaystyle M} if there is an automorphism of the monster model sending a to b fixing M {\displaystyle M} pointwise (note that types can be defined in a similar manner without using a monster model [ 2 ] ). Such types are called Galois types .
One can ask for such types to be determined by their restriction on a small domain. This gives rise to the notion of tameness:
Tame AECs are usually also assumed to satisfy amalgamation.
While (without the existence of large cardinals ) there are examples of non-tame AECs, [ 3 ] most of the known natural examples are tame. [ 4 ] In addition, the following sufficient conditions for a class to be tame are known:
Many results in the model theory of (general) AECs assume weak forms of the Generalized continuum hypothesis and rely on sophisticated combinatorial set-theoretic arguments. [ 8 ] On the other hand, the model theory of tame AECs is much easier to develop, as evidenced by the results presented below.
The following are some important results about tame AECs. | https://en.wikipedia.org/wiki/Tame_abstract_elementary_class |
In mathematical group theory , a tame group is a certain kind of group defined in model theory .
Formally, we define a bad field as a structure of the form ( K , T ), where K is an algebraically closed field and T is an infinite , proper, distinguished subgroup of K , such that ( K , T ) is of finite Morley rank in its full language. A group G is then called a tame group if no bad field is interpretable in G .
This group theory -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tame_group |
In mathematics, a tame topology is a hypothetical topology proposed by Alexander Grothendieck in his research program Esquisse d’un programme [ 1 ] under the French name topologie modérée (moderate topology). It is a topology in which the theory of dévissage can be applied to stratified structures such as semialgebraic or semianalytic sets , [ 2 ] and which excludes some pathological spaces that do not correspond to intuitive notions of spaces.
Some authors consider an o-minimal structure to be a candidate for realizing tame topology in the real case. [ 3 ] [ 4 ] There are also some other suggestions. [ 5 ]
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tame_topology |
In mathematics , the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame , in other words homeomorphic to the interior of a compact 3-manifold.
The tameness theorem was conjectured by Marden (1974) . It was proved by Agol (2004) and, independently, by Danny Calegari and David Gabai . It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem .
It also implies the Ahlfors measure conjecture .
Topological tameness may be viewed as a property of the ends of the manifold, namely, having a local product structure. An analogous statement is well known in two dimensions, that is, for surfaces . However, as the example of Alexander horned sphere shows, there are wild embeddings among 3-manifolds, so this property is not automatic.
The conjecture was raised in the form of a question by Albert Marden , who proved that any geometrically finite hyperbolic 3-manifold is topologically tame. The conjecture was also called the Marden conjecture or the tame ends conjecture .
There had been steady progress in understanding tameness before the conjecture was resolved. Partial results had been obtained by Thurston , Brock, Bromberg, Canary, Evans, Minsky, Ohshika. [ citation needed ] An important sufficient condition for tameness in terms of splittings of the fundamental group had been obtained by Bonahon . [ citation needed ]
The conjecture was proved in 2004 by Ian Agol , and independently, by Danny Calegari and David Gabai. Agol's proof relies on the use of manifolds of pinched negative curvature and on Canary's trick of "diskbusting" that allows to replace a compressible end with an incompressible end, for which the conjecture has already been proved. The Calegari–Gabai proof is centered on the existence of certain closed, non-positively curved surfaces that they call "shrinkwrapped". | https://en.wikipedia.org/wiki/Tameness_theorem |
A tamga [ a ] or tamgha (from Old Turkic : 𐱃𐰢𐰍𐰀 , romanized: tamga , lit. 'stamp, seal') was an abstract seal or brand used by Eurasian nomads initially as a livestock branding , and by cultures influenced by them. The tamga was used as a livestock branding for a particular tribe, clan or family. They were common among the Eurasian nomads throughout Classical Antiquity and the Middle Ages. As clan and family identifiers, the collection and systematic comparison of tamgas is regarded to provide insights into relations between families, individuals and ethnic groups in the steppe territory. [ 1 ] [ 2 ]
Similar tamga-like symbols were sometimes adopted by sedentary peoples adjacent to the Pontic–Caspian steppe both in Eastern Europe and Central Asia . [ 3 ] [ 4 ] [ 5 ] [ 6 ] Branding of livestock was a common practice across most sedentary populations, as far back as the ancient Egyptians.
It has been speculated that Turkic tamgas represent one of the sources of the Old Turkic script of the 6th–10th centuries, [ 7 ] but since the mid-20th century, this hypothesis is widely rejected as being unverifiable. [ 8 ]
Tamgas originate in pre-historic times, but their exact usage and development cannot be continuously traced over time. There are, however, symbols represented in rock art that are referred to as tamgas or tamga-like. [ 9 ] If they serve to record the presence of individuals at a particular place, they may be functionally equivalent with medieval tamgas.
In the later phases of the Bosporan Kingdom , the ruling dynasty applied personal tamgas, composed of a fragment representing the family and a fragment representing the individual king, apparently in continuation of steppe traditions and in an attempt to consolidate seditary and nomadic factions within the kingdom. [ 10 ]
According to Clauson (1972, p.504f.), Common Turkic tamga means "originally a `brand' or mark of ownership placed on horses, cattle, and other livestock; it became at a very early date something like a European coat of arms or crest, and as such appears at the head of several Türkü and many O[ld] Kir[giz] funary monuments". [ 11 ]
Among modern Turkic peoples , the tamga is a design identifying property or cattle belonging to a specific Turkic clan, usually as a cattle brand or stamp . [ 12 ] In Turkestan , it has remained what it originally was: a cattle brand and clan identifier. The Turks who remained pastoral nomad kings in eastern Anatolia and Iran, continued to use their clan tamgas and in fact, they became high-strung nationalistic imagery. The Aq Qoyunlu and Qara Qoyunlu , like many other royal dynasties in Eurasia, put their tamga on their flags and stamped their coinage with it.
When Turkish clans took over more urban or rural areas , tamgas dropped out of use as pastoral ways of life became forgotten. That is most evident in the Turkish clans that took over western and eastern Anatolia after the Battle of Manzikert . The Turks who took over western Anatolia founded the Sultanate of Rûm and became Roman-style aristocrats. Most of them adopted the then-Muslim symbol of the Seal of Solomon after the Sultanate disintegrated into a mass of feuding ghazi states (see Isfendiyarids , Karamanids ). Only the Ottoman ghazi state (later to become the Ottoman Empire ) kept its tamga , which was so highly stylized that the bow was stylized down eventually to a crescent moon.
Tamgas of the 21 Oghuz tribes (as Charuklug had none) according to Mahmud al-Kashgari in Dīwān Lughāt al-Turk : [ 13 ]
"Tamga", or "tamag'a", literally means "stamp" or "seal" in Mongolian and designates emblematic symbols which were historically used by various Mongolic tribes or clans in Central Asia. According to Clauson (1972, p.504), it was originally a Turkic word also "used for a Chinese 'seal' and passed into Mong[olian] in this meaning as tamaga ". [ 11 ]
In the Mongol Empire , a tamgha was a seal placed on taxed items and, by extension, a tax on commerce (see Eastern Europe below). [ 14 ]
Over a hundred different Mongolian tamga are known. Certain tamga were adopted by individual medieval Mongolic and Turkic rulers, and were consequently used on coins and seals issued by these rulers. Tamga are most widely found on Islamic coins issued by the descendants of Chinggis Khan in the various khanates of Central Asia during the 13th and 14th centuries, in particular the Chaghatai Khanate . Tamga are of immense interest to numismatists, and are discussed in many academic works relating to the medieval Islamic coins of Central Asia. However, numismatists and historians currently have limited options for representing tamga symbols in text, and cannot reliably interchange text including tamga symbols because they are either represented as images, or are handdrawn, or use an ad hoc font. Doctor Nyamaa [ 15 ] identifies nearly a hundred tamga signs used on coins, although only about half of them can be assigned to a specific ruler, and some of them are variant forms or presentation forms of the same tamga. [ 16 ]
Tamgas are also stamped using hot irons on domesticated animals such as horses in present-day Mongolia and others to identify that the livestock belongs to a certain family, since livestock is allowed to roam during the day. Each family has their own tamga markings for easier identification. Tamga marks are not very elaborate, since they are made from curved pieces of iron by the individual families.
A tamag'a is also used as the "state seal" of Mongolia , which is handed over by the President of Mongolia as part of the transition to a new president. In the presidential case, the tamag'a is a little more elaborate and is contained in a wooden box.
From Turkic, the term "tamga" has also been loaned into Caucasian languages, e.g., Adyghe : тамыгъэ , romanized: tamığə ; Kabardian : дамыгъэ , romanized: damığə . Among the Circassians , almost every family has a tamga to this day. [ 17 ]
Throughout the early Middle Ages, the Rurikid nobles of Rus' used Tamga-like symbols to denote property rights over various items ( Rurikid symbols ). [ 18 ] Very likely, these are of Khazar (Turkic) origin [ 19 ] and have been adopted along with the expansion of the Rus into steppe territory. [ 20 ] A similar process of acculturation of steppe elements can also be suspected for (or before) the Second Bulgarian Empire (1185–1396), as its flags closely resemble the Rurikid symbols in taking the shape of a trident.
In East Slavic languages, the term tamga (Russian тамга) survived in state institution of border customs, with associated cluster of terms: rastamozhit ' (Russian растаможить, Belarusian растаможыць, pay customs duties), tamozhnya (Russian таможня, customs), tamozhennik (Russian таможенник, customs officer), derived from the use of tamga as a certificate of State . In East Slavic, the steppe term competes with forms assumed to originate from Germanic (Old Church Slavonic мꙑто toll, Russian (historical) мы́то "customs duty", Ukrainian мито "toll, customs duty", and Belarusian мытня, Ukrainian митниця "customs"; cf. German Maut "street toll" and Medieval Latin mūta "toll").
In the 20th century, the Rurikid trident, colloquially called tryzub (тризуб), has been adopted as national symbol and the coat of arms of Ukraine . The modern version has been designed by Vasyl Krychevsky (1918) and Andriy Grechylo , Oleksii Kokhan, and Ivan Turetskyi (1992).
In the late medieval Turco-Mongol states, the term tamga was used for any kind of official stamp or seal. This usage persisted in the early modern Islamic Empires ( Ottoman Empire , Mughal Empire ), and in some of their modern successor states.
In the Urdu language (which absorbed Turkic vocabulary), Tamgha is used as medal . Tamgha-i-Jurat is the fourth highest Military medal of Pakistan . It is admissible to all ranks for gallantry and distinguished services in combat. Tamgha-i-Imtiaz or Tamgha-e-Imtiaz (Urdu: تمغہ امتیاز ), which translates as "medal of excellence", is fourth highest honour given by the Government of Pakistan to both the military and civilians. Tamgha-i-Khidmat or Tamgha-e-Khidmat ( تمغۂ خدمت ), which translates as "medal of services", is seventh highest honour given by the Government of Pakistan to both the military and civilians. It is admissible to non-commissioned officers and other ranks for long meritorious or distinguished services of a non-operational nature.
In Egypt , the term damgha ( Arabic : دمغة ) or tamgha ( تمغة ) is still used in two contexts. One is a tax or fee when dealing with the government. It is normally in the form of stamps that have to be purchased and affixed to government forms, such as a driver license or a registration deed for a contract. The term is derived from the Ottoman damga resmi . Another is a stamp put on every piece of jewelry made from gold or silver to indicate it is genuine, and not made of baser metals. | https://en.wikipedia.org/wiki/Tamga |
The Tammann Commemorative Medal is awarded once a year and was established in remembrance of Gustav Heinrich Johann Apollon Tammann . [ 1 ] It honors members of the Deutsche Gesellschaft für Materialkunde , who have made outstanding contributions to the field of materials research . [ 1 ]
This article about materials science is a stub . You can help Wikipedia by expanding it .
This science awards article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tammann_Commemorative_Medal |
The Tammann temperature (also spelled Tamman temperature ) and the Hüttig temperature of a given solid material are approximations to the absolute temperatures at which atoms in a bulk crystal lattice (Tammann) or on the surface (Hüttig) of the solid material become sufficiently mobile to diffuse readily, and are consequently more chemically reactive and susceptible to recrystallization, agglomeration or sintering . [ 1 ] [ 2 ] These temperatures are equal to one-half (Tammann) or one-third (Hüttig) of the absolute temperature of the compound's melting point . The absolute temperatures are usually measured in Kelvin .
Tammann and Hüttig temperatures are important for considerations in catalytic activity , segregation and sintering of solid materials. The Tammann temperature is important for reactive compounds like explosives and fuel oxiders , such as potassium chlorate ( KClO 3 , T Tammann = 42 °C), potassium nitrate ( KNO 3 , T Tammann = 31 °C), and sodium nitrate (NaNO 3 , T Tammann = 17 °C), which may unexpectedly react at much lower temperatures than their melting or decomposition temperatures. [ 1 ] : 152 [ 3 ] : 502
The bulk compounds should be contrasted with nanoparticles which exhibit melting-point depression , meaning that they have significantly lower melting points than the bulk material, and correspondingly lower Tammann and Hüttig temperatures. [ 4 ] For instance, 2 nm gold nanoparticles melt at only about 327 °C, in contrast to 1065 °C for a bulk gold. [ 4 ]
Tammann temperature was pioneered by German astronomer, solid-state chemistry , and physics professor Gustav Tammann in the first half of the 20th century. [ 1 ] : 152 He had considered a lattice motion very important for the reactivity of matter and quantified his theory by calculating a ratio of the given material temperatures at solid-liquid phases at absolute temperatures. The division of a solid's temperature by a melting point would yield a Tammann temperature. The value is usually measured in Kelvins (K): [ 1 ] : 152
where β {\displaystyle {\beta }} is a constant dimensionless number.
The threshold temperature for activation and diffusion of atoms at surfaces was studied by de:Gustav Franz Hüttig , physical chemist on the faculty of Graz University of Technology , who wrote in 1948 (translated from German): [ 6 ] [ 7 ]
In the solid state the atoms oscillate about their position in the lattice. ... There are always some atoms which happen to be highly energized. Such an atom may become dislodged and switch places with another one (exchange reaction) or it may, for a time, travel about aimlessly. ... the number of diffusing atoms increases with rising temperature, first slowly, and in the higher temperature ranges more rapidly. For every metal there is a definite temperature at which the exchange process is suddenly accelerated. The relationship between this temperature and the melting point in degrees K is constant for all metals. ... On the basis of these elementary processes, sintering is analyzed in relation to the coefficient α which is the fraction of the melting point in degrees K ... When α is between 0.23 and 0.36, activation as a result of the surface diffusion takes place. Loosening or release of adsorbed gasses occurs simultaneously.
The Hüttig temperature for a given material is
where T mp {\displaystyle T_{\text{mp}}} is the absolute temperature of the material's bulk melting point (usually specified in Kelvin units) and α {\displaystyle \alpha } is a unitless constant that is independent of the material, having the value α = 0.3 {\displaystyle \alpha =0.3} according to some sources, [ 4 ] [ 8 ] or α = 1 / 3 {\displaystyle \alpha =1/3} according to other sources. [ 9 ] [ 10 ] [ 11 ] It is an approximation to the temperature necessary for a metal or metal oxide surfaces to show significant atomic diffusion along the surface, sintering, and surface recrystallization. Desorption of adsorbed gasses and chemical reactivity of the surface often increase markedly as the temperature is increases above the Hüttig temperature.
The Tammann temperature for a given material is
where β {\displaystyle \beta } is a unitless constant usually taken to be 0.5 {\displaystyle 0.5} , regardless of the material. [ 4 ] [ 8 ] [ 9 ] [ 10 ] It is an approximation to the temperature necessary for mobility and diffusion of atoms, ions, and defects within a bulk crystal. Bulk chemical reactivity often increase markedly as the temperature is increased above the Tammann temperature.
The following table gives an example Tammann and Hüttig temperatures calculated from each compound's melting point T mp according to: | https://en.wikipedia.org/wiki/Tammann_and_Hüttig_temperatures |
TamoGraph Site Survey is an application for performing Wi-Fi site surveys and RF planning. It supports 802.11be, 802.11ax, 802.11ac, 802.11n, 802.11a, 802.11b, and 802.11g wireless networks. TamoGraph is developed by TamoSoft, a privately held New Zealand company founded in 1998 [ 1 ] that specializes in network analysis software. [ 2 ] [ 3 ] [ 4 ]
TamoGraph is used for measuring and visualizing such WLAN characteristics [ 5 ] as signal strength, signal-to-noise ratio, signal-to-interference ratio, TCP and UDP throughput rates , access point vendor, encryption type, [ citation needed ] etc. [ 6 ] Visualizations are overlaid on floor plans [ 7 ] or, in case of outdoor surveys, on site maps that can be imported from one of the online map services. Data is collected by a portable computer using a compatible Wi-Fi adapter. [ 8 ] [ unreliable source? ]
When performing planning of Wi-Fi networks, the tool can be used for creating a virtual model of a future network, [ 9 ] where walls and other obstructions are drawn using different material types (drywall, glass, brick, etc.). [ 10 ] The tool will then calculate the approximate locations of where the access points should be placed. At the post-deployment stage, TamoGraph is used to validate the deployed wireless network, as well as to measure interference from both Wi-Fi and non-Wi-Fi sources with the help of Wi-Spy or WiPry, USB-based spectrum analyzers. [ 10 ] | https://en.wikipedia.org/wiki/TamoGraph_Site_Survey |
In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) [ 1 ] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among the simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis .
The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function , and an analog among the hyperbolic functions .
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are " sin " for sine, " cos " for cosine, " tan " or " tg " for tangent, " sec " for secant, " csc " or " cosec " for cosecant, and " cot " or " ctg " for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean ( sin x ) + y , {\displaystyle (\sin x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).}
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote ( sin x ) 2 , {\displaystyle (\sin x)^{2},} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from the (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
In contrast, the superscript − 1 {\displaystyle -1} is commonly used to denote the inverse function , not the reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case, the superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use.
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. [ 2 ] [ 3 ]
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics ).
However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, [ 5 ] or as solutions to differential equations given particular initial values [ 6 ] ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. [ 7 ] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), [ 8 ] and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. [ 9 ] For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. [ 10 ]
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects the unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and the line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is,
In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity .
The other trigonometric functions can be found along the unit circle as
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change the position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities
hold for any angle θ and any integer k . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after a rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities
hold for any angle θ and any integer k .
The algebraic expressions for the most important angles are as follows:
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. [ 13 ]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. [ 14 ] Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
Sine and cosine can be defined as the unique solution to the initial value problem : [ 17 ]
Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of the same ordinary differential equation
Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 .
One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi } is then a definition of the real number π {\displaystyle \pi } which is independent of geometry.
Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} ,
so the tangent function satisfies the ordinary differential equation
It is the unique solution with y (0) = 0 .
The basic trigonometric functions can be defined by the following power series expansions. [ 18 ] These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane .
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. [ 19 ]
More precisely, defining
one has the following series expansions: [ 20 ]
The following continued fractions are valid in the whole complex plane:
The last one was used in the historically first proof that π is irrational . [ 21 ]
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [ 22 ]
This identity can be proved with the Herglotz trick. [ 23 ] Combining the (– n ) th with the n th term lead to absolutely convergent series:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: [ 24 ]
This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which is the logarithmic derivative of sin z {\displaystyle \sin z} . [ 25 ] From this, it can be deduced also that
Euler's formula relates sine and cosine to the exponential function :
This formula is commonly considered for real values of x , but it remains true for all complex values.
Proof : Let f 1 ( x ) = cos x + i sin x , {\displaystyle f_{1}(x)=\cos x+i\sin x,} and f 2 ( x ) = e i x . {\displaystyle f_{2}(x)=e^{ix}.} One has d f j ( x ) / d x = i f j ( x ) {\displaystyle df_{j}(x)/dx=if_{j}(x)} for j = 1, 2 . The quotient rule implies thus that d / d x ( f 1 ( x ) / f 2 ( x ) ) = 0 {\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0} . Therefore, f 1 ( x ) / f 2 ( x ) {\displaystyle f_{1}(x)/f_{2}(x)} is a constant function, which equals 1 , as f 1 ( 0 ) = f 2 ( 0 ) = 1. {\displaystyle f_{1}(0)=f_{2}(0)=1.} This proves the formula.
One has
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
When x is real, this may be rewritten as
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e a + b = e a e b {\displaystyle e^{a+b}=e^{a}e^{b}} for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups . [ 26 ] The set U {\displaystyle U} of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , via an isomorphism e : R / Z → U . {\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.} In pedestrian terms e ( t ) = exp ( 2 π i t ) {\displaystyle e(t)=\exp(2\pi it)} , and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number a {\displaystyle a} (the base ), the function t ↦ e ( t / a ) {\displaystyle t\mapsto e(t/a)} defines an isomorphism of the group R / a Z → U {\displaystyle \mathbb {R} /a\mathbb {Z} \to U} . The real and imaginary parts of e ( t / a ) {\displaystyle e(t/a)} are the cosine and sine, where a {\displaystyle a} is used as the base for measuring angles. For example, when a = 2 π {\displaystyle a=2\pi } , we get the measure in radians, and the usual trigonometric functions. When a = 360 {\displaystyle a=360} , we get the sine and cosine of angles measured in degrees.
Note that a = 2 π {\displaystyle a=2\pi } is the unique value at which the derivative d d t e ( t / a ) {\displaystyle {\frac {d}{dt}}e(t/a)} becomes a unit vector with positive imaginary part at t = 0 {\displaystyle t=0} . This fact can, in turn, be used to define the constant 2 π {\displaystyle 2\pi } .
Another way to define the trigonometric functions in analysis is using integration. [ 14 ] [ 27 ] For a real number t {\displaystyle t} , put θ ( t ) = ∫ 0 t d τ 1 + τ 2 = arctan t {\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t} where this defines this inverse tangent function. Also, π {\displaystyle \pi } is defined by 1 2 π = ∫ 0 ∞ d τ 1 + τ 2 {\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}} a definition that goes back to Karl Weierstrass . [ 28 ]
On the interval − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} , the trigonometric functions are defined by inverting the relation θ = arctan t {\displaystyle \theta =\arctan t} . Thus we define the trigonometric functions by tan θ = t , cos θ = ( 1 + t 2 ) − 1 / 2 , sin θ = t ( 1 + t 2 ) − 1 / 2 {\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}} where the point ( t , θ ) {\displaystyle (t,\theta )} is on the graph of θ = arctan t {\displaystyle \theta =\arctan t} and the positive square root is taken.
This defines the trigonometric functions on ( − π / 2 , π / 2 ) {\displaystyle (-\pi /2,\pi /2)} . The definition can be extended to all real numbers by first observing that, as θ → π / 2 {\displaystyle \theta \to \pi /2} , t → ∞ {\displaystyle t\to \infty } , and so cos θ = ( 1 + t 2 ) − 1 / 2 → 0 {\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0} and sin θ = t ( 1 + t 2 ) − 1 / 2 → 1 {\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1} . Thus cos θ {\displaystyle \cos \theta } and sin θ {\displaystyle \sin \theta } are extended continuously so that cos ( π / 2 ) = 0 , sin ( π / 2 ) = 1 {\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1} . Now the conditions cos ( θ + π ) = − cos ( θ ) {\displaystyle \cos(\theta +\pi )=-\cos(\theta )} and sin ( θ + π ) = − sin ( θ ) {\displaystyle \sin(\theta +\pi )=-\sin(\theta )} define the sine and cosine as periodic functions with period 2 π {\displaystyle 2\pi } , for all real numbers.
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, arctan s + arctan t = arctan s + t 1 − s t {\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}} holds, provided arctan s + arctan t ∈ ( − π / 2 , π / 2 ) {\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)} , since arctan s + arctan t = ∫ − s t d τ 1 + τ 2 = ∫ 0 s + t 1 − s t d τ 1 + τ 2 {\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}} after the substitution τ → s + τ 1 − s τ {\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}} . In particular, the limiting case as s → ∞ {\displaystyle s\to \infty } gives arctan t + π 2 = arctan ( − 1 / t ) , t ∈ ( − ∞ , 0 ) . {\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).} Thus we have sin ( θ + π 2 ) = − 1 t 1 + ( − 1 / t ) 2 = − 1 1 + t 2 = − cos ( θ ) {\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )} and cos ( θ + π 2 ) = 1 1 + ( − 1 / t ) 2 = t 1 + t 2 = sin ( θ ) . {\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).} So the sine and cosine functions are related by translation over a quarter period π / 2 {\displaystyle \pi /2} .
One can also define the trigonometric functions using various functional equations .
For example, [ 29 ] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
and the added condition
The sine and cosine of a complex number z = x + i y {\displaystyle z=x+iy} can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z {\displaystyle z} becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
sin z {\displaystyle \sin z\,}
cos z {\displaystyle \cos z\,}
tan z {\displaystyle \tan z\,}
cot z {\displaystyle \cot z\,}
sec z {\displaystyle \sec z\,}
csc z {\displaystyle \csc z\,}
The sine and cosine functions are periodic , with period 2 π {\displaystyle 2\pi } , which is the smallest positive period: sin ( z + 2 π ) = sin ( z ) , cos ( z + 2 π ) = cos ( z ) . {\displaystyle \sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).} Consequently, the cosecant and secant also have 2 π {\displaystyle 2\pi } as their period.
The functions sine and cosine also have semiperiods π {\displaystyle \pi } , and sin ( z + π ) = − sin ( z ) , cos ( z + π ) = − cos ( z ) {\displaystyle \sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)} and consequently tan ( z + π ) = tan ( z ) , cot ( z + π ) = cot ( z ) . {\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).} Also, sin ( x + π / 2 ) = cos ( x ) , cos ( x + π / 2 ) = − sin ( x ) {\displaystyle \sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)} (see Complementary angles ).
The function sin ( z ) {\displaystyle \sin(z)} has a unique zero (at z = 0 {\displaystyle z=0} ) in the strip − π < ℜ ( z ) < π {\displaystyle -\pi <\Re (z)<\pi } . The function cos ( z ) {\displaystyle \cos(z)} has the pair of zeros z = ± π / 2 {\displaystyle z=\pm \pi /2} in the same strip. Because of the periodicity, the zeros of sine are π Z = { … , − 2 π , − π , 0 , π , 2 π , … } ⊂ C . {\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .} There zeros of cosine are π 2 + π Z = { … , − 3 π 2 , − π 2 , π 2 , 3 π 2 , … } ⊂ C . {\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .} All of the zeros are simple zeros, and both functions have derivative ± 1 {\displaystyle \pm 1} at each of the zeros.
The tangent function tan ( z ) = sin ( z ) / cos ( z ) {\displaystyle \tan(z)=\sin(z)/\cos(z)} has a simple zero at z = 0 {\displaystyle z=0} and vertical asymptotes at z = ± π / 2 {\displaystyle z=\pm \pi /2} , where it has a simple pole of residue − 1 {\displaystyle -1} . Again, owing to the periodicity, the zeros are all the integer multiples of π {\displaystyle \pi } and the poles are odd multiples of π / 2 {\displaystyle \pi /2} , all having the same residue. The poles correspond to vertical asymptotes lim x → π − tan ( x ) = + ∞ , lim x → π + tan ( x ) = − ∞ . {\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
The cotangent function cot ( z ) = cos ( z ) / sin ( z ) {\displaystyle \cot(z)=\cos(z)/\sin(z)} has a simple pole of residue 1 at the integer multiples of π {\displaystyle \pi } and simple zeros at odd multiples of π / 2 {\displaystyle \pi /2} . The poles correspond to vertical asymptotes lim x → 0 − cot ( x ) = − ∞ , lim x → 0 + cot ( x ) = + ∞ . {\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities . These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π /2] , see Proofs of trigonometric identities ). For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
The cosine and the secant are even functions ; the other trigonometric functions are odd functions . That is:
All trigonometric functions are periodic functions of period 2 π . This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k , one has
See Periodicity and asymptotes .
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
Dividing through by either cos 2 x {\displaystyle \cos ^{2}x} or sin 2 x {\displaystyle \sin ^{2}x} gives
and
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy (see Angle sum and difference identities ). One can also produce them algebraically using Euler's formula .
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae .
These identities can be used to derive the product-to-sum identities .
By setting t = tan 1 2 θ , {\displaystyle t=\tan {\tfrac {1}{2}}\theta ,} all trigonometric functions of θ {\displaystyle \theta } can be expressed as rational fractions of t {\displaystyle t} :
Together with
this is the tangent half-angle substitution , which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule . The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration .
Note: For 0 < x < π {\displaystyle 0<x<\pi } the integral of csc x {\displaystyle \csc x} can also be written as − arsinh ( cot x ) , {\displaystyle -\operatorname {arsinh} (\cot x),} and for the integral of sec x {\displaystyle \sec x} for − π / 2 < x < π / 2 {\displaystyle -\pi /2<x<\pi /2} as arsinh ( tan x ) , {\displaystyle \operatorname {arsinh} (\tan x),} where arsinh {\displaystyle \operatorname {arsinh} } is the inverse hyperbolic sine .
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function . However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions . To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
The notations sin −1 , cos −1 , etc. are often used for arcsin and arccos , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms .
In this section A , B , C denote the three (interior) angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C : sin A a = sin B b = sin C c = 2 Δ a b c , {\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},} where Δ is the area of the triangle,
or, equivalently, a sin A = b sin B = c sin C = 2 R , {\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,} where R is the triangle's circumradius .
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem : c 2 = a 2 + b 2 − 2 a b cos C , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,} or equivalently, cos C = a 2 + b 2 − c 2 2 a b . {\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
In this formula the angle at C is opposite to the side c . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem .
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
The law of tangents says that:
If s is the triangle's semiperimeter, ( a + b + c )/2, and r is the radius of the triangle's incircle , then rs is the triangle's area. Therefore Heron's formula implies that:
The law of cotangents says that: [ 30 ]
It follows that
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion , which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion .
Trigonometric functions also prove to be useful in the study of general periodic functions . The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves . [ 31 ]
Under rather general conditions, a periodic function f ( x ) can be expressed as a sum of sine waves or cosine waves in a Fourier series . [ 32 ] Denoting the sine or cosine basis functions by φ k , the expansion of the periodic function f ( t ) takes the form: f ( t ) = ∑ k = 1 ∞ c k φ k ( t ) . {\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
For example, the square wave can be written as the Fourier series f square ( t ) = 4 π ∑ k = 1 ∞ sin ( ( 2 k − 1 ) t ) 2 k − 1 . {\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy ( Aryabhatiya , Surya Siddhanta ), via translation from Sanskrit to Arabic and then from Arabic to Latin. [ 33 ] (See Aryabhata's sine table .)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines , used in solving triangles . [ 34 ] Al-Khwārizmī (c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables. [ 35 ] [ 36 ] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. [ 36 ] The trigonometric functions were later studied by mathematicians including Omar Khayyám , Bhāskara II , Nasir al-Din al-Tusi , Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus , and Rheticus' student Valentinus Otho .
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series . [ 37 ] (See Madhava series and Madhava's sine table .)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates. [ 38 ]
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). [ 39 ]
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin , cos , and tan in his book Trigonométrie . [ 40 ]
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x . [ 41 ] Though defined as ratios of sides of a right triangle , and thus appearing to be rational functions , Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series . He presented " Euler's formula ", as well as near-modern abbreviations ( sin. , cos. , tang. , cot. , sec. , and cosec. ). [ 33 ]
A few functions were common historically, but are now seldom used, such as the chord , versine (which appeared in the earliest tables [ 33 ] ), haversine , coversine , [ 42 ] half-tangent (tangent of half an angle), and exsecant . List of trigonometric identities shows more relations between these functions.
Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent. [ 43 ] [ 44 ] [ 45 ] [ 46 ]
The word sine derives [ 47 ] from Latin sinus , meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga ", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib , meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin . [ 48 ] The choice was based on a misreading of the Arabic written form j-y-b ( جيب ), which itself originated as a transliteration from Sanskrit jīvā , which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". [ 49 ]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans —"cutting"—since the line cuts the circle. [ 50 ]
The prefix " co- " (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter 's Canon triangulorum (1620), which defines the cosinus as an abbreviation of the sinus complementi (sine of the complementary angle ) and proceeds to define the cotangens similarly. [ 51 ] [ 52 ] | https://en.wikipedia.org/wiki/Tan_(trigonometry) |
In coordination chemistry , Tanabe–Sugano diagrams are used to predict absorptions in the ultraviolet (UV), visible and infrared (IR) electromagnetic spectrum of coordination compounds . The results from a Tanabe–Sugano diagram analysis of a metal complex can also be compared to experimental spectroscopic data. They are qualitatively useful and can be used to approximate the value of 10Dq, the ligand field splitting energy. Tanabe–Sugano diagrams can be used for both high spin and low spin complexes, unlike Orgel diagrams , which apply only to high spin complexes. Tanabe–Sugano diagrams can also be used to predict the size of the ligand field necessary to cause high-spin to low-spin transitions.
In a Tanabe–Sugano diagram, the ground state is used as a constant reference, in contrast to Orgel diagrams. The energy of the ground state is taken to be zero for all field strengths, and the energies of all other terms and their components are plotted with respect to the ground term.
Until Yukito Tanabe and Satoru Sugano published their paper "On the absorption spectra of complex ions", in 1954, little was known about the excited electronic states of complex metal ions . They used Hans Bethe 's crystal field theory and Giulio Racah 's linear combinations of Slater integrals , [ 1 ] now called Racah parameters , to explain the absorption spectra of octahedral complex ions in a more quantitative way than had been achieved previously. [ 2 ] Many spectroscopic experiments later, they estimated the values for two of Racah's parameters, B and C, for each d-electron configuration based on the trends in the absorption spectra of isoelectronic first-row transition metals . The plots of the energies calculated for the electronic states of each electron configuration are now known as Tanabe–Sugano diagrams. [ 3 ] [ 4 ] Number must be fit for each octahedral coordination complex because the C / B can deviate strongly from the theoretical value of 4.0. This ratio changes the relative energies of the levels in the Tanabe–Sugano diagrams, and thus the diagrams may vary slightly between sources depending on what C / B ratio was selected when plotting.
The x-axis of a Tanabe–Sugano diagram is expressed in terms of the ligand field splitting parameter , Δ, or Dq (for "differential of quanta" [ 5 ] [ 6 ] ), divided by the Racah parameter B. The y-axis is in terms of energy, E, also scaled by B. Three Racah parameters exist, A, B, and C, which describe various aspects of interelectronic repulsion. A is an average total interelectron repulsion. B and C correspond with individual d-electron repulsions. A is constant among d-electron configuration, and it is not necessary for calculating relative energies, hence its absence from Tanabe and Sugano's studies of complex ions. C is necessary only in certain cases. B is the most important of Racah's parameters in this case. [ 7 ] One line corresponds to each electronic state. The bending of certain lines is due to the mixing of terms with the same symmetry. Although electronic transitions are only "allowed" if the spin multiplicity remains the same (i.e. electrons do not change from spin up to spin down or vice versa when moving from one energy level to another), energy levels for "spin-forbidden" electronic states are included in the diagrams, which are also not included in Orgel diagrams. [ 8 ] Each state is given its molecular-symmetry label (e.g. A 1g , T 2g , etc.), but "g" and "u" subscripts are usually left off because it is understood that all the states are gerade . Labels for each state are usually written on the right side of the table, though for more complicated diagrams (e.g. d 6 ) labels may be written in other locations for clarity. Term symbols (e.g. 3 P, 1 S, etc.) for a specific d n free ion are listed, in order of increasing energy, on the y-axis of the diagram. The relative order of energies is determined using Hund's rules . For an octahedral complex, the spherical, free ion term symbols split accordingly: [ 9 ]
Certain Tanabe–Sugano diagrams (d 4 , d 5 , d 6 , and d 7 ) also have a vertical line drawn at a specific Dq/B value, which is accompanied by a discontinuity in the slopes of the excited states' energy levels. This pucker in the lines occurs when the identity of the ground state changes, shown in the diagram below. The left depicts the relative energies of the d 7 ion states as functions of crystal field strength ( Dq ), showing an intersection of the 4 T 1 and the 2 E states near Dq/B ~ 2.1. Subtracting the ground state energy produces the standard Tanabe–Sugano diagram shown on the right.
This change in identity generally happens when the spin pairing energy, P, is equal to the ligand field splitting energy, Dq. Complexes to the left of this line (lower Dq/B values) are high-spin, while complexes to the right (higher Dq/B values) are low-spin. There is no low-spin or high-spin designation for d 2 , d 3 , or d 8 because none of the states cross at reasonable crystal field energies. [ 10 ]
The seven Tanabe–Sugano diagrams for octahedral complexes are shown below. [ 7 ] [ 11 ] [ 12 ]
There is no electron repulsion in a d 1 complex, and the single electron resides in the t 2g orbital ground state. A d 1 octahedral metal complex, such as [Ti(H 2 O) 6 ] 3+ , shows a single absorption band in a UV-vis experiment. [ 7 ] The term symbol for d 1 is 2 D, which splits into the 2 T 2g and 2 E g states. The t 2g orbital set holds the single electron and has a 2 T 2g state energy of -4Dq. When that electron is promoted to an e g orbital, it is excited to the 2 E g state energy, +6Dq. This is in accordance with the single absorption band in a UV-vis experiment. The prominent shoulder in this absorption band is due to a Jahn–Teller distortion which removes the degeneracy of the two 2 E g states. However, since these two transitions overlap in a UV-vis spectrum, this transition from 2 T 2g to 2 E g does not require a Tanabe–Sugano diagram.
Similar to d 1 metal complexes, d 9 octahedral metal complexes have 2 D spectral term. The transition is from the (t 2g ) 6 (e g ) 3 configuration ( 2 E g state) to the (t 2g ) 5 (e g ) 4 configuration ( 2 T 2g state). This could also be described as a positive "hole" that moves from the e g to the t 2g orbital set. The sign of Dq is opposite that for d 1 , with a 2 E g ground state and a 2 T 2g excited state. Like the d 1 case, d 9 octahedral complexes do not require the Tanabe–Sugano diagram to predict their absorption spectra.
There are no d-d electron transitions in d 10 metal complexes because the d orbitals are completely filled. Thus, UV-vis absorption bands are not observed and a Tanabe–Sugano diagram does not exist.
Tetrahedral Tanabe–Sugano diagrams are generally not found in textbooks because the diagram for a d n tetrahedral will be similar to that for d (10-n) octahedral, remembering that Δ T for tetrahedral complexes is approximately 4/9 of Δ O for an octahedral complex. A consequence of the much smaller size of Δ T results in (almost) all tetrahedral complexes being high spin and therefore the change in the ground state term seen on the X-axis for octahedral d 4 -d 7 diagrams is not required for interpreting spectra of tetrahedral complexes.
In Orgel diagrams , the magnitude of the splitting energy exerted by the ligands on d orbitals, as a free ion approach a ligand field, is compared to the electron-repulsion energy, which are both sufficient at providing the placement of electrons. However, if the ligand field splitting energy, 10Dq, is greater than the electron-repulsion energy, then Orgel diagrams fail in determining electron placement. In this case, Orgel diagrams are restricted to only high spin complexes. [ 8 ]
Tanabe–Sugano diagrams do not have this restriction, and can be applied to situations when 10Dq is significantly greater than electron repulsion. Thus, Tanabe–Sugano diagrams are utilized in determining electron placements for high spin and low spin metal complexes. However, they are limited in that they have only qualitative significance. Even so, Tanabe–Sugano diagrams are useful in interpreting UV-vis spectra and determining the value of 10Dq. [ 8 ]
In a centrosymmetric ligand field, such as in octahedral complexes of transition metals, the arrangement of electrons in the d-orbital is not only limited by electron repulsion energy, but it is also related to the splitting of the orbitals due to the ligand field. This leads to many more electron configuration states than is the case for the free ion. The relative energy of the repulsion energy and splitting energy defines the high-spin and low-spin states .
Considering both weak and strong ligand fields, a Tanabe–Sugano diagram shows the energy splitting of the spectral terms with the increase of the ligand field strength. It is possible for us to understand how the energy of the different configuration states is distributed at certain ligand strengths. The restriction of the spin selection rule makes it even easier to predict the possible transitions and their relative intensity. Although they are qualitative, Tanabe–Sugano diagrams are very useful tools for analyzing UV-vis spectra: they are used to assign bands and calculate Dq values for ligand field splitting. [ 13 ] [ 14 ]
In the [Mn(H 2 O) 6 ] 2+ metal complex, manganese has an oxidation state of +2, thus it is a d 5 ion. H 2 O is a weak field ligand (spectrum shown below), and according to the Tanabe–Sugano diagram for d 5 ions, the ground state is 6 A 1 . Note that there is no sextet spin multiplicity in any excited state, hence the transitions from this ground state are expected to be spin-forbidden and the band intensities should be low. From the spectra, only very low intensity bands are observed (low molar absorptivity (ε) values on y-axis). [ 13 ]
Another example is [Co(H 2 O) 6 ] 2+ . [ 14 ] Note that the ligand is the same as the last example. Here the cobalt ion has the oxidation state of +2, and it is a d 7 ion. From the high-spin (left) side of the d 7 Tanabe–Sugano diagram, the ground state is 4 T 1 (F), and the spin multiplicity is a quartet. The diagram shows that there are three quartet excited states: 4 T 2 , 4 A 2 , and 4 T 1 (P). From the diagram one can predict that there are three spin-allowed transitions. However, the spectrum of [Co(H 2 O) 6 ] 2+ does not show three distinct peaks that correspond to the three predicted excited states. Instead, the spectrum has a broad peak (spectrum shown below). Based on the T–S diagram, the lowest energy transition is 4 T 1 to 4 T 2 , which is seen in the near IR and is not observed in the visible spectrum. The main peak is the energy transition 4 T 1 (F) to 4 T 1 (P), and the slightly higher energy transition (the shoulder) is predicted to be 4 T 1 to 4 A 2 . The small energy difference leads to the overlap of the two peaks, which explains the broad peak observed in the visible spectrum. | https://en.wikipedia.org/wiki/Tanabe–Sugano_diagram |
In the stochastic calculus , Tanaka's formula for the Brownian motion states that
where B t is the standard Brownian motion, sgn denotes the sign function
and L t is its local time at 0 (the local time spent by B at 0 before time t ) given by the L 2 -limit
One can also extend the formula to semimartingales .
Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale | B t | into the martingale part (the integral on the right-hand side, which is a Brownian motion [ 1 ] ), and a continuous increasing process (local time). It can also be seen as the analogue of Itō's lemma for the (nonsmooth) absolute value function f ( x ) = | x | {\displaystyle f(x)=|x|} , with f ′ ( x ) = sgn ( x ) {\displaystyle f'(x)=\operatorname {sgn}(x)} and f ″ ( x ) = 2 δ ( x ) {\displaystyle f''(x)=2\delta (x)} ; see local time for a formal explanation of the Itō term.
The function | x | is not C 2 in x at x = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero (i.e. in [− ε , ε ]) by parabolas
and use Itō's formula , we can then take the limit as ε → 0, leading to Tanaka's formula. | https://en.wikipedia.org/wiki/Tanaka's_formula |
Tandem affinity purification ( TAP ) is an immunoprecipitation -based purification technique for studying protein–protein interactions . The goal is to extract from a cell only the protein of interest, in complex with any other proteins it interacted with. TAP uses two types of agarose beads that bind to the protein of interest and that can be separated from the cell lysate by centrifugation, without disturbing, denaturing or contaminating the involved complexes. To enable the protein of interest to bind to the beads, it is tagged with a designed piece, the TAP tag.
The original TAP method involves the fusion of the TAP tag to the C-terminus of the protein under study. The TAP tag consists of three components: a calmodulin binding peptide (CBP), TEV protease cleavage site , and two Protein A domains, which bind tightly to IgG (making a TAP tag a type of epitope tag). [ 1 ]
Many other tag/bead/eluent combinations have been proposed since the TAP principle was first published.
This tag is also known as the C-terminal TAP tag because an N-terminal version is also available. However, the method to be described assumes the use of a C-terminal tag, although the principle behind the method is still the same.
TAP tagging was invented by a research team working in the European Molecular Biology Laboratory in the late 1990s (Rigaut et al., 1999, [ 2 ] Puig et al.,2001 [ 3 ] ) and proposed as a new tool for proteome exploration. It was used by the team to characterize several protein complexes (Rigaut et al., 1999, [ 2 ] Caspary et al. 1999, [ 4 ] Bouveret et al., 2000, [ 5 ] Puig et al., 2001 [ 3 ] ). The first large-scale application of this technique was in 2002, in which the research team worked in collaboration with scientists of the proteomics company Cellzome to develop a visual map of the interaction of more than 230 multi-protein complexes in a yeast cell by systematically tagging the TAP tag to each protein. The first successful report of using TAP tag technology in plants came in 2004 (Rohila et al., 2004, [ 6 ] )
There are a few methods in which the fusion protein can be introduced into the host cells. If the host is yeast , then one of the methods may be the use of plasmids that will eventually translate the fusion protein within the host. Whichever method that is being used, it is preferable to maintain expression of the fusion protein as close as possible to its natural level. Once the fusion protein is translated within the host, it will interact with other proteins, ideally in a manner unaffected by the TAP tag.
Subsequently, the tagged protein (with its binding partners) is retrieved using an affinity selection process.
The first type of bead added is coated with Immunoglobulin G , which binds to the TAP tag's outermost end. The beads, with the proteins of interest, are separated from the lysate via centrifugation. The proteins are then released from the beads by an enzyme ( TEV protease ) which breaks the tag at the TEV cleavage site in the middle.
After this first purification step, a second type of bead (coated with calmodulin ) is added to the released proteins which binds reversibly to the remaining piece of the TAP tag still on the proteins. The beads are again separated by centrifugation, further removing contaminants as well as the TEV protease. [ 3 ] Finally, the beads are released by EGTA , leaving behind the native eluate containing only the protein of interest, its bound protein partners and the remaining CBP piece of the TAP tag.
The native eluate can then be analyzed using gel electrophoresis and mass spectrometry to identify the protein's binding partners.
An advantage of this method is that there can be real determination of protein partners quantitatively in vivo without prior knowledge of complex composition. It is also simple to execute and often provides high yield. [ 3 ] One of the obstacles of studying protein protein interaction is the contamination of the target protein especially when we don’t have any prior knowledge of it. TAP offers an effective, and highly specific means to purify target protein. After 2 successive affinity purifications, the chance for contaminants to be retained in the eluate reduces significantly.
However, there is also the possibility that a tag added to a protein might obscure binding of the new protein to its interacting partners. In addition, the tag may also affect protein expression levels. On the other hand, the tag may also not be sufficiently exposed to the affinity beads, hence skewing the results.
There may also be a possibility of a cleavage of the proteins by the TEV protease , although this is unlikely to be frequent given the high specificity of the TEV protease. [ 7 ]
As this method involves at least 2 rounds of washing, it may not be suitable for screening transient protein interactions , unlike the yeast two-hybrid method or in vivo crosslinking with photo-reactive amino acid analogs . However, it is a good method for testing stable protein interactions and allows various degrees of investigation by controlling the number of times the protein complex is purified. [ citation needed ]
In 2002, the TAP tag was first used with mass spectrometry in a large-scale approach to systematically analyse the proteomics of yeast by characterizing multiprotein complexes. [ 8 ] The study revealed 491 complexes, 257 of them wholly new. The rest were familiar from other research, but now virtually all of them were found to have new components. They drew up a map relating all the protein components functionally in a complex network.
Many other proteomic analyses also involve the use of TAP tag. A research by EMBO (Dziembowski, 2004) identified a new complex required for nuclear pre-mRNA retention and splicing. They have purified a novel trimeric complex composed of 3 other subunits (Snu17p, Bud13p and Pml1p) and find that these subunits are not essential for viability but required for efficient splicing (removal of introns) of pre-mRNA. In 2006, Fleischer et al. systematically identified proteins associated with eukaryotic ribosomal complexes. [ 9 ] They used multifaceted mass spectrometry proteomic screens to identify yeast ribosomal complexes and then used TAP tagging to functionally link up all these proteins.
The principle of tandem-affinity purification of multiprotein complexes is not limited to the combination of CBP and Protein A tags used in the original work by Rigaut et al. (1999). For example, the combination of FLAG- and HA-tags has been used since 2000 by the group of Nakatani [ 10 ] [ 11 ] to purify numerous protein complexes from mammalian cells. Many other tag combinations have been proposed since the TAP principle was published. | https://en.wikipedia.org/wiki/Tandem_affinity_purification |
Tandem exon duplication is defined as duplication of exons within the same gene to give rise to the subsequent exon. A complete exon analysis of all genes in Homo sapiens , Drosophila melanogaster , and Caenorhabditis elegans has shown 12,291 instances of tandem duplication in exons in human , fly , and worm . Analysis of the intronic region has produced further 4,660 unidentified duplicated exons referred to as unannotated exons. 1,578 of these unannotated exons contained stop codons thus not considered potential exons. 35.1% of the unannotated exons were found in the EST sequence thus confirming the potential of the presence of these exons in protein transcripts. [ 1 ]
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tandem_exon_duplication |
In analytical chemistry , a tandem mass tag ( TMT ) is a chemical label that facilitates sample multiplexing in mass spectrometry (MS)-based quantification and identification of biological macromolecules such as proteins , peptides and nucleic acids . TMT belongs to a family of reagents referred to as isobaric mass tags which are a set of molecules with the same mass, but yield reporter ions of differing mass after fragmentation. The relative ratio of the measured reporter ions represents the relative abundance of the tagged molecule, although ion suppression has a detrimental effect on accuracy. [ 1 ] [ 2 ] Despite these complications, TMT-based proteomics has been shown to afford higher precision than label-free quantification . [ 3 ] In addition to aiding in protein quantification, TMT tags can also increase the detection sensitivity of certain highly hydrophilic analytes, such as phosphopeptides, in RPLC -MS analyses. [ 4 ]
There are currently six varieties of TMT available: TMTzero, a non-isotopically substituted core structure; TMTduplex, an isobaric pair of mass tags with a single isotopic substitution; [ 5 ] TMTsixplex, an isobaric set of six mass tags with five isotopic substitutions; [ 6 ] [ non-primary source needed ] TMT 10-plex – a set of 10 isotopic mass tags which use the TMTsixplex reporter region, but use different elemental isotope to create a mass difference of 0.0063 Da, [ 7 ] [ non-primary source needed ] TMTpro a 16 plex version with a different reporter and mass normalizer than the original TMT, and TMTpro Zero.
The tags contain four regions, namely a mass reporter region (M), a cleavable linker region (F), a mass normalization region (N) and a protein reactive group (R). The chemical structures of all the tags are identical but each contains isotopes substituted at various positions, such that the mass reporter and mass normalization regions have different molecular masses in each tag. The combined M-F-N-R regions of the tags have the same total molecular weights and structure so that during chromatographic or electrophoretic separation and in single MS mode, molecules labelled with different tags are indistinguishable. Upon fragmentation in MS/MS mode, sequence information is obtained from fragmentation of the peptide back bone and quantification data are simultaneously obtained from fragmentation of the tags, giving rise to mass reporter ions.
The structures of TMT tags are publicly available through the unimod database at unimod.org and hence, mass spectrometry software such as Mascot are able to account for the tag masses. Additionally, as of version 2.2, Mascot has the capability to quantify using TMT and other isobaric mass tags without the use of additional software. Intuitively, the trust associated with a protein measurement depends on the similarity of ratios from different peptides and the signal level of these measurements. A mathematically rigorous approach called BACIQ, that integrates peptide intensities and peptide-measurement agreement into confidence intervals for protein ratios has emerged. [ 8 ] The TKO standard can be used to assess interference [ 9 ] [ non-primary source needed ]
TMT tags are commonly used to label samples of equal abundance. If one of the labeled samples is more abundant, however, it may increase the sensitivity of the analysis for all samples. [ 10 ] Such isobarically labeled samples are referred to as isobaric carriers. They were introduced for single-cell protein analysis by mass spectrometry, [ 11 ] and have found many other applications. [ 12 ] | https://en.wikipedia.org/wiki/Tandem_mass_tag |
Variable number of tandem repeat locus ( VNTR locus) is any DNA sequence that exist in multiple copies strung together in a variety of tandem lengths. [ 1 ] The number of repeat copies present at a locus can be visualized by means of a Multi-locus or Multiple Loci VNTR Analysis ( MLVA ). In short, oligonucleotide primers are developed for each specific tandem repeat locus, followed by PCR and agarose gel electrophoresis . When the length of the repeat and the size of the flanking regions is known, the number of repeats can be calculated. Analysis of multiple loci will result in a genotype . [ 2 ]
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tandem_repeat_locus |
The Tandy Memorex Video Information System ( VIS ) is an interactive, multimedia CD-ROM player produced by the Tandy Corporation starting in 1992. [ 1 ] It is similar in function to the Philips CD-i and Commodore CDTV systems (particularly the CDTV, since both the VIS and CDTV were adaptations of existing computer platforms and operating systems to the set-top-box design). The VIS systems were sold only at Radio Shack , under the Memorex brand, both of which Tandy owned at the time.
Modular Windows is a special version of Microsoft Windows 3.1 , designed to run on the Tandy Video Information System. [ 2 ] Microsoft intended Modular Windows to be an embedded operating system for various devices, especially those designed to be connected to televisions . However, the VIS is the only known product that actually used this Windows version. [ 3 ] It has been claimed that Microsoft created a new, incompatible version of Modular Windows ("1.1") shortly after the VIS shipped. [ 4 ] No products are known to have actually used Modular Windows 1.1.
The VIS was not a successful product; by some reports Radio Shack only sold 11,000 units during the lifetime of the product. [ 5 ] Radio Shack store employees jokingly referred to the VIS as "Virtually Impossible to Sell". [ 1 ] [ 6 ] Tandy discontinued the product in early 1994 and all remaining units were sold to a liquidator.
Details of the system include: [ 5 ]
Additional details: [ 8 ] | https://en.wikipedia.org/wiki/Tandy_Video_Information_System |
Tang Aoqing ( Chinese : 唐敖庆 ; Wade–Giles : T'ang Ao-ch'ing ; 18 November 1915 – 15 July 2008), or Au-Chin Tang , was a Chinese theoretical chemist and educator, known as the "Father of Quantum Chemistry " in China. [ 1 ] He established the Department of Chemistry of Jilin University , and served as President of the university from 1978 to 1986. He was a founding member of the Chinese Academy of Sciences and a member of the International Academy of Quantum Molecular Science (IAQMS). He established the National Natural Science Foundation of China in 1986 and served as its first president.
Tang was born on 18 November 1915 in Yixing , Jiangsu , Republic of China . [ 2 ] He entered the Department of Chemistry of Peking University in the summer of 1936. When the Second Sino-Japanese War broke out in 1937, Beijing came under Japanese attack and Peking University, together with Tsinghua and Nankai universities, evacuated to Kunming in Southwest China. In Kunming, the universities combined their diminished resources to form the temporary National Southwestern Associated University , where Tang continued his studies. After graduating in 1940, he was hired by the university as a faculty member. [ 2 ]
After the end of World War II , Tang was sent to the United States in 1946 to study nuclear physics , together with Tsung-Dao Lee , who would win the Nobel Prize in 1957, and other distinguished scientists. [ 2 ] However, Sino-American relations deteriorated after the Chinese Civil War broke out, and Tang studied chemistry at Columbia University instead of nuclear physics. [ 2 ]
After earning his Ph.D. from Columbia in 1949, Tang returned to the newly established People's Republic of China in early 1950 and became a professor of Peking University. [ 2 ] In 1952, he moved to Changchun to help establish Jilin University (initially called the Northeast People's University). [ 2 ] He founded the university's Department of Chemistry, [ 3 ] and served as Vice President of the university from 1956. [ 2 ] After the Cultural Revolution , he served as President of Jilin University from 1978 to 1986, [ 3 ] and as President Emeritus afterwards until his death. [ 2 ] He was elected a founding member of the Chinese Academy of Sciences in 1955 and a member of the International Academy of Quantum Molecular Science (IAQMS) in 1981. [ 4 ]
Tang's research was mainly focused on quantum chemistry , polymer chemistry, and polymer physics. [ 3 ] In the 1950s, he pioneered a method to calculate the "potential function of molecular internal rotation". [ 2 ] He later made contributions to the ligand field theory and developed three graph theorems of molecular orbital . [ 2 ] [ 3 ] He co-authored eight monographs and was conferred four consecutive State Natural Science Awards (including two first-class awards), an unprecedented achievement. [ 3 ]
Tang is widely considered the "Father of Quantum Chemistry" in China. [ 2 ] Five of his students were elected academicians of the Chinese Academy of Sciences in 1991 and 1993, [ 3 ] and many of his students became leaders of theoretical chemistry in major Chinese universities, including Peking, Nanjing , Xiamen , and Beijing Normal University . [ 2 ]
In 1986, Tang established the National Natural Science Foundation of China and served as its first president. [ 3 ]
Tang died on 15 July 2008 in Beijing, at the age of 92. [ 2 ] [ 4 ] In 2011, the Chinese Chemical Society established the Tang Au-Chin Youth Award on Theoretical Chemistry in his memory. [ 5 ] Asteroid 218914 Tangauchin , discovered by astronomers with the PMO NEO Survey Program at Purple Mountain Observatory in 2007, was named in his memory. [ 1 ] The official naming citation was published by the Minor Planet Center on 9 January 2020 ( M.P.C. 120069 ). [ 6 ]
He was a delegate to the 2nd and 3rd National People's Congress . He was a member of 6th National Committee of the Chinese People's Political Consultative Conference and a member of the 7th and 8th Standing Committee of the Chinese People's Political Consultative Conference . He was a deputy to the 10th , 11th and 12th National Congress of the Chinese Communist Party . | https://en.wikipedia.org/wiki/Tang_Aoqing |
In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) [ 1 ] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among the simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis .
The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function , and an analog among the hyperbolic functions .
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are " sin " for sine, " cos " for cosine, " tan " or " tg " for tangent, " sec " for secant, " csc " or " cosec " for cosecant, and " cot " or " ctg " for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean ( sin x ) + y , {\displaystyle (\sin x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).}
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote ( sin x ) 2 , {\displaystyle (\sin x)^{2},} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from the (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
In contrast, the superscript − 1 {\displaystyle -1} is commonly used to denote the inverse function , not the reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case, the superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use.
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. [ 2 ] [ 3 ]
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics ).
However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, [ 5 ] or as solutions to differential equations given particular initial values [ 6 ] ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. [ 7 ] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), [ 8 ] and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. [ 9 ] For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. [ 10 ]
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects the unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and the line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is,
In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity .
The other trigonometric functions can be found along the unit circle as
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change the position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities
hold for any angle θ and any integer k . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after a rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities
hold for any angle θ and any integer k .
The algebraic expressions for the most important angles are as follows:
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. [ 13 ]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. [ 14 ] Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
Sine and cosine can be defined as the unique solution to the initial value problem : [ 17 ]
Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of the same ordinary differential equation
Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 .
One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi } is then a definition of the real number π {\displaystyle \pi } which is independent of geometry.
Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} ,
so the tangent function satisfies the ordinary differential equation
It is the unique solution with y (0) = 0 .
The basic trigonometric functions can be defined by the following power series expansions. [ 18 ] These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane .
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. [ 19 ]
More precisely, defining
one has the following series expansions: [ 20 ]
The following continued fractions are valid in the whole complex plane:
The last one was used in the historically first proof that π is irrational . [ 21 ]
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [ 22 ]
This identity can be proved with the Herglotz trick. [ 23 ] Combining the (– n ) th with the n th term lead to absolutely convergent series:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: [ 24 ]
This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which is the logarithmic derivative of sin z {\displaystyle \sin z} . [ 25 ] From this, it can be deduced also that
Euler's formula relates sine and cosine to the exponential function :
This formula is commonly considered for real values of x , but it remains true for all complex values.
Proof : Let f 1 ( x ) = cos x + i sin x , {\displaystyle f_{1}(x)=\cos x+i\sin x,} and f 2 ( x ) = e i x . {\displaystyle f_{2}(x)=e^{ix}.} One has d f j ( x ) / d x = i f j ( x ) {\displaystyle df_{j}(x)/dx=if_{j}(x)} for j = 1, 2 . The quotient rule implies thus that d / d x ( f 1 ( x ) / f 2 ( x ) ) = 0 {\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0} . Therefore, f 1 ( x ) / f 2 ( x ) {\displaystyle f_{1}(x)/f_{2}(x)} is a constant function, which equals 1 , as f 1 ( 0 ) = f 2 ( 0 ) = 1. {\displaystyle f_{1}(0)=f_{2}(0)=1.} This proves the formula.
One has
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
When x is real, this may be rewritten as
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e a + b = e a e b {\displaystyle e^{a+b}=e^{a}e^{b}} for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups . [ 26 ] The set U {\displaystyle U} of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , via an isomorphism e : R / Z → U . {\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.} In pedestrian terms e ( t ) = exp ( 2 π i t ) {\displaystyle e(t)=\exp(2\pi it)} , and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number a {\displaystyle a} (the base ), the function t ↦ e ( t / a ) {\displaystyle t\mapsto e(t/a)} defines an isomorphism of the group R / a Z → U {\displaystyle \mathbb {R} /a\mathbb {Z} \to U} . The real and imaginary parts of e ( t / a ) {\displaystyle e(t/a)} are the cosine and sine, where a {\displaystyle a} is used as the base for measuring angles. For example, when a = 2 π {\displaystyle a=2\pi } , we get the measure in radians, and the usual trigonometric functions. When a = 360 {\displaystyle a=360} , we get the sine and cosine of angles measured in degrees.
Note that a = 2 π {\displaystyle a=2\pi } is the unique value at which the derivative d d t e ( t / a ) {\displaystyle {\frac {d}{dt}}e(t/a)} becomes a unit vector with positive imaginary part at t = 0 {\displaystyle t=0} . This fact can, in turn, be used to define the constant 2 π {\displaystyle 2\pi } .
Another way to define the trigonometric functions in analysis is using integration. [ 14 ] [ 27 ] For a real number t {\displaystyle t} , put θ ( t ) = ∫ 0 t d τ 1 + τ 2 = arctan t {\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t} where this defines this inverse tangent function. Also, π {\displaystyle \pi } is defined by 1 2 π = ∫ 0 ∞ d τ 1 + τ 2 {\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}} a definition that goes back to Karl Weierstrass . [ 28 ]
On the interval − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} , the trigonometric functions are defined by inverting the relation θ = arctan t {\displaystyle \theta =\arctan t} . Thus we define the trigonometric functions by tan θ = t , cos θ = ( 1 + t 2 ) − 1 / 2 , sin θ = t ( 1 + t 2 ) − 1 / 2 {\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}} where the point ( t , θ ) {\displaystyle (t,\theta )} is on the graph of θ = arctan t {\displaystyle \theta =\arctan t} and the positive square root is taken.
This defines the trigonometric functions on ( − π / 2 , π / 2 ) {\displaystyle (-\pi /2,\pi /2)} . The definition can be extended to all real numbers by first observing that, as θ → π / 2 {\displaystyle \theta \to \pi /2} , t → ∞ {\displaystyle t\to \infty } , and so cos θ = ( 1 + t 2 ) − 1 / 2 → 0 {\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0} and sin θ = t ( 1 + t 2 ) − 1 / 2 → 1 {\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1} . Thus cos θ {\displaystyle \cos \theta } and sin θ {\displaystyle \sin \theta } are extended continuously so that cos ( π / 2 ) = 0 , sin ( π / 2 ) = 1 {\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1} . Now the conditions cos ( θ + π ) = − cos ( θ ) {\displaystyle \cos(\theta +\pi )=-\cos(\theta )} and sin ( θ + π ) = − sin ( θ ) {\displaystyle \sin(\theta +\pi )=-\sin(\theta )} define the sine and cosine as periodic functions with period 2 π {\displaystyle 2\pi } , for all real numbers.
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, arctan s + arctan t = arctan s + t 1 − s t {\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}} holds, provided arctan s + arctan t ∈ ( − π / 2 , π / 2 ) {\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)} , since arctan s + arctan t = ∫ − s t d τ 1 + τ 2 = ∫ 0 s + t 1 − s t d τ 1 + τ 2 {\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}} after the substitution τ → s + τ 1 − s τ {\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}} . In particular, the limiting case as s → ∞ {\displaystyle s\to \infty } gives arctan t + π 2 = arctan ( − 1 / t ) , t ∈ ( − ∞ , 0 ) . {\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).} Thus we have sin ( θ + π 2 ) = − 1 t 1 + ( − 1 / t ) 2 = − 1 1 + t 2 = − cos ( θ ) {\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )} and cos ( θ + π 2 ) = 1 1 + ( − 1 / t ) 2 = t 1 + t 2 = sin ( θ ) . {\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).} So the sine and cosine functions are related by translation over a quarter period π / 2 {\displaystyle \pi /2} .
One can also define the trigonometric functions using various functional equations .
For example, [ 29 ] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
and the added condition
The sine and cosine of a complex number z = x + i y {\displaystyle z=x+iy} can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z {\displaystyle z} becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
sin z {\displaystyle \sin z\,}
cos z {\displaystyle \cos z\,}
tan z {\displaystyle \tan z\,}
cot z {\displaystyle \cot z\,}
sec z {\displaystyle \sec z\,}
csc z {\displaystyle \csc z\,}
The sine and cosine functions are periodic , with period 2 π {\displaystyle 2\pi } , which is the smallest positive period: sin ( z + 2 π ) = sin ( z ) , cos ( z + 2 π ) = cos ( z ) . {\displaystyle \sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).} Consequently, the cosecant and secant also have 2 π {\displaystyle 2\pi } as their period.
The functions sine and cosine also have semiperiods π {\displaystyle \pi } , and sin ( z + π ) = − sin ( z ) , cos ( z + π ) = − cos ( z ) {\displaystyle \sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)} and consequently tan ( z + π ) = tan ( z ) , cot ( z + π ) = cot ( z ) . {\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).} Also, sin ( x + π / 2 ) = cos ( x ) , cos ( x + π / 2 ) = − sin ( x ) {\displaystyle \sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)} (see Complementary angles ).
The function sin ( z ) {\displaystyle \sin(z)} has a unique zero (at z = 0 {\displaystyle z=0} ) in the strip − π < ℜ ( z ) < π {\displaystyle -\pi <\Re (z)<\pi } . The function cos ( z ) {\displaystyle \cos(z)} has the pair of zeros z = ± π / 2 {\displaystyle z=\pm \pi /2} in the same strip. Because of the periodicity, the zeros of sine are π Z = { … , − 2 π , − π , 0 , π , 2 π , … } ⊂ C . {\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .} There zeros of cosine are π 2 + π Z = { … , − 3 π 2 , − π 2 , π 2 , 3 π 2 , … } ⊂ C . {\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .} All of the zeros are simple zeros, and both functions have derivative ± 1 {\displaystyle \pm 1} at each of the zeros.
The tangent function tan ( z ) = sin ( z ) / cos ( z ) {\displaystyle \tan(z)=\sin(z)/\cos(z)} has a simple zero at z = 0 {\displaystyle z=0} and vertical asymptotes at z = ± π / 2 {\displaystyle z=\pm \pi /2} , where it has a simple pole of residue − 1 {\displaystyle -1} . Again, owing to the periodicity, the zeros are all the integer multiples of π {\displaystyle \pi } and the poles are odd multiples of π / 2 {\displaystyle \pi /2} , all having the same residue. The poles correspond to vertical asymptotes lim x → π − tan ( x ) = + ∞ , lim x → π + tan ( x ) = − ∞ . {\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
The cotangent function cot ( z ) = cos ( z ) / sin ( z ) {\displaystyle \cot(z)=\cos(z)/\sin(z)} has a simple pole of residue 1 at the integer multiples of π {\displaystyle \pi } and simple zeros at odd multiples of π / 2 {\displaystyle \pi /2} . The poles correspond to vertical asymptotes lim x → 0 − cot ( x ) = − ∞ , lim x → 0 + cot ( x ) = + ∞ . {\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities . These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π /2] , see Proofs of trigonometric identities ). For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
The cosine and the secant are even functions ; the other trigonometric functions are odd functions . That is:
All trigonometric functions are periodic functions of period 2 π . This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k , one has
See Periodicity and asymptotes .
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
Dividing through by either cos 2 x {\displaystyle \cos ^{2}x} or sin 2 x {\displaystyle \sin ^{2}x} gives
and
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy (see Angle sum and difference identities ). One can also produce them algebraically using Euler's formula .
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae .
These identities can be used to derive the product-to-sum identities .
By setting t = tan 1 2 θ , {\displaystyle t=\tan {\tfrac {1}{2}}\theta ,} all trigonometric functions of θ {\displaystyle \theta } can be expressed as rational fractions of t {\displaystyle t} :
Together with
this is the tangent half-angle substitution , which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule . The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration .
Note: For 0 < x < π {\displaystyle 0<x<\pi } the integral of csc x {\displaystyle \csc x} can also be written as − arsinh ( cot x ) , {\displaystyle -\operatorname {arsinh} (\cot x),} and for the integral of sec x {\displaystyle \sec x} for − π / 2 < x < π / 2 {\displaystyle -\pi /2<x<\pi /2} as arsinh ( tan x ) , {\displaystyle \operatorname {arsinh} (\tan x),} where arsinh {\displaystyle \operatorname {arsinh} } is the inverse hyperbolic sine .
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function . However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions . To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
The notations sin −1 , cos −1 , etc. are often used for arcsin and arccos , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms .
In this section A , B , C denote the three (interior) angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C : sin A a = sin B b = sin C c = 2 Δ a b c , {\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},} where Δ is the area of the triangle,
or, equivalently, a sin A = b sin B = c sin C = 2 R , {\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,} where R is the triangle's circumradius .
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem : c 2 = a 2 + b 2 − 2 a b cos C , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,} or equivalently, cos C = a 2 + b 2 − c 2 2 a b . {\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
In this formula the angle at C is opposite to the side c . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem .
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
The law of tangents says that:
If s is the triangle's semiperimeter, ( a + b + c )/2, and r is the radius of the triangle's incircle , then rs is the triangle's area. Therefore Heron's formula implies that:
The law of cotangents says that: [ 30 ]
It follows that
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion , which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion .
Trigonometric functions also prove to be useful in the study of general periodic functions . The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves . [ 31 ]
Under rather general conditions, a periodic function f ( x ) can be expressed as a sum of sine waves or cosine waves in a Fourier series . [ 32 ] Denoting the sine or cosine basis functions by φ k , the expansion of the periodic function f ( t ) takes the form: f ( t ) = ∑ k = 1 ∞ c k φ k ( t ) . {\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
For example, the square wave can be written as the Fourier series f square ( t ) = 4 π ∑ k = 1 ∞ sin ( ( 2 k − 1 ) t ) 2 k − 1 . {\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy ( Aryabhatiya , Surya Siddhanta ), via translation from Sanskrit to Arabic and then from Arabic to Latin. [ 33 ] (See Aryabhata's sine table .)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines , used in solving triangles . [ 34 ] Al-Khwārizmī (c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables. [ 35 ] [ 36 ] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. [ 36 ] The trigonometric functions were later studied by mathematicians including Omar Khayyám , Bhāskara II , Nasir al-Din al-Tusi , Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus , and Rheticus' student Valentinus Otho .
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series . [ 37 ] (See Madhava series and Madhava's sine table .)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates. [ 38 ]
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). [ 39 ]
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin , cos , and tan in his book Trigonométrie . [ 40 ]
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x . [ 41 ] Though defined as ratios of sides of a right triangle , and thus appearing to be rational functions , Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series . He presented " Euler's formula ", as well as near-modern abbreviations ( sin. , cos. , tang. , cot. , sec. , and cosec. ). [ 33 ]
A few functions were common historically, but are now seldom used, such as the chord , versine (which appeared in the earliest tables [ 33 ] ), haversine , coversine , [ 42 ] half-tangent (tangent of half an angle), and exsecant . List of trigonometric identities shows more relations between these functions.
Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent. [ 43 ] [ 44 ] [ 45 ] [ 46 ]
The word sine derives [ 47 ] from Latin sinus , meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga ", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib , meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin . [ 48 ] The choice was based on a misreading of the Arabic written form j-y-b ( جيب ), which itself originated as a transliteration from Sanskrit jīvā , which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". [ 49 ]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans —"cutting"—since the line cuts the circle. [ 50 ]
The prefix " co- " (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter 's Canon triangulorum (1620), which defines the cosinus as an abbreviation of the sinus complementi (sine of the complementary angle ) and proceeds to define the cotangens similarly. [ 51 ] [ 52 ] | https://en.wikipedia.org/wiki/Tangens |
In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) [ 1 ] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among the simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis .
The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function , and an analog among the hyperbolic functions .
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are " sin " for sine, " cos " for cosine, " tan " or " tg " for tangent, " sec " for secant, " csc " or " cosec " for cosecant, and " cot " or " ctg " for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean ( sin x ) + y , {\displaystyle (\sin x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).}
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote ( sin x ) 2 , {\displaystyle (\sin x)^{2},} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from the (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
In contrast, the superscript − 1 {\displaystyle -1} is commonly used to denote the inverse function , not the reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case, the superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use.
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. [ 2 ] [ 3 ]
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics ).
However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, [ 5 ] or as solutions to differential equations given particular initial values [ 6 ] ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. [ 7 ] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), [ 8 ] and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. [ 9 ] For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. [ 10 ]
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects the unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and the line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is,
In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity .
The other trigonometric functions can be found along the unit circle as
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change the position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities
hold for any angle θ and any integer k . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after a rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities
hold for any angle θ and any integer k .
The algebraic expressions for the most important angles are as follows:
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. [ 13 ]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. [ 14 ] Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
Sine and cosine can be defined as the unique solution to the initial value problem : [ 17 ]
Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of the same ordinary differential equation
Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 .
One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi } is then a definition of the real number π {\displaystyle \pi } which is independent of geometry.
Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} ,
so the tangent function satisfies the ordinary differential equation
It is the unique solution with y (0) = 0 .
The basic trigonometric functions can be defined by the following power series expansions. [ 18 ] These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane .
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. [ 19 ]
More precisely, defining
one has the following series expansions: [ 20 ]
The following continued fractions are valid in the whole complex plane:
The last one was used in the historically first proof that π is irrational . [ 21 ]
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [ 22 ]
This identity can be proved with the Herglotz trick. [ 23 ] Combining the (– n ) th with the n th term lead to absolutely convergent series:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: [ 24 ]
This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which is the logarithmic derivative of sin z {\displaystyle \sin z} . [ 25 ] From this, it can be deduced also that
Euler's formula relates sine and cosine to the exponential function :
This formula is commonly considered for real values of x , but it remains true for all complex values.
Proof : Let f 1 ( x ) = cos x + i sin x , {\displaystyle f_{1}(x)=\cos x+i\sin x,} and f 2 ( x ) = e i x . {\displaystyle f_{2}(x)=e^{ix}.} One has d f j ( x ) / d x = i f j ( x ) {\displaystyle df_{j}(x)/dx=if_{j}(x)} for j = 1, 2 . The quotient rule implies thus that d / d x ( f 1 ( x ) / f 2 ( x ) ) = 0 {\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0} . Therefore, f 1 ( x ) / f 2 ( x ) {\displaystyle f_{1}(x)/f_{2}(x)} is a constant function, which equals 1 , as f 1 ( 0 ) = f 2 ( 0 ) = 1. {\displaystyle f_{1}(0)=f_{2}(0)=1.} This proves the formula.
One has
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
When x is real, this may be rewritten as
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e a + b = e a e b {\displaystyle e^{a+b}=e^{a}e^{b}} for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups . [ 26 ] The set U {\displaystyle U} of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , via an isomorphism e : R / Z → U . {\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.} In pedestrian terms e ( t ) = exp ( 2 π i t ) {\displaystyle e(t)=\exp(2\pi it)} , and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number a {\displaystyle a} (the base ), the function t ↦ e ( t / a ) {\displaystyle t\mapsto e(t/a)} defines an isomorphism of the group R / a Z → U {\displaystyle \mathbb {R} /a\mathbb {Z} \to U} . The real and imaginary parts of e ( t / a ) {\displaystyle e(t/a)} are the cosine and sine, where a {\displaystyle a} is used as the base for measuring angles. For example, when a = 2 π {\displaystyle a=2\pi } , we get the measure in radians, and the usual trigonometric functions. When a = 360 {\displaystyle a=360} , we get the sine and cosine of angles measured in degrees.
Note that a = 2 π {\displaystyle a=2\pi } is the unique value at which the derivative d d t e ( t / a ) {\displaystyle {\frac {d}{dt}}e(t/a)} becomes a unit vector with positive imaginary part at t = 0 {\displaystyle t=0} . This fact can, in turn, be used to define the constant 2 π {\displaystyle 2\pi } .
Another way to define the trigonometric functions in analysis is using integration. [ 14 ] [ 27 ] For a real number t {\displaystyle t} , put θ ( t ) = ∫ 0 t d τ 1 + τ 2 = arctan t {\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t} where this defines this inverse tangent function. Also, π {\displaystyle \pi } is defined by 1 2 π = ∫ 0 ∞ d τ 1 + τ 2 {\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}} a definition that goes back to Karl Weierstrass . [ 28 ]
On the interval − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} , the trigonometric functions are defined by inverting the relation θ = arctan t {\displaystyle \theta =\arctan t} . Thus we define the trigonometric functions by tan θ = t , cos θ = ( 1 + t 2 ) − 1 / 2 , sin θ = t ( 1 + t 2 ) − 1 / 2 {\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}} where the point ( t , θ ) {\displaystyle (t,\theta )} is on the graph of θ = arctan t {\displaystyle \theta =\arctan t} and the positive square root is taken.
This defines the trigonometric functions on ( − π / 2 , π / 2 ) {\displaystyle (-\pi /2,\pi /2)} . The definition can be extended to all real numbers by first observing that, as θ → π / 2 {\displaystyle \theta \to \pi /2} , t → ∞ {\displaystyle t\to \infty } , and so cos θ = ( 1 + t 2 ) − 1 / 2 → 0 {\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0} and sin θ = t ( 1 + t 2 ) − 1 / 2 → 1 {\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1} . Thus cos θ {\displaystyle \cos \theta } and sin θ {\displaystyle \sin \theta } are extended continuously so that cos ( π / 2 ) = 0 , sin ( π / 2 ) = 1 {\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1} . Now the conditions cos ( θ + π ) = − cos ( θ ) {\displaystyle \cos(\theta +\pi )=-\cos(\theta )} and sin ( θ + π ) = − sin ( θ ) {\displaystyle \sin(\theta +\pi )=-\sin(\theta )} define the sine and cosine as periodic functions with period 2 π {\displaystyle 2\pi } , for all real numbers.
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, arctan s + arctan t = arctan s + t 1 − s t {\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}} holds, provided arctan s + arctan t ∈ ( − π / 2 , π / 2 ) {\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)} , since arctan s + arctan t = ∫ − s t d τ 1 + τ 2 = ∫ 0 s + t 1 − s t d τ 1 + τ 2 {\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}} after the substitution τ → s + τ 1 − s τ {\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}} . In particular, the limiting case as s → ∞ {\displaystyle s\to \infty } gives arctan t + π 2 = arctan ( − 1 / t ) , t ∈ ( − ∞ , 0 ) . {\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).} Thus we have sin ( θ + π 2 ) = − 1 t 1 + ( − 1 / t ) 2 = − 1 1 + t 2 = − cos ( θ ) {\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )} and cos ( θ + π 2 ) = 1 1 + ( − 1 / t ) 2 = t 1 + t 2 = sin ( θ ) . {\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).} So the sine and cosine functions are related by translation over a quarter period π / 2 {\displaystyle \pi /2} .
One can also define the trigonometric functions using various functional equations .
For example, [ 29 ] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
and the added condition
The sine and cosine of a complex number z = x + i y {\displaystyle z=x+iy} can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z {\displaystyle z} becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
sin z {\displaystyle \sin z\,}
cos z {\displaystyle \cos z\,}
tan z {\displaystyle \tan z\,}
cot z {\displaystyle \cot z\,}
sec z {\displaystyle \sec z\,}
csc z {\displaystyle \csc z\,}
The sine and cosine functions are periodic , with period 2 π {\displaystyle 2\pi } , which is the smallest positive period: sin ( z + 2 π ) = sin ( z ) , cos ( z + 2 π ) = cos ( z ) . {\displaystyle \sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).} Consequently, the cosecant and secant also have 2 π {\displaystyle 2\pi } as their period.
The functions sine and cosine also have semiperiods π {\displaystyle \pi } , and sin ( z + π ) = − sin ( z ) , cos ( z + π ) = − cos ( z ) {\displaystyle \sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)} and consequently tan ( z + π ) = tan ( z ) , cot ( z + π ) = cot ( z ) . {\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).} Also, sin ( x + π / 2 ) = cos ( x ) , cos ( x + π / 2 ) = − sin ( x ) {\displaystyle \sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)} (see Complementary angles ).
The function sin ( z ) {\displaystyle \sin(z)} has a unique zero (at z = 0 {\displaystyle z=0} ) in the strip − π < ℜ ( z ) < π {\displaystyle -\pi <\Re (z)<\pi } . The function cos ( z ) {\displaystyle \cos(z)} has the pair of zeros z = ± π / 2 {\displaystyle z=\pm \pi /2} in the same strip. Because of the periodicity, the zeros of sine are π Z = { … , − 2 π , − π , 0 , π , 2 π , … } ⊂ C . {\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .} There zeros of cosine are π 2 + π Z = { … , − 3 π 2 , − π 2 , π 2 , 3 π 2 , … } ⊂ C . {\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .} All of the zeros are simple zeros, and both functions have derivative ± 1 {\displaystyle \pm 1} at each of the zeros.
The tangent function tan ( z ) = sin ( z ) / cos ( z ) {\displaystyle \tan(z)=\sin(z)/\cos(z)} has a simple zero at z = 0 {\displaystyle z=0} and vertical asymptotes at z = ± π / 2 {\displaystyle z=\pm \pi /2} , where it has a simple pole of residue − 1 {\displaystyle -1} . Again, owing to the periodicity, the zeros are all the integer multiples of π {\displaystyle \pi } and the poles are odd multiples of π / 2 {\displaystyle \pi /2} , all having the same residue. The poles correspond to vertical asymptotes lim x → π − tan ( x ) = + ∞ , lim x → π + tan ( x ) = − ∞ . {\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
The cotangent function cot ( z ) = cos ( z ) / sin ( z ) {\displaystyle \cot(z)=\cos(z)/\sin(z)} has a simple pole of residue 1 at the integer multiples of π {\displaystyle \pi } and simple zeros at odd multiples of π / 2 {\displaystyle \pi /2} . The poles correspond to vertical asymptotes lim x → 0 − cot ( x ) = − ∞ , lim x → 0 + cot ( x ) = + ∞ . {\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities . These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π /2] , see Proofs of trigonometric identities ). For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
The cosine and the secant are even functions ; the other trigonometric functions are odd functions . That is:
All trigonometric functions are periodic functions of period 2 π . This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k , one has
See Periodicity and asymptotes .
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
Dividing through by either cos 2 x {\displaystyle \cos ^{2}x} or sin 2 x {\displaystyle \sin ^{2}x} gives
and
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy (see Angle sum and difference identities ). One can also produce them algebraically using Euler's formula .
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae .
These identities can be used to derive the product-to-sum identities .
By setting t = tan 1 2 θ , {\displaystyle t=\tan {\tfrac {1}{2}}\theta ,} all trigonometric functions of θ {\displaystyle \theta } can be expressed as rational fractions of t {\displaystyle t} :
Together with
this is the tangent half-angle substitution , which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule . The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration .
Note: For 0 < x < π {\displaystyle 0<x<\pi } the integral of csc x {\displaystyle \csc x} can also be written as − arsinh ( cot x ) , {\displaystyle -\operatorname {arsinh} (\cot x),} and for the integral of sec x {\displaystyle \sec x} for − π / 2 < x < π / 2 {\displaystyle -\pi /2<x<\pi /2} as arsinh ( tan x ) , {\displaystyle \operatorname {arsinh} (\tan x),} where arsinh {\displaystyle \operatorname {arsinh} } is the inverse hyperbolic sine .
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function . However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions . To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
The notations sin −1 , cos −1 , etc. are often used for arcsin and arccos , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms .
In this section A , B , C denote the three (interior) angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C : sin A a = sin B b = sin C c = 2 Δ a b c , {\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},} where Δ is the area of the triangle,
or, equivalently, a sin A = b sin B = c sin C = 2 R , {\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,} where R is the triangle's circumradius .
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem : c 2 = a 2 + b 2 − 2 a b cos C , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,} or equivalently, cos C = a 2 + b 2 − c 2 2 a b . {\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
In this formula the angle at C is opposite to the side c . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem .
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
The law of tangents says that:
If s is the triangle's semiperimeter, ( a + b + c )/2, and r is the radius of the triangle's incircle , then rs is the triangle's area. Therefore Heron's formula implies that:
The law of cotangents says that: [ 30 ]
It follows that
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion , which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion .
Trigonometric functions also prove to be useful in the study of general periodic functions . The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves . [ 31 ]
Under rather general conditions, a periodic function f ( x ) can be expressed as a sum of sine waves or cosine waves in a Fourier series . [ 32 ] Denoting the sine or cosine basis functions by φ k , the expansion of the periodic function f ( t ) takes the form: f ( t ) = ∑ k = 1 ∞ c k φ k ( t ) . {\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
For example, the square wave can be written as the Fourier series f square ( t ) = 4 π ∑ k = 1 ∞ sin ( ( 2 k − 1 ) t ) 2 k − 1 . {\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy ( Aryabhatiya , Surya Siddhanta ), via translation from Sanskrit to Arabic and then from Arabic to Latin. [ 33 ] (See Aryabhata's sine table .)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines , used in solving triangles . [ 34 ] Al-Khwārizmī (c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables. [ 35 ] [ 36 ] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. [ 36 ] The trigonometric functions were later studied by mathematicians including Omar Khayyám , Bhāskara II , Nasir al-Din al-Tusi , Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus , and Rheticus' student Valentinus Otho .
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series . [ 37 ] (See Madhava series and Madhava's sine table .)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates. [ 38 ]
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). [ 39 ]
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin , cos , and tan in his book Trigonométrie . [ 40 ]
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x . [ 41 ] Though defined as ratios of sides of a right triangle , and thus appearing to be rational functions , Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series . He presented " Euler's formula ", as well as near-modern abbreviations ( sin. , cos. , tang. , cot. , sec. , and cosec. ). [ 33 ]
A few functions were common historically, but are now seldom used, such as the chord , versine (which appeared in the earliest tables [ 33 ] ), haversine , coversine , [ 42 ] half-tangent (tangent of half an angle), and exsecant . List of trigonometric identities shows more relations between these functions.
Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent. [ 43 ] [ 44 ] [ 45 ] [ 46 ]
The word sine derives [ 47 ] from Latin sinus , meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga ", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib , meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin . [ 48 ] The choice was based on a misreading of the Arabic written form j-y-b ( جيب ), which itself originated as a transliteration from Sanskrit jīvā , which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". [ 49 ]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans —"cutting"—since the line cuts the circle. [ 50 ]
The prefix " co- " (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter 's Canon triangulorum (1620), which defines the cosinus as an abbreviation of the sinus complementi (sine of the complementary angle ) and proceeds to define the cotangens similarly. [ 51 ] [ 52 ] | https://en.wikipedia.org/wiki/Tangens_complementi |
In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) [ 1 ] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among the simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis .
The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function , and an analog among the hyperbolic functions .
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are " sin " for sine, " cos " for cosine, " tan " or " tg " for tangent, " sec " for secant, " csc " or " cosec " for cosecant, and " cot " or " ctg " for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean ( sin x ) + y , {\displaystyle (\sin x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).}
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote ( sin x ) 2 , {\displaystyle (\sin x)^{2},} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from the (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
In contrast, the superscript − 1 {\displaystyle -1} is commonly used to denote the inverse function , not the reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case, the superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use.
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. [ 2 ] [ 3 ]
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics ).
However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, [ 5 ] or as solutions to differential equations given particular initial values [ 6 ] ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. [ 7 ] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), [ 8 ] and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. [ 9 ] For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. [ 10 ]
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects the unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and the line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is,
In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity .
The other trigonometric functions can be found along the unit circle as
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change the position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities
hold for any angle θ and any integer k . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after a rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities
hold for any angle θ and any integer k .
The algebraic expressions for the most important angles are as follows:
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. [ 13 ]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. [ 14 ] Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
Sine and cosine can be defined as the unique solution to the initial value problem : [ 17 ]
Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of the same ordinary differential equation
Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 .
One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi } is then a definition of the real number π {\displaystyle \pi } which is independent of geometry.
Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} ,
so the tangent function satisfies the ordinary differential equation
It is the unique solution with y (0) = 0 .
The basic trigonometric functions can be defined by the following power series expansions. [ 18 ] These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane .
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. [ 19 ]
More precisely, defining
one has the following series expansions: [ 20 ]
The following continued fractions are valid in the whole complex plane:
The last one was used in the historically first proof that π is irrational . [ 21 ]
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [ 22 ]
This identity can be proved with the Herglotz trick. [ 23 ] Combining the (– n ) th with the n th term lead to absolutely convergent series:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: [ 24 ]
This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which is the logarithmic derivative of sin z {\displaystyle \sin z} . [ 25 ] From this, it can be deduced also that
Euler's formula relates sine and cosine to the exponential function :
This formula is commonly considered for real values of x , but it remains true for all complex values.
Proof : Let f 1 ( x ) = cos x + i sin x , {\displaystyle f_{1}(x)=\cos x+i\sin x,} and f 2 ( x ) = e i x . {\displaystyle f_{2}(x)=e^{ix}.} One has d f j ( x ) / d x = i f j ( x ) {\displaystyle df_{j}(x)/dx=if_{j}(x)} for j = 1, 2 . The quotient rule implies thus that d / d x ( f 1 ( x ) / f 2 ( x ) ) = 0 {\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0} . Therefore, f 1 ( x ) / f 2 ( x ) {\displaystyle f_{1}(x)/f_{2}(x)} is a constant function, which equals 1 , as f 1 ( 0 ) = f 2 ( 0 ) = 1. {\displaystyle f_{1}(0)=f_{2}(0)=1.} This proves the formula.
One has
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
When x is real, this may be rewritten as
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e a + b = e a e b {\displaystyle e^{a+b}=e^{a}e^{b}} for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups . [ 26 ] The set U {\displaystyle U} of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , via an isomorphism e : R / Z → U . {\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.} In pedestrian terms e ( t ) = exp ( 2 π i t ) {\displaystyle e(t)=\exp(2\pi it)} , and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number a {\displaystyle a} (the base ), the function t ↦ e ( t / a ) {\displaystyle t\mapsto e(t/a)} defines an isomorphism of the group R / a Z → U {\displaystyle \mathbb {R} /a\mathbb {Z} \to U} . The real and imaginary parts of e ( t / a ) {\displaystyle e(t/a)} are the cosine and sine, where a {\displaystyle a} is used as the base for measuring angles. For example, when a = 2 π {\displaystyle a=2\pi } , we get the measure in radians, and the usual trigonometric functions. When a = 360 {\displaystyle a=360} , we get the sine and cosine of angles measured in degrees.
Note that a = 2 π {\displaystyle a=2\pi } is the unique value at which the derivative d d t e ( t / a ) {\displaystyle {\frac {d}{dt}}e(t/a)} becomes a unit vector with positive imaginary part at t = 0 {\displaystyle t=0} . This fact can, in turn, be used to define the constant 2 π {\displaystyle 2\pi } .
Another way to define the trigonometric functions in analysis is using integration. [ 14 ] [ 27 ] For a real number t {\displaystyle t} , put θ ( t ) = ∫ 0 t d τ 1 + τ 2 = arctan t {\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t} where this defines this inverse tangent function. Also, π {\displaystyle \pi } is defined by 1 2 π = ∫ 0 ∞ d τ 1 + τ 2 {\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}} a definition that goes back to Karl Weierstrass . [ 28 ]
On the interval − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} , the trigonometric functions are defined by inverting the relation θ = arctan t {\displaystyle \theta =\arctan t} . Thus we define the trigonometric functions by tan θ = t , cos θ = ( 1 + t 2 ) − 1 / 2 , sin θ = t ( 1 + t 2 ) − 1 / 2 {\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}} where the point ( t , θ ) {\displaystyle (t,\theta )} is on the graph of θ = arctan t {\displaystyle \theta =\arctan t} and the positive square root is taken.
This defines the trigonometric functions on ( − π / 2 , π / 2 ) {\displaystyle (-\pi /2,\pi /2)} . The definition can be extended to all real numbers by first observing that, as θ → π / 2 {\displaystyle \theta \to \pi /2} , t → ∞ {\displaystyle t\to \infty } , and so cos θ = ( 1 + t 2 ) − 1 / 2 → 0 {\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0} and sin θ = t ( 1 + t 2 ) − 1 / 2 → 1 {\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1} . Thus cos θ {\displaystyle \cos \theta } and sin θ {\displaystyle \sin \theta } are extended continuously so that cos ( π / 2 ) = 0 , sin ( π / 2 ) = 1 {\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1} . Now the conditions cos ( θ + π ) = − cos ( θ ) {\displaystyle \cos(\theta +\pi )=-\cos(\theta )} and sin ( θ + π ) = − sin ( θ ) {\displaystyle \sin(\theta +\pi )=-\sin(\theta )} define the sine and cosine as periodic functions with period 2 π {\displaystyle 2\pi } , for all real numbers.
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, arctan s + arctan t = arctan s + t 1 − s t {\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}} holds, provided arctan s + arctan t ∈ ( − π / 2 , π / 2 ) {\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)} , since arctan s + arctan t = ∫ − s t d τ 1 + τ 2 = ∫ 0 s + t 1 − s t d τ 1 + τ 2 {\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}} after the substitution τ → s + τ 1 − s τ {\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}} . In particular, the limiting case as s → ∞ {\displaystyle s\to \infty } gives arctan t + π 2 = arctan ( − 1 / t ) , t ∈ ( − ∞ , 0 ) . {\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).} Thus we have sin ( θ + π 2 ) = − 1 t 1 + ( − 1 / t ) 2 = − 1 1 + t 2 = − cos ( θ ) {\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )} and cos ( θ + π 2 ) = 1 1 + ( − 1 / t ) 2 = t 1 + t 2 = sin ( θ ) . {\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).} So the sine and cosine functions are related by translation over a quarter period π / 2 {\displaystyle \pi /2} .
One can also define the trigonometric functions using various functional equations .
For example, [ 29 ] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
and the added condition
The sine and cosine of a complex number z = x + i y {\displaystyle z=x+iy} can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z {\displaystyle z} becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
sin z {\displaystyle \sin z\,}
cos z {\displaystyle \cos z\,}
tan z {\displaystyle \tan z\,}
cot z {\displaystyle \cot z\,}
sec z {\displaystyle \sec z\,}
csc z {\displaystyle \csc z\,}
The sine and cosine functions are periodic , with period 2 π {\displaystyle 2\pi } , which is the smallest positive period: sin ( z + 2 π ) = sin ( z ) , cos ( z + 2 π ) = cos ( z ) . {\displaystyle \sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).} Consequently, the cosecant and secant also have 2 π {\displaystyle 2\pi } as their period.
The functions sine and cosine also have semiperiods π {\displaystyle \pi } , and sin ( z + π ) = − sin ( z ) , cos ( z + π ) = − cos ( z ) {\displaystyle \sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)} and consequently tan ( z + π ) = tan ( z ) , cot ( z + π ) = cot ( z ) . {\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).} Also, sin ( x + π / 2 ) = cos ( x ) , cos ( x + π / 2 ) = − sin ( x ) {\displaystyle \sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)} (see Complementary angles ).
The function sin ( z ) {\displaystyle \sin(z)} has a unique zero (at z = 0 {\displaystyle z=0} ) in the strip − π < ℜ ( z ) < π {\displaystyle -\pi <\Re (z)<\pi } . The function cos ( z ) {\displaystyle \cos(z)} has the pair of zeros z = ± π / 2 {\displaystyle z=\pm \pi /2} in the same strip. Because of the periodicity, the zeros of sine are π Z = { … , − 2 π , − π , 0 , π , 2 π , … } ⊂ C . {\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .} There zeros of cosine are π 2 + π Z = { … , − 3 π 2 , − π 2 , π 2 , 3 π 2 , … } ⊂ C . {\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .} All of the zeros are simple zeros, and both functions have derivative ± 1 {\displaystyle \pm 1} at each of the zeros.
The tangent function tan ( z ) = sin ( z ) / cos ( z ) {\displaystyle \tan(z)=\sin(z)/\cos(z)} has a simple zero at z = 0 {\displaystyle z=0} and vertical asymptotes at z = ± π / 2 {\displaystyle z=\pm \pi /2} , where it has a simple pole of residue − 1 {\displaystyle -1} . Again, owing to the periodicity, the zeros are all the integer multiples of π {\displaystyle \pi } and the poles are odd multiples of π / 2 {\displaystyle \pi /2} , all having the same residue. The poles correspond to vertical asymptotes lim x → π − tan ( x ) = + ∞ , lim x → π + tan ( x ) = − ∞ . {\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
The cotangent function cot ( z ) = cos ( z ) / sin ( z ) {\displaystyle \cot(z)=\cos(z)/\sin(z)} has a simple pole of residue 1 at the integer multiples of π {\displaystyle \pi } and simple zeros at odd multiples of π / 2 {\displaystyle \pi /2} . The poles correspond to vertical asymptotes lim x → 0 − cot ( x ) = − ∞ , lim x → 0 + cot ( x ) = + ∞ . {\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities . These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π /2] , see Proofs of trigonometric identities ). For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
The cosine and the secant are even functions ; the other trigonometric functions are odd functions . That is:
All trigonometric functions are periodic functions of period 2 π . This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k , one has
See Periodicity and asymptotes .
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
Dividing through by either cos 2 x {\displaystyle \cos ^{2}x} or sin 2 x {\displaystyle \sin ^{2}x} gives
and
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy (see Angle sum and difference identities ). One can also produce them algebraically using Euler's formula .
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae .
These identities can be used to derive the product-to-sum identities .
By setting t = tan 1 2 θ , {\displaystyle t=\tan {\tfrac {1}{2}}\theta ,} all trigonometric functions of θ {\displaystyle \theta } can be expressed as rational fractions of t {\displaystyle t} :
Together with
this is the tangent half-angle substitution , which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule . The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration .
Note: For 0 < x < π {\displaystyle 0<x<\pi } the integral of csc x {\displaystyle \csc x} can also be written as − arsinh ( cot x ) , {\displaystyle -\operatorname {arsinh} (\cot x),} and for the integral of sec x {\displaystyle \sec x} for − π / 2 < x < π / 2 {\displaystyle -\pi /2<x<\pi /2} as arsinh ( tan x ) , {\displaystyle \operatorname {arsinh} (\tan x),} where arsinh {\displaystyle \operatorname {arsinh} } is the inverse hyperbolic sine .
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function . However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions . To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
The notations sin −1 , cos −1 , etc. are often used for arcsin and arccos , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms .
In this section A , B , C denote the three (interior) angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C : sin A a = sin B b = sin C c = 2 Δ a b c , {\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},} where Δ is the area of the triangle,
or, equivalently, a sin A = b sin B = c sin C = 2 R , {\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,} where R is the triangle's circumradius .
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem : c 2 = a 2 + b 2 − 2 a b cos C , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,} or equivalently, cos C = a 2 + b 2 − c 2 2 a b . {\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
In this formula the angle at C is opposite to the side c . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem .
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
The law of tangents says that:
If s is the triangle's semiperimeter, ( a + b + c )/2, and r is the radius of the triangle's incircle , then rs is the triangle's area. Therefore Heron's formula implies that:
The law of cotangents says that: [ 30 ]
It follows that
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion , which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion .
Trigonometric functions also prove to be useful in the study of general periodic functions . The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves . [ 31 ]
Under rather general conditions, a periodic function f ( x ) can be expressed as a sum of sine waves or cosine waves in a Fourier series . [ 32 ] Denoting the sine or cosine basis functions by φ k , the expansion of the periodic function f ( t ) takes the form: f ( t ) = ∑ k = 1 ∞ c k φ k ( t ) . {\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
For example, the square wave can be written as the Fourier series f square ( t ) = 4 π ∑ k = 1 ∞ sin ( ( 2 k − 1 ) t ) 2 k − 1 . {\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy ( Aryabhatiya , Surya Siddhanta ), via translation from Sanskrit to Arabic and then from Arabic to Latin. [ 33 ] (See Aryabhata's sine table .)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines , used in solving triangles . [ 34 ] Al-Khwārizmī (c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables. [ 35 ] [ 36 ] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. [ 36 ] The trigonometric functions were later studied by mathematicians including Omar Khayyám , Bhāskara II , Nasir al-Din al-Tusi , Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus , and Rheticus' student Valentinus Otho .
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series . [ 37 ] (See Madhava series and Madhava's sine table .)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates. [ 38 ]
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). [ 39 ]
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin , cos , and tan in his book Trigonométrie . [ 40 ]
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x . [ 41 ] Though defined as ratios of sides of a right triangle , and thus appearing to be rational functions , Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series . He presented " Euler's formula ", as well as near-modern abbreviations ( sin. , cos. , tang. , cot. , sec. , and cosec. ). [ 33 ]
A few functions were common historically, but are now seldom used, such as the chord , versine (which appeared in the earliest tables [ 33 ] ), haversine , coversine , [ 42 ] half-tangent (tangent of half an angle), and exsecant . List of trigonometric identities shows more relations between these functions.
Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent. [ 43 ] [ 44 ] [ 45 ] [ 46 ]
The word sine derives [ 47 ] from Latin sinus , meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga ", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib , meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin . [ 48 ] The choice was based on a misreading of the Arabic written form j-y-b ( جيب ), which itself originated as a transliteration from Sanskrit jīvā , which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". [ 49 ]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans —"cutting"—since the line cuts the circle. [ 50 ]
The prefix " co- " (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter 's Canon triangulorum (1620), which defines the cosinus as an abbreviation of the sinus complementi (sine of the complementary angle ) and proceeds to define the cotangens similarly. [ 51 ] [ 52 ] | https://en.wikipedia.org/wiki/Tangent_(trigonometry) |
In mathematics , the trigonometric functions (also called circular functions , angle functions or goniometric functions ) [ 1 ] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry , such as navigation , solid mechanics , celestial mechanics , geodesy , and many others. They are among the simplest periodic functions , and as such are also widely used for studying periodic phenomena through Fourier analysis .
The trigonometric functions most widely used in modern mathematics are the sine , the cosine , and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function , and an analog among the hyperbolic functions .
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles . To extend the sine and cosine functions to functions whose domain is the whole real line , geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations . This allows extending the domain of sine and cosine functions to the whole complex plane , and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are " sin " for sine, " cos " for cosine, " tan " or " tg " for tangent, " sec " for secant, " csc " or " cosec " for cosecant, and " cot " or " ctg " for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation , for example sin( x ) . Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression sin x + y {\displaystyle \sin x+y} would typically be interpreted to mean ( sin x ) + y , {\displaystyle (\sin x)+y,} so parentheses are required to express sin ( x + y ) . {\displaystyle \sin(x+y).}
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation , not function composition . For example sin 2 x {\displaystyle \sin ^{2}x} and sin 2 ( x ) {\displaystyle \sin ^{2}(x)} denote ( sin x ) 2 , {\displaystyle (\sin x)^{2},} not sin ( sin x ) . {\displaystyle \sin(\sin x).} This differs from the (historically later) general functional notation in which f 2 ( x ) = ( f ∘ f ) ( x ) = f ( f ( x ) ) . {\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
In contrast, the superscript − 1 {\displaystyle -1} is commonly used to denote the inverse function , not the reciprocal . For example sin − 1 x {\displaystyle \sin ^{-1}x} and sin − 1 ( x ) {\displaystyle \sin ^{-1}(x)} denote the inverse trigonometric function alternatively written arcsin x . {\displaystyle \arcsin x\,.} The equation θ = sin − 1 x {\displaystyle \theta =\sin ^{-1}x} implies sin θ = x , {\displaystyle \sin \theta =x,} not θ ⋅ sin x = 1. {\displaystyle \theta \cdot \sin x=1.} In this case, the superscript could be considered as denoting a composed or iterated function , but negative superscripts other than − 1 {\displaystyle {-1}} are not in common use.
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ , and adjacent represents the side between the angle θ and the right angle. [ 2 ] [ 3 ]
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π / 2 radians . Therefore sin ( θ ) {\displaystyle \sin(\theta )} and cos ( 90 ∘ − θ ) {\displaystyle \cos(90^{\circ }-\theta )} represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
In geometric applications, the argument of a trigonometric function is generally the measure of an angle . For this purpose, any angular unit is convenient. One common unit is degrees , in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics ).
However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function , via power series, [ 5 ] or as solutions to differential equations given particular initial values [ 6 ] ( see below ), without reference to any geometric notions. The other four trigonometric functions ( tan , cot , sec , csc ) can be defined as quotients and reciprocals of sin and cos , except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. [ 7 ] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), [ 8 ] and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. [ 9 ] For real number x , the notation sin x , cos x , etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown ( sin x° , cos x° , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180 x / π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175. [ 10 ]
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 {\textstyle {\frac {\pi }{2}}} radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let L {\displaystyle {\mathcal {L}}} be the ray obtained by rotating by an angle θ the positive half of the x -axis ( counterclockwise rotation for θ > 0 , {\displaystyle \theta >0,} and clockwise rotation for θ < 0 {\displaystyle \theta <0} ). This ray intersects the unit circle at the point A = ( x A , y A ) . {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).} The ray L , {\displaystyle {\mathcal {L}},} extended to a line if necessary, intersects the line of equation x = 1 {\displaystyle x=1} at point B = ( 1 , y B ) , {\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),} and the line of equation y = 1 {\displaystyle y=1} at point C = ( x C , 1 ) . {\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).} The tangent line to the unit circle at the point A , is perpendicular to L , {\displaystyle {\mathcal {L}},} and intersects the y - and x -axes at points D = ( 0 , y D ) {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })} and E = ( x E , 0 ) . {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).} The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A . That is,
In the range 0 ≤ θ ≤ π / 2 {\displaystyle 0\leq \theta \leq \pi /2} , this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse . And since the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} holds for all points P = ( x , y ) {\displaystyle \mathrm {P} =(x,y)} on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity .
The other trigonometric functions can be found along the unit circle as
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
Since a rotation of an angle of ± 2 π {\displaystyle \pm 2\pi } does not change the position or size of a shape, the points A , B , C , D , and E are the same for two angles whose difference is an integer multiple of 2 π {\displaystyle 2\pi } . Thus trigonometric functions are periodic functions with period 2 π {\displaystyle 2\pi } . That is, the equalities
hold for any angle θ and any integer k . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π {\displaystyle 2\pi } is the smallest value for which they are periodic (i.e., 2 π {\displaystyle 2\pi } is the fundamental period of these functions). However, after a rotation by an angle π {\displaystyle \pi } , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π {\displaystyle \pi } . That is, the equalities
hold for any angle θ and any integer k .
The algebraic expressions for the most important angles are as follows:
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. [ 13 ]
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. [ 14 ] Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
Sine and cosine can be defined as the unique solution to the initial value problem : [ 17 ]
Differentiating again, d 2 d x 2 sin x = d d x cos x = − sin x {\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x} and d 2 d x 2 cos x = − d d x sin x = − cos x {\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x} , so both sine and cosine are solutions of the same ordinary differential equation
Sine is the unique solution with y (0) = 0 and y ′(0) = 1 ; cosine is the unique solution with y (0) = 1 and y ′(0) = 0 .
One can then prove, as a theorem, that solutions cos , sin {\displaystyle \cos ,\sin } are periodic, having the same period. Writing this period as 2 π {\displaystyle 2\pi } is then a definition of the real number π {\displaystyle \pi } which is independent of geometry.
Applying the quotient rule to the tangent tan x = sin x / cos x {\displaystyle \tan x=\sin x/\cos x} ,
so the tangent function satisfies the ordinary differential equation
It is the unique solution with y (0) = 0 .
The basic trigonometric functions can be defined by the following power series expansions. [ 18 ] These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane .
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles . Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 {\textstyle (2k+1){\frac {\pi }{2}}} for the tangent and the secant, or k π {\displaystyle k\pi } for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence . Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. [ 19 ]
More precisely, defining
one has the following series expansions: [ 20 ]
The following continued fractions are valid in the whole complex plane:
The last one was used in the historically first proof that π is irrational . [ 21 ]
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [ 22 ]
This identity can be proved with the Herglotz trick. [ 23 ] Combining the (– n ) th with the n th term lead to absolutely convergent series:
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
The following infinite product for the sine is due to Leonhard Euler , and is of great importance in complex analysis: [ 24 ]
This may be obtained from the partial fraction decomposition of cot z {\displaystyle \cot z} given above, which is the logarithmic derivative of sin z {\displaystyle \sin z} . [ 25 ] From this, it can be deduced also that
Euler's formula relates sine and cosine to the exponential function :
This formula is commonly considered for real values of x , but it remains true for all complex values.
Proof : Let f 1 ( x ) = cos x + i sin x , {\displaystyle f_{1}(x)=\cos x+i\sin x,} and f 2 ( x ) = e i x . {\displaystyle f_{2}(x)=e^{ix}.} One has d f j ( x ) / d x = i f j ( x ) {\displaystyle df_{j}(x)/dx=if_{j}(x)} for j = 1, 2 . The quotient rule implies thus that d / d x ( f 1 ( x ) / f 2 ( x ) ) = 0 {\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0} . Therefore, f 1 ( x ) / f 2 ( x ) {\displaystyle f_{1}(x)/f_{2}(x)} is a constant function, which equals 1 , as f 1 ( 0 ) = f 2 ( 0 ) = 1. {\displaystyle f_{1}(0)=f_{2}(0)=1.} This proves the formula.
One has
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
When x is real, this may be rewritten as
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e a + b = e a e b {\displaystyle e^{a+b}=e^{a}e^{b}} for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups . [ 26 ] The set U {\displaystyle U} of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , via an isomorphism e : R / Z → U . {\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.} In pedestrian terms e ( t ) = exp ( 2 π i t ) {\displaystyle e(t)=\exp(2\pi it)} , and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number a {\displaystyle a} (the base ), the function t ↦ e ( t / a ) {\displaystyle t\mapsto e(t/a)} defines an isomorphism of the group R / a Z → U {\displaystyle \mathbb {R} /a\mathbb {Z} \to U} . The real and imaginary parts of e ( t / a ) {\displaystyle e(t/a)} are the cosine and sine, where a {\displaystyle a} is used as the base for measuring angles. For example, when a = 2 π {\displaystyle a=2\pi } , we get the measure in radians, and the usual trigonometric functions. When a = 360 {\displaystyle a=360} , we get the sine and cosine of angles measured in degrees.
Note that a = 2 π {\displaystyle a=2\pi } is the unique value at which the derivative d d t e ( t / a ) {\displaystyle {\frac {d}{dt}}e(t/a)} becomes a unit vector with positive imaginary part at t = 0 {\displaystyle t=0} . This fact can, in turn, be used to define the constant 2 π {\displaystyle 2\pi } .
Another way to define the trigonometric functions in analysis is using integration. [ 14 ] [ 27 ] For a real number t {\displaystyle t} , put θ ( t ) = ∫ 0 t d τ 1 + τ 2 = arctan t {\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t} where this defines this inverse tangent function. Also, π {\displaystyle \pi } is defined by 1 2 π = ∫ 0 ∞ d τ 1 + τ 2 {\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}} a definition that goes back to Karl Weierstrass . [ 28 ]
On the interval − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} , the trigonometric functions are defined by inverting the relation θ = arctan t {\displaystyle \theta =\arctan t} . Thus we define the trigonometric functions by tan θ = t , cos θ = ( 1 + t 2 ) − 1 / 2 , sin θ = t ( 1 + t 2 ) − 1 / 2 {\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}} where the point ( t , θ ) {\displaystyle (t,\theta )} is on the graph of θ = arctan t {\displaystyle \theta =\arctan t} and the positive square root is taken.
This defines the trigonometric functions on ( − π / 2 , π / 2 ) {\displaystyle (-\pi /2,\pi /2)} . The definition can be extended to all real numbers by first observing that, as θ → π / 2 {\displaystyle \theta \to \pi /2} , t → ∞ {\displaystyle t\to \infty } , and so cos θ = ( 1 + t 2 ) − 1 / 2 → 0 {\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0} and sin θ = t ( 1 + t 2 ) − 1 / 2 → 1 {\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1} . Thus cos θ {\displaystyle \cos \theta } and sin θ {\displaystyle \sin \theta } are extended continuously so that cos ( π / 2 ) = 0 , sin ( π / 2 ) = 1 {\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1} . Now the conditions cos ( θ + π ) = − cos ( θ ) {\displaystyle \cos(\theta +\pi )=-\cos(\theta )} and sin ( θ + π ) = − sin ( θ ) {\displaystyle \sin(\theta +\pi )=-\sin(\theta )} define the sine and cosine as periodic functions with period 2 π {\displaystyle 2\pi } , for all real numbers.
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First, arctan s + arctan t = arctan s + t 1 − s t {\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}} holds, provided arctan s + arctan t ∈ ( − π / 2 , π / 2 ) {\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)} , since arctan s + arctan t = ∫ − s t d τ 1 + τ 2 = ∫ 0 s + t 1 − s t d τ 1 + τ 2 {\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}} after the substitution τ → s + τ 1 − s τ {\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}} . In particular, the limiting case as s → ∞ {\displaystyle s\to \infty } gives arctan t + π 2 = arctan ( − 1 / t ) , t ∈ ( − ∞ , 0 ) . {\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).} Thus we have sin ( θ + π 2 ) = − 1 t 1 + ( − 1 / t ) 2 = − 1 1 + t 2 = − cos ( θ ) {\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )} and cos ( θ + π 2 ) = 1 1 + ( − 1 / t ) 2 = t 1 + t 2 = sin ( θ ) . {\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).} So the sine and cosine functions are related by translation over a quarter period π / 2 {\displaystyle \pi /2} .
One can also define the trigonometric functions using various functional equations .
For example, [ 29 ] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
and the added condition
The sine and cosine of a complex number z = x + i y {\displaystyle z=x+iy} can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z {\displaystyle z} becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
sin z {\displaystyle \sin z\,}
cos z {\displaystyle \cos z\,}
tan z {\displaystyle \tan z\,}
cot z {\displaystyle \cot z\,}
sec z {\displaystyle \sec z\,}
csc z {\displaystyle \csc z\,}
The sine and cosine functions are periodic , with period 2 π {\displaystyle 2\pi } , which is the smallest positive period: sin ( z + 2 π ) = sin ( z ) , cos ( z + 2 π ) = cos ( z ) . {\displaystyle \sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).} Consequently, the cosecant and secant also have 2 π {\displaystyle 2\pi } as their period.
The functions sine and cosine also have semiperiods π {\displaystyle \pi } , and sin ( z + π ) = − sin ( z ) , cos ( z + π ) = − cos ( z ) {\displaystyle \sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)} and consequently tan ( z + π ) = tan ( z ) , cot ( z + π ) = cot ( z ) . {\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).} Also, sin ( x + π / 2 ) = cos ( x ) , cos ( x + π / 2 ) = − sin ( x ) {\displaystyle \sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)} (see Complementary angles ).
The function sin ( z ) {\displaystyle \sin(z)} has a unique zero (at z = 0 {\displaystyle z=0} ) in the strip − π < ℜ ( z ) < π {\displaystyle -\pi <\Re (z)<\pi } . The function cos ( z ) {\displaystyle \cos(z)} has the pair of zeros z = ± π / 2 {\displaystyle z=\pm \pi /2} in the same strip. Because of the periodicity, the zeros of sine are π Z = { … , − 2 π , − π , 0 , π , 2 π , … } ⊂ C . {\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .} There zeros of cosine are π 2 + π Z = { … , − 3 π 2 , − π 2 , π 2 , 3 π 2 , … } ⊂ C . {\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .} All of the zeros are simple zeros, and both functions have derivative ± 1 {\displaystyle \pm 1} at each of the zeros.
The tangent function tan ( z ) = sin ( z ) / cos ( z ) {\displaystyle \tan(z)=\sin(z)/\cos(z)} has a simple zero at z = 0 {\displaystyle z=0} and vertical asymptotes at z = ± π / 2 {\displaystyle z=\pm \pi /2} , where it has a simple pole of residue − 1 {\displaystyle -1} . Again, owing to the periodicity, the zeros are all the integer multiples of π {\displaystyle \pi } and the poles are odd multiples of π / 2 {\displaystyle \pi /2} , all having the same residue. The poles correspond to vertical asymptotes lim x → π − tan ( x ) = + ∞ , lim x → π + tan ( x ) = − ∞ . {\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
The cotangent function cot ( z ) = cos ( z ) / sin ( z ) {\displaystyle \cot(z)=\cos(z)/\sin(z)} has a simple pole of residue 1 at the integer multiples of π {\displaystyle \pi } and simple zeros at odd multiples of π / 2 {\displaystyle \pi /2} . The poles correspond to vertical asymptotes lim x → 0 − cot ( x ) = − ∞ , lim x → 0 + cot ( x ) = + ∞ . {\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities . These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π /2] , see Proofs of trigonometric identities ). For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
The cosine and the secant are even functions ; the other trigonometric functions are odd functions . That is:
All trigonometric functions are periodic functions of period 2 π . This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k , one has
See Periodicity and asymptotes .
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
Dividing through by either cos 2 x {\displaystyle \cos ^{2}x} or sin 2 x {\displaystyle \sin ^{2}x} gives
and
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy (see Angle sum and difference identities ). One can also produce them algebraically using Euler's formula .
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae .
These identities can be used to derive the product-to-sum identities .
By setting t = tan 1 2 θ , {\displaystyle t=\tan {\tfrac {1}{2}}\theta ,} all trigonometric functions of θ {\displaystyle \theta } can be expressed as rational fractions of t {\displaystyle t} :
Together with
this is the tangent half-angle substitution , which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule . The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration .
Note: For 0 < x < π {\displaystyle 0<x<\pi } the integral of csc x {\displaystyle \csc x} can also be written as − arsinh ( cot x ) , {\displaystyle -\operatorname {arsinh} (\cot x),} and for the integral of sec x {\displaystyle \sec x} for − π / 2 < x < π / 2 {\displaystyle -\pi /2<x<\pi /2} as arsinh ( tan x ) , {\displaystyle \operatorname {arsinh} (\tan x),} where arsinh {\displaystyle \operatorname {arsinh} } is the inverse hyperbolic sine .
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function . However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions . To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
The notations sin −1 , cos −1 , etc. are often used for arcsin and arccos , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms .
In this section A , B , C denote the three (interior) angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C : sin A a = sin B b = sin C c = 2 Δ a b c , {\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},} where Δ is the area of the triangle,
or, equivalently, a sin A = b sin B = c sin C = 2 R , {\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,} where R is the triangle's circumradius .
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem : c 2 = a 2 + b 2 − 2 a b cos C , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,} or equivalently, cos C = a 2 + b 2 − c 2 2 a b . {\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
In this formula the angle at C is opposite to the side c . This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem .
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
The law of tangents says that:
If s is the triangle's semiperimeter, ( a + b + c )/2, and r is the radius of the triangle's incircle , then rs is the triangle's area. Therefore Heron's formula implies that:
The law of cotangents says that: [ 30 ]
It follows that
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion , which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion .
Trigonometric functions also prove to be useful in the study of general periodic functions . The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves . [ 31 ]
Under rather general conditions, a periodic function f ( x ) can be expressed as a sum of sine waves or cosine waves in a Fourier series . [ 32 ] Denoting the sine or cosine basis functions by φ k , the expansion of the periodic function f ( t ) takes the form: f ( t ) = ∑ k = 1 ∞ c k φ k ( t ) . {\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
For example, the square wave can be written as the Fourier series f square ( t ) = 4 π ∑ k = 1 ∞ sin ( ( 2 k − 1 ) t ) 2 k − 1 . {\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy ( Aryabhatiya , Surya Siddhanta ), via translation from Sanskrit to Arabic and then from Arabic to Latin. [ 33 ] (See Aryabhata's sine table .)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines , used in solving triangles . [ 34 ] Al-Khwārizmī (c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables. [ 35 ] [ 36 ] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. [ 36 ] The trigonometric functions were later studied by mathematicians including Omar Khayyám , Bhāskara II , Nasir al-Din al-Tusi , Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus , and Rheticus' student Valentinus Otho .
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series . [ 37 ] (See Madhava series and Madhava's sine table .)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates. [ 38 ]
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). [ 39 ]
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin , cos , and tan in his book Trigonométrie . [ 40 ]
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x . [ 41 ] Though defined as ratios of sides of a right triangle , and thus appearing to be rational functions , Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series . He presented " Euler's formula ", as well as near-modern abbreviations ( sin. , cos. , tang. , cot. , sec. , and cosec. ). [ 33 ]
A few functions were common historically, but are now seldom used, such as the chord , versine (which appeared in the earliest tables [ 33 ] ), haversine , coversine , [ 42 ] half-tangent (tangent of half an angle), and exsecant . List of trigonometric identities shows more relations between these functions.
Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent. [ 43 ] [ 44 ] [ 45 ] [ 46 ]
The word sine derives [ 47 ] from Latin sinus , meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga ", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib , meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin . [ 48 ] The choice was based on a misreading of the Arabic written form j-y-b ( جيب ), which itself originated as a transliteration from Sanskrit jīvā , which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". [ 49 ]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans —"cutting"—since the line cuts the circle. [ 50 ]
The prefix " co- " (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter 's Canon triangulorum (1620), which defines the cosinus as an abbreviation of the sinus complementi (sine of the complementary angle ) and proceeds to define the cotangens similarly. [ 51 ] [ 52 ] | https://en.wikipedia.org/wiki/Tangent_complement |
In integral calculus , the tangent half-angle substitution is a change of variables used for evaluating integrals , which converts a rational function of trigonometric functions of x {\textstyle x} into an ordinary rational function of t {\textstyle t} by setting t = tan x 2 {\textstyle t=\tan {\tfrac {x}{2}}} . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line . The general [ 1 ] transformation formula is:
∫ f ( sin x , cos x ) d x = ∫ f ( 2 t 1 + t 2 , 1 − t 2 1 + t 2 ) 2 d t 1 + t 2 . {\displaystyle \int f(\sin x,\cos x)\,dx=\int f{\left({\frac {2t}{1+t^{2}}},{\frac {1-t^{2}}{1+t^{2}}}\right)}{\frac {2\,dt}{1+t^{2}}}.}
The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. [ 2 ] Leonhard Euler used it to evaluate the integral ∫ d x / ( a + b cos x ) {\textstyle \int dx/(a+b\cos x)} in his 1768 integral calculus textbook , [ 3 ] and Adrien-Marie Legendre described the general method in 1817. [ 4 ]
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [ 5 ] It is known in Russia as the universal trigonometric substitution , [ 6 ] and also known by variant names such as half-tangent substitution or half-angle substitution . It is sometimes misattributed as the Weierstrass substitution . [ 7 ] Michael Spivak called it the "world's sneakiest substitution". [ 8 ]
Introducing a new variable t = tan x 2 , {\textstyle t=\tan {\tfrac {x}{2}},} sines and cosines can be expressed as rational functions of t , {\displaystyle t,} and d x {\displaystyle dx} can be expressed as the product of d t {\displaystyle dt} and a rational function of t , {\displaystyle t,} as follows: sin x = 2 t 1 + t 2 , cos x = 1 − t 2 1 + t 2 , and d x = 2 1 + t 2 d t . {\displaystyle \sin x={\frac {2t}{1+t^{2}}},\quad \cos x={\frac {1-t^{2}}{1+t^{2}}},\quad {\text{and}}\quad dx={\frac {2}{1+t^{2}}}\,dt.}
Similar expressions can be written for tan x , cot x , sec x , and csc x .
Using the double-angle formulas sin x = 2 sin x 2 cos x 2 {\displaystyle \sin x=2\sin {\tfrac {x}{2}}\cos {\tfrac {x}{2}}} and cos x = cos 2 x 2 − sin 2 x 2 {\displaystyle \cos x=\cos ^{2}{\tfrac {x}{2}}-\sin ^{2}{\tfrac {x}{2}}} and introducing denominators equal to one by the Pythagorean identity 1 = cos 2 x 2 + sin 2 x 2 {\displaystyle 1=\cos ^{2}{\tfrac {x}{2}}+\sin ^{2}{\tfrac {x}{2}}} results in
sin x = 2 sin x 2 cos x 2 cos 2 x 2 + sin 2 x 2 = 2 tan x 2 1 + tan 2 x 2 = 2 t 1 + t 2 , cos x = cos 2 x 2 − sin 2 x 2 cos 2 x 2 + sin 2 x 2 = 1 − tan 2 x 2 1 + tan 2 x 2 = 1 − t 2 1 + t 2 . {\displaystyle {\begin{aligned}\sin x&={\frac {2\sin {\tfrac {x}{2}}\,\cos {\tfrac {x}{2}}}{\cos ^{2}{\tfrac {x}{2}}+\sin ^{2}{\tfrac {x}{2}}}}={\frac {2\tan {\tfrac {x}{2}}}{1+\tan ^{2}{\tfrac {x}{2}}}}={\frac {2t}{1+t^{2}}},\\[18mu]\cos x&={\frac {\cos ^{2}{\tfrac {x}{2}}-\sin ^{2}{\tfrac {x}{2}}}{\cos ^{2}{\tfrac {x}{2}}+\sin ^{2}{\tfrac {x}{2}}}}={\frac {1-\tan ^{2}{\tfrac {x}{2}}}{1+\tan ^{2}{\tfrac {x}{2}}}}={\frac {1-t^{2}}{1+t^{2}}}.\end{aligned}}}
Finally, since t = tan x 2 {\textstyle t=\tan {\tfrac {x}{2}}} , differentiation rules imply
d t = 1 2 ( 1 + tan 2 x 2 ) d x = 1 + t 2 2 d x , {\displaystyle dt={\tfrac {1}{2}}\left(1+\tan ^{2}{\tfrac {x}{2}}\right)dx={\frac {1+t^{2}}{2}}\,dx,} and thus d x = 2 1 + t 2 d t . {\displaystyle dx={\frac {2}{1+t^{2}}}\,dt.}
∫ csc x d x = ∫ d x sin x = ∫ ( 1 + t 2 2 t ) ( 2 1 + t 2 ) d t t = tan x 2 = ∫ d t t = ln | t | + C = ln | tan x 2 | + C . {\displaystyle {\begin{aligned}\int \csc x\,dx&=\int {\frac {dx}{\sin x}}\\[6pt]&=\int \left({\frac {1+t^{2}}{2t}}\right)\left({\frac {2}{1+t^{2}}}\right)dt&&t=\tan {\tfrac {x}{2}}\\[6pt]&=\int {\frac {dt}{t}}\\[6pt]&=\ln |t|+C\\[6pt]&=\ln \left|\tan {\tfrac {x}{2}}\right|+C.\end{aligned}}}
We can confirm the above result using a standard method of evaluating the cosecant integral by multiplying the numerator and denominator by csc x − cot x {\textstyle \csc x-\cot x} and performing the substitution u = csc x − cot x , {\textstyle u=\csc x-\cot x,} d u = ( − csc x cot x + csc 2 x ) d x {\textstyle du=\left(-\csc x\cot x+\csc ^{2}x\right)\,dx} . ∫ csc x d x = ∫ csc x ( csc x − cot x ) csc x − cot x d x = ∫ ( csc 2 x − csc x cot x ) d x csc x − cot x u = csc x − cot x = ∫ d u u = ln | u | + C = ln | csc x − cot x | + C . {\displaystyle {\begin{aligned}\int \csc x\,dx&=\int {\frac {\csc x(\csc x-\cot x)}{\csc x-\cot x}}\,dx\\[6pt]&=\int {\frac {\left(\csc ^{2}x-\csc x\cot x\right)\,dx}{\csc x-\cot x}}\qquad u=\csc x-\cot x\\[6pt]&=\int {\frac {du}{u}}\\[6pt]&=\ln |u|+C\\[6pt]&=\ln \left|\csc x-\cot x\right|+C.\end{aligned}}}
These two answers are the same because csc x − cot x = tan x 2 : {\textstyle \csc x-\cot x=\tan {\tfrac {x}{2}}\colon }
csc x − cot x = 1 sin x − cos x sin x = 1 + t 2 2 t − 1 − t 2 1 + t 2 1 + t 2 2 t t = tan x 2 = 2 t 2 2 t = t = tan x 2 {\displaystyle {\begin{aligned}\csc x-\cot x&={\frac {1}{\sin x}}-{\frac {\cos x}{\sin x}}\\[6pt]&={\frac {1+t^{2}}{2t}}-{\frac {1-t^{2}}{1+t^{2}}}{\frac {1+t^{2}}{2t}}\qquad \qquad t=\tan {\tfrac {x}{2}}\\[6pt]&={\frac {2t^{2}}{2t}}=t\\[6pt]&=\tan {\tfrac {x}{2}}\end{aligned}}}
The secant integral may be evaluated in a similar manner.
∫ 0 2 π d x 2 + cos x = ∫ 0 π d x 2 + cos x + ∫ π 2 π d x 2 + cos x = ∫ 0 ∞ 2 d t 3 + t 2 + ∫ − ∞ 0 2 d t 3 + t 2 t = tan x 2 = ∫ − ∞ ∞ 2 d t 3 + t 2 = 2 3 ∫ − ∞ ∞ d u 1 + u 2 t = u 3 = 2 π 3 . {\displaystyle {\begin{aligned}\int _{0}^{2\pi }{\frac {dx}{2+\cos x}}&=\int _{0}^{\pi }{\frac {dx}{2+\cos x}}+\int _{\pi }^{2\pi }{\frac {dx}{2+\cos x}}\\[6pt]&=\int _{0}^{\infty }{\frac {2\,dt}{3+t^{2}}}+\int _{-\infty }^{0}{\frac {2\,dt}{3+t^{2}}}&t&=\tan {\tfrac {x}{2}}\\[6pt]&=\int _{-\infty }^{\infty }{\frac {2\,dt}{3+t^{2}}}\\[6pt]&={\frac {2}{\sqrt {3}}}\int _{-\infty }^{\infty }{\frac {du}{1+u^{2}}}&t&=u{\sqrt {3}}\\[6pt]&={\frac {2\pi }{\sqrt {3}}}.\end{aligned}}}
In the first line, one cannot simply substitute t = 0 {\textstyle t=0} for both limits of integration . The singularity (in this case, a vertical asymptote ) of t = tan x 2 {\textstyle t=\tan {\tfrac {x}{2}}} at x = π {\textstyle x=\pi } must be taken into account. Alternatively, first evaluate the indefinite integral, then apply the boundary values. ∫ d x 2 + cos x = ∫ 1 2 + 1 − t 2 1 + t 2 2 d t t 2 + 1 t = tan x 2 = ∫ 2 d t 2 ( t 2 + 1 ) + ( 1 − t 2 ) = ∫ 2 d t t 2 + 3 = 2 3 ∫ d t ( t / 3 ) 2 + 1 u = t / 3 = 2 3 ∫ d u u 2 + 1 tan θ = u = 2 3 ∫ cos 2 θ sec 2 θ d θ = 2 3 ∫ d θ = 2 3 θ + C = 2 3 arctan ( t 3 ) + C = 2 3 arctan ( tan x 2 3 ) + C . {\displaystyle {\begin{aligned}\int {\frac {dx}{2+\cos x}}&=\int {\frac {1}{2+{\frac {1-t^{2}}{1+t^{2}}}}}{\frac {2\,dt}{t^{2}+1}}&&t=\tan {\tfrac {x}{2}}\\[6pt]&=\int {\frac {2\,dt}{2(t^{2}+1)+(1-t^{2})}}=\int {\frac {2\,dt}{t^{2}+3}}\\[6pt]&={\frac {2}{3}}\int {\frac {dt}{{\bigl (}t{\big /}{\sqrt {3}}{\bigr )}^{2}+1}}&&u=t{\big /}{\sqrt {3}}\\[6pt]&={\frac {2}{\sqrt {3}}}\int {\frac {du}{u^{2}+1}}&&\tan \theta =u\\[6pt]&={\frac {2}{\sqrt {3}}}\int \cos ^{2}\theta \sec ^{2}\theta \,d\theta ={\frac {2}{\sqrt {3}}}\int d\theta \\[6pt]&={\frac {2}{\sqrt {3}}}\theta +C={\frac {2}{\sqrt {3}}}\arctan \left({\frac {t}{\sqrt {3}}}\right)+C\\[6pt]&={\frac {2}{\sqrt {3}}}\arctan \left({\frac {\tan {\tfrac {x}{2}}}{\sqrt {3}}}\right)+C.\end{aligned}}} By symmetry, ∫ 0 2 π d x 2 + cos x = 2 ∫ 0 π d x 2 + cos x = lim b → π 4 3 arctan ( tan x 2 3 ) | 0 b = 4 3 [ lim b → π arctan ( tan b 2 3 ) − arctan ( 0 ) ] = 4 3 ( π 2 − 0 ) = 2 π 3 , {\displaystyle {\begin{aligned}\int _{0}^{2\pi }{\frac {dx}{2+\cos x}}&=2\int _{0}^{\pi }{\frac {dx}{2+\cos x}}=\lim _{b\rightarrow \pi }{\frac {4}{\sqrt {3}}}\arctan \left({\frac {\tan {\tfrac {x}{2}}}{\sqrt {3}}}\right){\Biggl |}_{0}^{b}\\[6pt]&={\frac {4}{\sqrt {3}}}{\Biggl [}\lim _{b\rightarrow \pi }\arctan \left({\frac {\tan {\tfrac {b}{2}}}{\sqrt {3}}}\right)-\arctan(0){\Biggl ]}={\frac {4}{\sqrt {3}}}\left({\frac {\pi }{2}}-0\right)={\frac {2\pi }{\sqrt {3}}},\end{aligned}}} which is the same as the previous answer.
∫ d x a cos x + b sin x + c = ∫ 2 d t a ( 1 − t 2 ) + 2 b t + c ( t 2 + 1 ) = ∫ 2 d t ( c − a ) t 2 + 2 b t + a + c = 2 c 2 − ( a 2 + b 2 ) arctan ( ( c − a ) tan x 2 + b c 2 − ( a 2 + b 2 ) ) + C {\displaystyle {\begin{aligned}\int {\frac {dx}{a\cos x+b\sin x+c}}&=\int {\frac {2\,dt}{a(1-t^{2})+2bt+c(t^{2}+1)}}\\[6pt]&=\int {\frac {2\,dt}{(c-a)t^{2}+2bt+a+c}}\\[6pt]&={\frac {2}{\sqrt {c^{2}-(a^{2}+b^{2})}}}\arctan \left({\frac {(c-a)\tan {\tfrac {x}{2}}+b}{\sqrt {c^{2}-(a^{2}+b^{2})}}}\right)+C\end{aligned}}} if c 2 − ( a 2 + b 2 ) > 0. {\textstyle c^{2}-(a^{2}+b^{2})>0.}
As x varies, the point (cos x , sin x ) winds repeatedly around the unit circle centered at (0, 0). The point
( 1 − t 2 1 + t 2 , 2 t 1 + t 2 ) {\displaystyle \left({\frac {1-t^{2}}{1+t^{2}}},{\frac {2t}{1+t^{2}}}\right)}
goes only once around the circle as t goes from −∞ to +∞, and never reaches the point (−1, 0), which is approached as a limit as t approaches ±∞. As t goes from −∞ to −1, the point determined by t goes through the part of the circle in the third quadrant, from (−1, 0) to (0, −1). As t goes from −1 to 0, the point follows the part of the circle in the fourth quadrant from (0, −1) to (1, 0). As t goes from 0 to 1, the point follows the part of the circle in the first quadrant from (1, 0) to (0, 1). Finally, as t goes from 1 to +∞, the point follows the part of the circle in the second quadrant from (0, 1) to (−1, 0).
Here is another geometric point of view. Draw the unit circle, and let P be the point (−1, 0) . A line through P (except the vertical line) is determined by its slope. Furthermore, each of the lines (except the vertical line) intersects the unit circle in exactly two points, one of which is P . This determines a function from points on the unit circle to slopes. The trigonometric functions determine a function from angles to points on the unit circle, and by combining these two functions we have a function from angles to slopes.
As with other properties shared between the trigonometric functions and the hyperbolic functions, it is possible to use hyperbolic identities to construct a similar form of the substitution, t = tanh x 2 {\textstyle t=\tanh {\tfrac {x}{2}}} :
sinh x = 2 t 1 − t 2 , cosh x = 1 + t 2 1 − t 2 , and d x = 2 1 − t 2 d t . {\displaystyle \sinh x={\frac {2t}{1-t^{2}}},\quad \cosh x={\frac {1+t^{2}}{1-t^{2}}},\quad {\text{and}}\quad dx={\frac {2}{1-t^{2}}}\,dt.}
Similar expressions can be written for tanh x , coth x , sech x , and csch x . Geometrically, this change of variables is a one-dimensional stereographic projection of the hyperbolic line onto the real interval, analogous to the Poincaré disk model of the hyperbolic plane.
There are other approaches to integrating trigonometric functions. For example, it can be helpful to rewrite trigonometric functions in terms of e ix and e − ix using Euler's formula .
Later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution , for instance:
Weierstrass, Karl (1915) [1875]. "8. Bestimmung des Integrals ..." . Mathematische Werke von Karl Weierstrass (in German). Vol. 6. Mayer & Müller. pp. 89– 99. | https://en.wikipedia.org/wiki/Tangent_half-angle_substitution |
Tangential speed is the speed of an object undergoing circular motion , i.e., moving along a circular path . [ 1 ] A point on the outside edge of a merry-go-round or turntable travels a greater distance in one complete rotation than a point nearer the center. Travelling a greater distance in the same time means a greater speed, and so linear speed is greater on the outer edge of a rotating object than it is closer to the axis. This speed along a circular path is known as tangential speed because the direction of motion is tangent to the circumference of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, and is measured in SI units as meters per second (m/s).
Rotational speed (or rotational frequency) measures the number of revolutions per unit of time. All parts of a rigid merry-go-round or turntable turn about the axis of rotation in the same amount of time. Thus, all parts share the same rate of rotation, or the same number of rotations or revolutions per unit of time.
When a direction is assigned to rotational speed, it is known as rotational velocity , a vector whose magnitude is the rotational speed.
( Angular speed and angular velocity are related to the rotational speed and velocity by a factor of 2 π , the number of radians turned in a full rotation.)
Tangential speed and rotational speed are related: the faster an object rotates around an axis, the larger the speed. Tangential speed is directly proportional to rotational speed at any fixed distance from the axis of rotation. [ 1 ] However, tangential speed, unlike rotational speed, depends on radial distance (the distance from the axis). For a platform rotating with a fixed rotational speed, the tangential speed in the centre is zero. Towards the edge of the platform the tangential speed increases proportional to the distance from the axis. [ 2 ] In equation form: v ∝ r ω , {\displaystyle v\propto \!\,r\omega \,,}
where v is tangential speed and ω (Greek letter omega ) is rotational speed. One moves faster if the rate of rotation increases (a larger value for ω ), and one also moves faster if movement farther from the axis occurs (a larger value for r ). Move twice as far from the rotational axis at the centre and you move twice as fast. Move out three times as far, and you have three times as much tangential speed. In any kind of rotating system, tangential speed depends on how far you are from the axis of rotation.
When proper units are used for tangential speed v , rotational speed ω , and radial distance r , the direct proportion of v to both r and ω becomes the exact equation v = r ω . {\displaystyle v=r\omega \,.} This comes from the following: the linear (tangential) velocity of an object in rotation is the rate at which it covers the circumference's length:
The angular velocity ω {\textstyle \omega } is defined as 2 π / T {\displaystyle 2\pi /T} , where T is the rotation period , hence v = ω r {\displaystyle v=\omega r} .
Thus, tangential speed will be directly proportional to r when all parts of a system simultaneously have the same ω , as for a wheel, disk, or rigid wand.
For tangential velocity vector (rapidity or speed is his norm or module) is the vector product : v → = ω → × r → = | | ω → | | | | r → | | sin ( | Δ θ | ) ⋅ u ^ n = v → = r → ˙ = d r → d t {\displaystyle {\vec {v}}={\vec {\omega }}\times {\vec {r}}=||{\vec {\omega }}||||{\vec {r}}||\sin(|\Delta \theta |)\cdot {\hat {u}}_{n}={\vec {v}}={\dot {\vec {r}}}={d{\vec {r}} \over dt}} Because of the right hand rule linear tangential velocity vector points tangential to the rotation.
Where ω → = d β d t u ^ β {\displaystyle {\vec {\omega }}={d\beta \over dt}{\hat {u}}_{\beta }} is the angular velocity (angular frequency) vector normal to the plane of rotation of the body, where β {\displaystyle \beta } is the angle (scalar in radians) of the rotational movement (similar to r that is the norm (scalar) of the translational movement position vector).
r → {\displaystyle {\vec {r}}} is the position vector (equivalent to radio) to the rotating puntual particle or distributed continuous body or where is measured the tangential velocity in a body.
u ^ n {\displaystyle {\hat {u}}_{n}} is the normal (to the plane of ω → {\displaystyle {\vec {\omega }}} and r → {\displaystyle {\vec {r}}} ) unit vector .
θ {\displaystyle \theta } are the angles of the vectors ω → {\displaystyle {\vec {\omega }}} and r → {\displaystyle {\vec {r}}} in their common plane where they are, form or describe.
Rapidity or speed v {\displaystyle v} is the norm or module of velocity vector v → {\displaystyle {\vec {v}}} :
v = | | v → | | = | | ω → × r → | | = | | ω → | | | | r → | | sin ( | Δ θ | ) = v {\displaystyle v=||{\vec {v}}||=||{\vec {\omega }}\times {\vec {r}}||=||{\vec {\omega }}||||{\vec {r}}||\sin(|\Delta \theta |)=v}
v = | | v → | | = | | ω → × r → | | = | | ω → | | | | r → | | = ω r = v {\displaystyle v=||{\vec {v}}||=||{\vec {\omega }}\times {\vec {r}}||=||{\vec {\omega }}||||{\vec {r}}||=\omega r=v}
Only if: sin ( | Δ θ | ) = 1 {\displaystyle \sin(|\Delta \theta |)=1} , when: | Δ θ | = | θ r → − θ ω → | = π 2 = 90 o {\displaystyle |\Delta \theta |=|\theta _{\vec {r}}-\theta _{\vec {\omega }}|={\pi \over 2}=90^{o}} , when: ω → ⊥ r → {\displaystyle {\vec {\omega }}\perp {\vec {r}}} which means that angular velocity vector is orthogonal (perpendicular) to the position vector.
Tangential aceleration a → {\displaystyle {\vec {a}}} is:
a → = ω → × ( ω → × r → ) = r → ¨ {\displaystyle {\vec {a}}={\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}})={\ddot {\vec {r}}}} | https://en.wikipedia.org/wiki/Tangential_speed |
Tangible symbols are a type of augmentative and alternative communication (AAC) that uses objects or pictures that share a perceptual relationship with the items they represent as symbols . A tangible symbol's relation to the item it represents is perceptually obvious and concrete – the visual or tactile properties of the symbol resemble the intended item. Tangible Symbols can easily be manipulated and are most strongly associated with the sense of touch. These symbols can be used by individuals who are not able to communicate using speech or other abstract symbol systems, such as sign language . [ 1 ] However, for those who have the ability to communicate using speech, learning to use tangible symbols does not hinder further developing acquisition of natural speech and/or language development , and may even facilitate it. [ 2 ]
The term tangible symbols was first developed by Charity Rowland and Philip Schweigert, [ 2 ] [ 3 ] and refers to two-dimensional pictures or three-dimensional objects used as symbols to convey meaning. The items are termed "tangible" because they are concrete items that can be manipulated by the user and communication partner. Symbols can be used individually or combined with other symbols in order to create new messages. Tangible symbols are used as a means of communication for individuals who are unable to understand or communicate using abstract systems, such as speech or sign language . [ 2 ] [ 4 ]
Properties of tangible symbols include permanency, capacity to be manipulated by both the user and the communication partner, and an obvious relationship between the symbol and the referent. They can represent items, people, activities and/or events, and look or feel similar to what they refer to. [ 2 ] [ 3 ] [ 5 ] [ 6 ] For example, a cup can be used as three-dimensional tangible symbol to represent the action: "drink". [ 6 ] A photograph of a cup can be used as a two-dimensional tangible symbol to also represent the action: "drink". Two- and three-dimensional symbols are used to fit the cognitive and sensory abilities of the individual, as well as the individual's unique experiences. [ 2 ] [ 7 ]
Rowland and Schweigert use the term tangible symbols to refer to conceptually tangible items like two-dimensional pictures or three-dimensional objects. [ 2 ] [ 3 ] However, other authors, such as Beukelman and Mirenda, use the term to exclusively describe three-dimensional physical objects that display concrete properties such as shape or texture. [ 4 ]
According to Rowland and Schweigert, "for some individuals, the use of tangible symbols may be used to bridge the gap between gestural communication and the use of formal language systems. For others, tangible symbols may represent an ultimate level of communicative competence." [ 2 ]
[ 2 ]
Historically, objects and pictures have frequently been used as communication devices. [ 3 ] [ 5 ] [ 6 ] Many authors have also used picture symbols, such as line drawings and photographs to develop language in individuals with little or no speech and/or cognitive disabilities . [ 2 ] Tangible symbols emerged from Van Dijk’s work in the 1960s using objects as symbols to develop language in deaf-blind children. [ 3 ] [ 8 ] [ 9 ] In turn, Van Dijk’s work was based on the concept "symbol formation" developed by Werner and Kaplan (1963), who theorized that "symbol formation" referred to the process of developing language by creating symbols in our minds. [ 3 ] [ 10 ]
Rowland and Schweigert propose that tangible symbols can be divided into hierarchical categories, ranging from most concrete to most abstract symbols: [ 2 ] [ 3 ]
The type of tangible symbol used is chosen based on the cognitive and sensory abilities of the learner/user. [ 2 ] [ 3 ] The meaning behind each symbol is not universal, but by using a symbol the individual is familiar with, a meaningful symbol is created. Tangible symbols should be constructed by meaningful and motivating symbols that will provide the individual with the most opportunities to practice using the new system. [ 2 ] [ 3 ]
Individuals who can benefit from using tangible symbols include those who may lack the skills to communicate using verbal speech or other various communication systems such as sign language. Users of tangible symbols may include individuals with cognitive disabilities (including developmental delay and intellectual disability), sensory and/or visual impairments (blindness and/or deafblindness), developmental disabilities (such as autism spectrum disorder), and orthopedic impairments. [ 2 ] [ 4 ] Rowland and Schweigert claim that tangible symbols do not require the use of high demands on the learner’s cognitive abilities , memory , visual perception , and motor abilities because they are:
Furthermore, simple behavioral responses can be used with tangible items. For example, learners that are unable to speak can simply point, touch, pick up, or look (in cases of severe motoric impairment) at the object to answer a question or make a request. [ 2 ] [ 3 ] Finally, three-dimensional objects can be distinguished from one another using touch, and therefore they are suitable for people with visual impairments or blindness. [ 2 ] [ 3 ] A study by Rowland and Schweigert found individuals who were already able to communicate using gestures or vocalizations more readily learned to use tangible symbols than those who did not have intentional pre-symbolic communication skills.
Presentation format depends on the users visual scanning and motoric ability. The tangible symbols can placed in front of the user within reach, placed on a board for visual scanning, or placed in a book for access.
Typically, tangible symbols are custom made and tailored to the individual child. If pre-made sets are used, it is assumed that the symbols are familiar and motivating for the user. It is important to utilize frequently occurring and highly motivating symbols in order to optimize opportunities for use. [ 2 ]
Tangible symbol system offers a manual and DVD as well as an online course. For more information on tangible symbol system instructional strategies, please reference: http://designtolearn.com/products/tangible_symbol_systems
In 2009, Ellen Trief, Susan M. Bruce, Paul W. Cascella, and Sarah Ivy created a Universal Tangible Symbol System. They began by developing a survey to determine which tangible symbols were already in use, new activities and concepts for which tangible systems are needed, and participant preferences for tangible symbols from a pilot study. Participants included teachers and speech–language pathologists from four New York City schools. Following the survey, an advisory board consisting of directors of the New York City schools, speech-language pathologists, the designer and manufacturer of the symbols, a representative from the Perkins School for the Blind, college professors, and a graduate research assistant reviewed and discussed the results. This resulted in the establishment of the 55 universal tangible symbols seen in the chart below. However, this universal tangible symbol system should not replace a system already established for an individual. [ 13 ] | https://en.wikipedia.org/wiki/Tangible_symbol_systems |
Tango (named Project Tango while in testing) was an augmented reality computing platform , developed and authored by the Advanced Technology and Projects (ATAP), a skunkworks division of Google . It used computer vision to enable mobile devices , such as smartphones and tablets , to detect their position relative to the world around them without using GPS or other external signals. This allowed application developers to create user experiences that include indoor navigation , 3D mapping, physical space measurement, environmental recognition, augmented reality , and windows into a virtual world .
The first product to emerge from ATAP, [ 1 ] Tango was developed by a team led by computer scientist Johnny Lee , a core contributor to Microsoft 's Kinect . In an interview in June 2015, Lee said, "We're developing the hardware and software technologies to help everything and everyone understand precisely where they are, anywhere." [ 2 ]
Google produced two devices to demonstrate the Tango technology: the Peanut phone and the Yellowstone 7-inch tablet. More than 3,000 of these devices had been sold as of June 2015, [ 3 ] chiefly to researchers and software developers interested in building applications for the platform. In the summer of 2015, Qualcomm and Intel both announced that they were developing Tango reference devices as models for device manufacturers who use their mobile chipsets . [ 4 ] [ 5 ]
At CES, in January 2016, Google announced a partnership with Lenovo to release a consumer smartphone during the summer of 2016 to feature Tango technology marketed at consumers, noting a less than $500 price-point and a small form factor below 6.5 inches. At the same time, both companies also announced an application incubator to get applications developed to be on the device on launch.
On 15 December 2017, Google announced that they would be ending support for Tango on March 1, 2018, in favor of ARCore . [ 6 ]
Tango was different from other contemporary 3D-sensing computer vision products, in that it was designed to run on a standalone mobile phone or tablet and was chiefly concerned with determining the device's position and orientation within the environment.
The software worked by integrating three types of functionality:
Together, these generate data about the device in " six degrees of freedom " (3 axes of orientation plus 3 axes of position) and detailed three-dimensional information about the environment.
Project Tango was also the first project to graduate from Google X in 2012 [ 7 ]
Applications on mobile devices use Tango's C and Java APIs to access this data in real time. In addition, an API was also provided for integrating Tango with the Unity game engine ; this enabled the conversion or creation of games that allow the user to interact and navigate in the game space by moving and rotating a Tango device in real space. These APIs were documented on the Google developer website. [ 8 ]
Tango enabled apps to track a device's position and orientation within a detailed 3D environment, and to recognize known environments. This allowed the creations of applications such as in-store navigation, visual measurement and mapping utilities, presentation and design tools, [ 9 ] and a variety of immersive games. At Augmented World Expo 2015, [ 10 ] Johnny Lee demonstrated a construction game that builds a virtual structure in real space, an AR showroom app that allows users to view a full-size virtual automobile and customize its features, a hybrid Nerf gun with mounted Tango screen for dodging and shooting AR monsters superimposed on reality, and a multiplayer VR app that lets multiple players converse in a virtual space where their avatar movements match their real-life movements. [ 11 ]
Tango apps are distributed through Play . Google has encouraged the development of more apps with hackathons , an app contest, and promotional discounts on the development tablet. [ 12 ]
As a platform for software developers and a model for device manufacturers, Google created two Tango devices.
"Peanut" was the first production Tango device, released in the first quarter of 2014. It was a small Android phone with a Qualcomm MSM8974 quad-core processor and additional special hardware including a fisheye motion camera , "RGB-IR" camera for color image and infrared depth detection, and Movidius Vision processing units . A high-performance accelerometer and gyroscope were added after testing several competing models in the MARS lab at the University of Minnesota .
Several hundred Peanut devices were distributed to early-access partners including university researchers in computer vision and robotics , as well as application developers and technology startups . Google stopped supporting the Peanut device in September 2015, as by then the Tango software stack had evolved beyond the versions of Android that run on the device.
"Yellowstone" was a 7-inch tablet with full Tango functionality, released in June 2014, and sold as the Project Tango Tablet Development Kit. [ 13 ] It featured a 2.3 GHz quad-core Nvidia Tegra K1 processor, 128GB flash memory, 1920x1200-pixel touchscreen, 4MP color camera, fisheye-lens (motion-tracking) camera, an IR projector with RGB-IR camera for integrated depth sensing, and 4G LTE connectivity. [ 14 ] [ 15 ] As of May 27, 2017, the Tango tablet is considered officially unsupported by Google. [ 16 ]
In May 2014, two Peanut phones were delivered to the International Space Station to be part of a NASA project to develop autonomous robots that navigate in a variety of environments, including outer space. The soccer-ball-sized, 18-sided polyhedral SPHERES robots were developed at the NASA Ames Research Center , adjacent to the Google campus in Mountain View, California . Andres Martinez, SPHERES manager at NASA, said "We are researching how effective [Tango's] vision-based navigation abilities are for performing localization and navigation of a mobile free flyer on ISS. [ 17 ]
Announced at Intel's Developer Forum in August 2015, [ 18 ] and offered to public through a Developer Kit since January 2016. [ 19 ] It incorporated a RealSense ZR300 camera [ 20 ] which had optical features required for Tango, such as the fisheye camera. [ 21 ]
Lenovo Phab 2 Pro was the first commercial smartphone with the Tango Technology, the device was announced at the beginning of 2016, launched in August, and available for purchase in the US in November. The Phab 2 Pro had a 6.4 inch screen, a Snapdragon 652 processor, and 64 GB of internal storage, with a rear facing 16 Megapixels camera and 8 MP front camera.
Asus Zenfone AR, announced at CES 2017, [ 22 ] was the second commercial smartphone with the Tango Technology. It ran Tango AR & Daydream VR on Snapdragon 821 , with 6GB or 8GB of RAM and 128 or 256GB of internal memory depending on the configuration. | https://en.wikipedia.org/wiki/Tango_(platform) |
Taniyama's problems are a set of 36 mathematical problems posed by Japanese mathematician Yutaka Taniyama in 1955. The problems primarily focused on algebraic geometry , number theory , and the connections between modular forms and elliptic curves . [ 1 ] [ 2 ] [ 3 ]
In the 1950s post-World War II period of mathematics, there was renewed interest in the theory of modular curves due to the work of Taniyama and Goro Shimura . [ 3 ] During the 1955 international symposium on algebraic number theory at Tokyo and Nikkō —the first symposium of its kind to be held in Japan that was attended by international mathematicians including Jean-Pierre Serre , Emil Artin , Andre Weil , Richard Brauer , K. G. Ramanathan , and Daniel Zelinsky [ 4 ] —Taniyama compiled his 36 problems in a document titled "Problems of Number Theory" and distributed mimeographs of his collection to the symposium's participants. These problems would become well known in mathematical folklore . [ 2 ] [ 5 ] Serre later brought attention to these problems in the early 1970s. [ 3 ]
The most famous of Taniyama's problems are his twelfth and thirteenth problems. [ 3 ] [ 2 ] These problems led to the formulation of the Taniyama–Shimura conjecture (now known as the modularity theorem ), which states that every elliptic curve over the rational numbers is modular . This conjecture became central to modern number theory and played a crucial role in Andrew Wiles ' proof of Fermat's Last Theorem in 1995. [ 2 ] [ 5 ]
Taniyama's problems influenced the development of modern number theory and algebraic geometry , including the Langlands program , the theory of modular forms , and the study of elliptic curves . [ 2 ]
Taniyama's tenth problem addressed Dedekind zeta functions and Hecke L-series , and while distributed in English at the 1955 Tokyo - Nikkō conference attended by both Serre and André Weil , it was only formally published in Japanese in Taniyama's collected works. [ 3 ]
Let k {\displaystyle k} be a totally real number field , and F ( τ ) {\displaystyle F(\tau )} be a Hilbert modular form to the field k {\displaystyle k} . Then, choosing F ( τ ) {\displaystyle F(\tau )} in a suitable manner, we can obtain a system of Erich Hecke 's L -series with Größencharakter λ {\displaystyle \lambda } , which corresponds one-to-one to this F ( τ ) {\displaystyle F(\tau )} by the process of Mellin transformation . This can be proved by a generalization of the theory of operator T {\displaystyle T} of Hecke to Hilbert modular functions (cf. Hermann Weyl ). [ 3 ]
According to Serge Lang , Taniyama's eleventh problem deals with elliptic curves with complex multiplication, but is unrelated to Taniyama's twelfth and thirteenth problems. [ 3 ]
Let C {\displaystyle C} be an elliptic curve defined over an algebraic number field k {\displaystyle k} , and L C ( s ) {\displaystyle L_{C}(s)} the L -function of C {\displaystyle C} over k {\displaystyle k} in the sense that ζ C ( s ) = ζ k ( s ) ζ k ( s − 1 ) / L C ( s ) {\displaystyle \zeta _{C}(s)=\zeta _{k}(s)\zeta _{k}(s-1)/L_{C}(s)} is the zeta function of C {\displaystyle C} over k {\displaystyle k} . If the Hasse–Weil conjecture is true for ζ C ( s ) {\displaystyle \zeta _{C}(s)} , then the Fourier series obtained from L C ( s ) {\displaystyle L_{C}(s)} by the inverse Mellin transformation must be an automorphic form of dimension −2 of a special type (see Hecke [ a ] ). If so, it is very plausible that this form is an ellipic differential of the field of associated automorphic functions. Now, going through these observations backward, is it possible to prove the Hasse–Weil conjecture by finding a suitable automorphic form from which L C ( s ) {\displaystyle L_{C}(s)} can be obtained? [ 6 ] [ 3 ]
Taniyama's twelfth problem's significance lies in its suggestion of a deep connection between elliptic curves and modular forms . While Taniyama's original formulation was somewhat imprecise, it captured a profound insight that would later be refined into the modularity theorem . [ 1 ] [ 2 ] The problem specifically proposed that the L -functions of elliptic curves could be identified with those of certain modular forms, a connection that seemed surprising at the time.
Fellow Japanese mathematician Goro Shimura noted that Taniyama's formulation in his twelfth problem was unclear: the proposed Mellin transform method would only work for elliptic curves over rational numbers . [ 1 ] For curves over number fields , the situation is substantially more complex and remains unclear even at a conjectural level today. [ 2 ]
To characterize the field of elliptic modular functions of level N {\displaystyle N} , and especially to decompose the Jacobian variety J {\displaystyle J} of this function field into simple factors up to isogeny . Also it is well known that if N = q {\displaystyle N=q} , a prime , and q ≡ 3 ( mod 4 ) {\displaystyle q\equiv 3{\pmod {4}}} , then J {\displaystyle J} contains elliptic curves with complex multiplication. What can one say for general N {\displaystyle N} ? [ 1 ] | https://en.wikipedia.org/wiki/Taniyama's_problems |
In number theory , the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves , which was enough to imply Fermat's Last Theorem . Later, a series of papers by Wiles's former students Brian Conrad , Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil , extended Wiles's techniques to prove the full modularity theorem in 2001. Before that, the statement was known as the Taniyama–Shimura conjecture , Taniyama–Shimura–Weil conjecture , or the modularity conjecture for elliptic curves .
The theorem states that any elliptic curve over Q {\displaystyle \mathbb {Q} } can be obtained via a rational map with integer coefficients from the classical modular curve X 0 ( N ) for some integer N ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level N . If N is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor ), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N , a normalized newform with integer q -expansion, followed if need be by an isogeny .
The modularity theorem implies a closely related analytic statement:
To each elliptic curve E over Q {\displaystyle \mathbb {Q} } we may attach a corresponding L -series . The L -series is a Dirichlet series , commonly written
The generating function of the coefficients a n is then
If we make the substitution
we see that we have written the Fourier expansion of a function f ( E , τ ) of the complex variable τ , so the coefficients of the q -series are also thought of as the Fourier coefficients of f . The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators ); this is the Hasse–Weil conjecture , which follows from the modularity theorem.
Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties , corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it).
Yutaka Taniyama [ 1 ] stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō as the twelfth of his set of 36 unsolved problems . Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil [ 2 ] rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted L -series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the Langlands program . [ 3 ] [ 4 ]
The conjecture attracted considerable interest when Gerhard Frey [ 5 ] suggested in 1986 that it implies Fermat's Last Theorem . He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre [ 6 ] identified a missing link (now known as the epsilon conjecture or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture. [ 7 ]
Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to prove. [ 8 ] For example, Wiles's Ph.D. supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".
In 1995, Andrew Wiles, with some help from Richard Taylor , proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves . Wiles used this to prove Fermat's Last Theorem, [ 9 ] and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond, [ 10 ] Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999. [ 11 ] [ 12 ] Once fully proven, the conjecture became known as the modularity theorem.
Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime n th powers, n ≥ 3 . [ a ]
The modularity theorem is a special case of more general conjectures due to Robert Langlands . The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry , such as to every elliptic curve over a number field . Most cases of these extended conjectures have not yet been proved.
In 2013, Freitas, Le Hung, and Siksek proved that elliptic curves defined over real quadratic fields are modular. [ 13 ]
For example, [ 14 ] [ 15 ] [ 16 ] the elliptic curve y 2 − y = x 3 − x , with discriminant (and conductor) 37, is associated to the form
For prime numbers l not equal to 37, one can verify the property about the coefficients. Thus, for l = 3 , there are 6 solutions of the equation modulo 3: (0, 0) , (0, 1) , (1, 0) , (1, 1) , (2, 0) , (2, 1) ; thus a (3) = 3 − 6 = −3 .
The conjecture, going back to the 1950s, was completely proven by 1999 using the ideas of Andrew Wiles , who proved it in 1994 for a large family of elliptic curves. [ 17 ]
There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ , from the modular curve X 0 ( N ) to E . In particular, the points of E can be parametrized by modular functions .
For example, a modular parametrization of the curve y 2 − y = x 3 − x is given by [ 18 ]
where, as above, q = e 2 πiz . The functions x ( z ) and y ( z ) are modular of weight 0 and level 37; in other words they are meromorphic , defined on the upper half-plane Im( z ) > 0 and satisfy
and likewise for y ( z ) , for all integers a , b , c , d with ad − bc = 1 and 37 | c .
Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.
The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5 , the Fermat equation
has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch [ fr ] was the first to notice, [ 19 ] the elliptic curve
of discriminant
cannot be modular. [ 7 ] Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987. [ 20 ] | https://en.wikipedia.org/wiki/Taniyama-Shimura_conjecture |
In number theory , the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves , which was enough to imply Fermat's Last Theorem . Later, a series of papers by Wiles's former students Brian Conrad , Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil , extended Wiles's techniques to prove the full modularity theorem in 2001. Before that, the statement was known as the Taniyama–Shimura conjecture , Taniyama–Shimura–Weil conjecture , or the modularity conjecture for elliptic curves .
The theorem states that any elliptic curve over Q {\displaystyle \mathbb {Q} } can be obtained via a rational map with integer coefficients from the classical modular curve X 0 ( N ) for some integer N ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level N . If N is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor ), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N , a normalized newform with integer q -expansion, followed if need be by an isogeny .
The modularity theorem implies a closely related analytic statement:
To each elliptic curve E over Q {\displaystyle \mathbb {Q} } we may attach a corresponding L -series . The L -series is a Dirichlet series , commonly written
The generating function of the coefficients a n is then
If we make the substitution
we see that we have written the Fourier expansion of a function f ( E , τ ) of the complex variable τ , so the coefficients of the q -series are also thought of as the Fourier coefficients of f . The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators ); this is the Hasse–Weil conjecture , which follows from the modularity theorem.
Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties , corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it).
Yutaka Taniyama [ 1 ] stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō as the twelfth of his set of 36 unsolved problems . Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil [ 2 ] rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted L -series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the Langlands program . [ 3 ] [ 4 ]
The conjecture attracted considerable interest when Gerhard Frey [ 5 ] suggested in 1986 that it implies Fermat's Last Theorem . He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre [ 6 ] identified a missing link (now known as the epsilon conjecture or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture. [ 7 ]
Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to prove. [ 8 ] For example, Wiles's Ph.D. supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".
In 1995, Andrew Wiles, with some help from Richard Taylor , proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves . Wiles used this to prove Fermat's Last Theorem, [ 9 ] and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond, [ 10 ] Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999. [ 11 ] [ 12 ] Once fully proven, the conjecture became known as the modularity theorem.
Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime n th powers, n ≥ 3 . [ a ]
The modularity theorem is a special case of more general conjectures due to Robert Langlands . The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry , such as to every elliptic curve over a number field . Most cases of these extended conjectures have not yet been proved.
In 2013, Freitas, Le Hung, and Siksek proved that elliptic curves defined over real quadratic fields are modular. [ 13 ]
For example, [ 14 ] [ 15 ] [ 16 ] the elliptic curve y 2 − y = x 3 − x , with discriminant (and conductor) 37, is associated to the form
For prime numbers l not equal to 37, one can verify the property about the coefficients. Thus, for l = 3 , there are 6 solutions of the equation modulo 3: (0, 0) , (0, 1) , (1, 0) , (1, 1) , (2, 0) , (2, 1) ; thus a (3) = 3 − 6 = −3 .
The conjecture, going back to the 1950s, was completely proven by 1999 using the ideas of Andrew Wiles , who proved it in 1994 for a large family of elliptic curves. [ 17 ]
There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ , from the modular curve X 0 ( N ) to E . In particular, the points of E can be parametrized by modular functions .
For example, a modular parametrization of the curve y 2 − y = x 3 − x is given by [ 18 ]
where, as above, q = e 2 πiz . The functions x ( z ) and y ( z ) are modular of weight 0 and level 37; in other words they are meromorphic , defined on the upper half-plane Im( z ) > 0 and satisfy
and likewise for y ( z ) , for all integers a , b , c , d with ad − bc = 1 and 37 | c .
Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.
The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5 , the Fermat equation
has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch [ fr ] was the first to notice, [ 19 ] the elliptic curve
of discriminant
cannot be modular. [ 7 ] Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987. [ 20 ] | https://en.wikipedia.org/wiki/Taniyama–Shimura_conjecture |
Tanja Stadler is a mathematician and professor of computational evolution at the Swiss Federal Institute of Technology ( ETH Zurich ). She’s the current president of the Swiss Scientific Advisory Panel COVID-19 and Vize-Chair of the Department of Biosystems Science and Engineering at ETH Zürich .
Tanja Stadler studied applied mathematics and statistics at the Technical University of Munich , University of Cardiff , and the University of Canterbury . [ 1 ] She continued at the Technical University of Munich to obtain a PhD in 2008 on the topic 'Evolving Trees – Models for Speciation and Extinction in Phylogenetics' (with Prof. Anusch Taraz and Prof. Mike Steel). [ 2 ] After a postdoctoral period with Prof. Sebastian Bonhoeffer in the Department of Environmental Systems Sciences at ETH Zürich, she was promoted to Junior Group Leader at ETH Zürich in 2011. In 2014, she became an assistant professor at the Department of Biosystems Science and Engineering of ETH Zürich , where she was promoted to associate professor in 2017 and to full professor in 2021. [ 3 ]
Tanja's research addresses core questions in the life sciences through an evolutionary perspective, in particular in macroevolution , epidemiology , developmental biology and immunology . Her research questions include fundamental aspects such as how speciation processes led to the current biodiversity , as well as questions directly relevant to human societies, such as the spread of pathogens like COVID-19 or Ebola . Tanja assesses these questions by developing and applying statistical phylodynamic tools [ 4 ] to estimate evolutionary and population dynamics from genomic sequencing data while in parallel leading consortia to produce such data. Her unique approach is an innovative mix of mathematics , computer science and biology .
Tanja made major theoretical contributions to the field of phylodynamics by developing statistical frameworks to use birth-death processes in the context of phylogenetic trees. [ 5 ] In particular, she laid the foundations to account for sampling through time in birth-death models – enabling coherent analysis of genetic sequencing data collected through time during epidemics as well as coherent analysis of fossil (collected sequentially through time) and present-day species data. [ 6 ]
Tanja used this framework for example to quantify HCV spread, the spread of Ebola during the 2014 outbreak, assess Zika spread, to show that influenza waves in a city are majorly driven by travel patterns, and to provide real-time information during the COVID-19 pandemic . [ 7 ] [ 8 ] [ 9 ] In macroevolution, Tanja explored in particular the impact of dinosaur extinction on mammal diversification. [ 10 ] Most recently, she is introducing statistical tree thinking into developmental biology . [ 11 ]
Her group founded “Taming the BEAST”, in 2016. BEAST 2 is a widely used Bayesian phylogenetic software platform allowing to infer evolutionary and population dynamics from genomic sequencing data to which Tanja’s team contributed many package. “Taming the BEAST” is both an international workshop series and an online resource, to teach the usage of BEAST 2. [ 12 ] [ 13 ]
In the field of epidemiology , Tanja is currently spear-heading the use of wastewater information to understand pathogen spread. She is principal investigator of a project between ETH Zürich and Eawag . Her team is estimating the reproductive number for SARS-CoV-2 and influenza from wastewater and contributes to understanding variant dynamics. [ 14 ]
During the COVID-19 pandemic , Tanja was president of the Swiss National COVID-19 Science task force advising the authorities and decision makers of Switzerland from August 2021 until the termination of the task force in March 2022. [ 15 ] She started the presidency after having been a member and later chaired the data & modelling group of the task force. She was responsible for the weekly communication of the pandemic situation to the Swiss Federal Government and the corresponding authorities. In addition, she presented scientific insights in briefings with the complete Federal Government and with members of the executive branches of the Federal and Cantonal Governments, as well as with different divisions of the Swiss Parliament .
Tanja actively contributed core scientific insights to the Task Force. Her daily calculations of the reproductive number became a key part of the epidemic monitoring. [ 16 ] The reproductive numbers were employed in the national „Ordinance of 19 June 2020 on Measures during the Special Situation to combat the COVID-19 Epidemic”. Further, the reproductive number dashboard was highlighted when the South Africa Health department informed the world about the new variant Omicron. Tanja also lead the most extensive Swiss-based SARS-CoV-2 sequencing effort providing results on the emergence and spread of new variants. [ 17 ] Through this effort, the first beta, gamma, and delta variants in Switzerland were detected. [ 18 ] [ 19 ]
The platform cov-spectrum is developed by Tanja’s team and became essential in SARS-CoV-2 variant tracking. [ 20 ] It is widely used to facilitate SARS-CoV-2 lineage designation and used in policy such as the FDA advisory committee meeting discussing possible SARS-CoV-2 strains for a vaccine update. During the mpox outbreak , the team launched mpox-spectrum within days to track the newly spreading virus. [ 21 ]
In addition to advising the government and informing policy makers, she became actively involved in informing the public about the situation of the pandemic. Tanja communicated the scientific insights often on national news and national TV shows in Switzerland, as well as through Federal press conferences. [ 22 ]
Stadler lives with her partner and their two daughters in Basel. [ 23 ] | https://en.wikipedia.org/wiki/Tanja_Stadler |
Tank blanketing , also called gas sealing or tank padding , is the process of applying a gas to the empty space in a storage container . The term storage container here refers to any container that is used to store products, regardless of its size. Though tank blanketing is used for a variety of reasons, it typically involves using a buffer gas to protect products inside the storage container. A few of the benefits of blanketing include a longer product life in the container, reduced hazards, and longer equipment life cycles.
In 1970, Appalachian Controls Environmental (ACE) was the world’s first company to introduce a tank blanketing valve . There are now many ready-made systems available for purchase from a variety of process equipment companies. It is also possible to piece together your own system using a variety of different equipment. Regardless of which method is used, the basic requirements are the same. There must be a way of allowing the blanketing gas into the system, and a way to vent the gas should the pressure get too high.
Since ACE introduced its valve many companies have engineered their own versions. Though many of the products available vary in features and applicability, the fundamental design is the same. When the pressure inside the container drops below a set point, a valve opens and allows the blanketing gas to enter. Once the pressure reaches the set point, the valve closes. As a safety feature, many systems include a pressure vent that opens when the pressure inside exceeds a maximum pressure set point. This helps to prevent the container from rupturing due to high pressure. Since most blanketing gas sources will provide gas at a much higher than desired pressure, a blanketing system will also use a pressure reducing valve to decrease the inlet pressure to the tank.
Although it varies from application to application, blanketing systems usually operate at a slightly higher than atmospheric pressure (a few inches of water column above atmospheric ). Higher pressures than this are generally not used as they often yield only marginal increases in results while wasting large amounts of expensive blanketing gas.
Some systems also utilize inert gases to agitate the liquid contents of the container. This is desirable because products, such as citric acid, are added to food oils the tank will begin to settle over time with the heavier contents sinking to the bottom. However, a system that utilizes nitrogen sparging (and then subsequently tank blanketing once the nitrogen reaches the vapor space) may have negative impact on the products involved. Nitrogen sparging creates a significantly higher amount of surface contact between the gas and the product, which in turn creates a much larger opportunity for undesired oxidation to occur. It is possible for nitrogen that is as much 99.9% free of oxygen to increase the amount of oxidation within the product due to the high amount of surface contact.
The most common gas used in blanketing is nitrogen . Nitrogen is widely used due to its inert properties, as well as its availability and relatively low cost. Tank blanketing is used for a variety of products including cooking oils , volatile combustible products, and purified water . These applications also cover a wide variety of storage containers, ranging from as large as a tank containing millions of gallons of vegetable oil down to a quart-size container or smaller. Nitrogen is appropriate for use at any of these scales.
The use of an inert blanketing gas for food products helps to keep oxygen levels low in and around the product. Low levels of oxygen surrounding the product help to reduce the amount of oxidation that may occur, and increases shelf life. In the case of cooking oils, lipid oxidation can cause the oil to change its color , flavor , or aroma . It also decreases the nutrient levels in the food and can even generate toxic substances. Tank blanketing strategies are also implemented to prepare the product for transit ( railcar or truck ) and for final packaging before sealing the product.
When considering the application for combustible products, the greatest benefit is process safety. Since fuels require oxygen to combust, reduced oxygen content in the vapor space lowers the risk of unwanted combustion.
Tank blanketing is also used to keep contaminants out of a storage space. This is accomplished by creating positive pressure inside the container. This positive pressure ensures that if a leak should occur, the gas will leak out rather than having the contaminants infiltrate the container. Some examples include its use on purified water to keep unwanted minerals out and its use on food products to keep contaminants out.
To ensure their safety, gas-blanketing systems for food use are regulated by the U.S. Food and Drug Administration (FDA) and must adhere to strict maintenance schedules and follow all product-contact regulations with regards to purity, toxicity, and filter specs. As with any use of inert gases, care must be taken to ensure that workers are not exposed to large quantities of nitrogen or other non-breathable substances, which can quickly result in asphyxiation and death. [ 1 ] Use of them in commercial applications is subject to the regulation of OSHA in the USA and similar regulatory bodies elsewhere.
Author unavailable (2000), Fisher Controls becomes an “ACE” in tank blanketing [Electronic version]. Control Engineering Europe, July 2000 , 12.
Kanner, J., Rosenthal, I. (1992), An Assessment of Lipid Oxidation in Foods [Electronic version]. Pure Appl. Chem., Vol. 64, No. 12 , 1959-1964. Retrieved February 15, 2007, from http://www.iupac.org/publications/pac/1992/pdf/6412x1959.pdf
Amos, Kenna (1999). Leakless vapor-space valve controls unveiled. InTech, January 1999 . Retrieved February 15, 2007, from http://findarticles.com/p/articles/mi_qa3739/is_199901/ai_n8840650 | https://en.wikipedia.org/wiki/Tank_blanketing |
In metallurgical processes tank leaching is a hydrometallurgical method of extracting valuable material (usually metals) from ore.
Tank leaching is usually differentiated from vat leaching on the following factors:
In a tank leach the slurry is moved, while in a vat leach the solids remain in the vat, and solution is moved.
Tank and vat leaching involves placing ore, usually after size reduction and classification, into large tanks or vats at ambient operating conditions containing a leaching solution and allowing the valuable material to leach from the ore into solution.
In tank leaching the ground, classified solids are already mixed with water to form a slurry or pulp, and this is pumped into the tanks. Leaching reagents are added to the tanks to achieve the leaching reaction. In a continuous system the slurry will then either overflow from one tank to the next, or be pumped to the next tank. Ultimately the “pregnant” solution is separated from the slurry using some form of liquid/solid separation process, and the solution passes on to the next phase of recovery.
In vat leaching the solids are loaded into the vat, once full the vat is flooded with a leaching solution. The solution drains from the tank, and is either recycled back into the vat or is pumped to the next step of the recovery process. Vat leach units are rectangular containers (drums, barrels, tanks or vats), usually very big and made of wood or concrete, lined with material resistant to the leaching media. The treated ore is usually coarse.
The vats are usually run sequentially to maximize the contact time between the ore and the reagent. In such a series the leachate collected from one container is added to another vat with fresher ore
As mentioned previously tanks are equipped with agitators to keep the solids in suspension in the vats and improve the solid to liquid to gas contact. [ 1 ] Agitation is further assisted by the use of tank baffles to increase the efficiency of agitation and prevent centrifuging of slurries in circular tanks...
Aside from chemical requirements several key factors influence extraction efficiency:
The tank leaching method is commonly used to extract gold and silver from ore, such as with the Sepro Leach Reactor . | https://en.wikipedia.org/wiki/Tank_leaching |
The tank services industry exists to assist companies in maintaining their tanks.
Regular maintenance, as well as other services are required for many types of above ground storage tank systems used in the energy and petro-chemical industry.
Some of the areas that tank service companies provide assistance with are:
Companies that specialise in Tank Services, oil tank removals, replacement and installations in the UK follow detailed rules & regulations. OFTEC governs a comprehensive set of guidelines and regulations on fuel tank installations. Tank Servicing companies follow a combination of both Building Regulations and OFTEC Guidelines to provide a fully compliant safe installation.
At the start of any project, an API certified above ground storage tank inspector should be on hand to provide inspection and consultation services, and to ensure compliance with applicable codes and standards in the region. Each tank upgrading project has specific requirements and presents unique challenges. Proper procedure will ensure project safety, tank integrity, and the best utilization of resources and materials. A complete engineering analysis should be performed to ensure that tank components will not be overstressed during lifting and remedial operations. A professional engineer will develop project-specific drawings and procedures, and will supervise each stage of the work.
An analysis of the situation will result in recommendations for the appropriate remedial action, and what repair work should be performed, possibly including but not limited to:
Storage tanks should be updated to meet the highest local environmental standards, including the installation of release prevention barriers and leak detection.
Several different technologies exist for lifting tanks. The conventional methods involve hydraulic jacking equipment, while another method utilizes an airbag lifting system to elevate the tank from its base to allow for remedial action.
Some companies choose to relocate existing tanks, rather than embark on new tank construction. Tank relocation services can be utilized on land or via navigable waterways. There can be cost benefits to relocating an existing tank, depending on variables such as distance and the condition of the tank.
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tank_services |
Tanla Platforms Ltd , previously known as Tanla Solutions Ltd , is an Indian multinational communications platform as a service (CPaaS) company based in Hyderabad . The company provides value-added services in the cloud communications space. Tanla has more than 600 employees across its offices, including two overseas locations– Singapore and Dubai . The company is listed on BSE and NSE in India. [ 3 ]
Tanla was founded by Dasari Uday Kumar Reddy in May 1999 during the dot-com bubble . [ 4 ]
In December 2006, Tanla was listed on BSE and NSE after its initial public offering . [ 5 ]
Tanla acquired Finland -based mobile payments company OpenBit Oy (now called Tanla Oy) in 2008. In June 2008, Tanla picked up 85% in the company, followed by an acquisition of 5% in June 2009 and the remaining 10% in April 2010. [ 6 ] [ 7 ]
Tanla Platforms, in August 2018, acquired 100% of Karix Mobile (formerly known as mGage India) and its wholly-owned subsidiary Unicel from GSO Capital Partners, a Blackstone company, at an enterprise value of ₹ 340 crore. [ 8 ] The closure of this acquisition was announced in April 2019. [ 9 ]
In 2023, Tanla acquired 100% stake in ValueFirst Group from Twilio for $42 million ( ₹ 346 crore). [ 10 ]
Tanla Platforms Limited initially started as a Bulk SMS provider in Hyderabad catering to SME . As the team grew, the company evolved into a Cloud communication provider with services and products with aggregators and Telcos across the globe.
This article about an Indian company is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tanla_Platforms |
Tanmay A. M. Bharat is a programme leader in the Structural Studies Division of the MRC Laboratory of Molecular Biology . [ 1 ] He and his group use electron tomography, together with several structural and cell biology methods to study the cell surfaces of bacteria and archaea. His work has increased the understanding of how surface molecules help in the formation of multicellular communities of prokaryotes, examples of which include biofilms and microbiomes. [ 2 ] He has been awarded several prizes and fellowships for his work. [ 3 ] [ 4 ]
Bharat graduated with a BA in Biological Sciences from the University of Oxford, UK. His studies were supported by a Rhodes Scholarship . He then undertook research at the European Molecular Biology Laboratory in Heidelberg, Germany for his PhD working with John A. G. Briggs. [ 5 ] He studied the structure and assembly of pathogenic viruses using cryogenic electron microscopy and tomography . His work on several viral capsid proteins improved understanding of how viruses are assembled within infected cells. [ 6 ] [ 7 ] [ 8 ] [ 9 ]
He subsequently joined the MRC Laboratory of Molecular Biology (LMB) in Cambridge to pursue post-doctoral research with Jan Löwe using cryo-EM to study proteins within bacterial cells. [ 5 ] [ 10 ] [ 11 ] [ 12 ] After his post-doctoral appointment concluded, he was recruited to the Sir William Dunn School of Pathology , University of Oxford as a Wellcome Trust and Royal Society Sir Henry Dale Fellow. [ 13 ] After obtaining tenure at Oxford, he moved back to the LMB as a programme leader in 2022. [ 14 ] His research investigates how bacteria and archaea use their surface molecules to form multicellular communities. For instance, during human infections bacteria form biofilms that help them evade antibiotics . The group also use electron tomography . [ 6 ]
Bharat is the author or co-author of over 65 scientific publications. These include:
Bharat has been awarded many prizes and fellowships. [ 5 ] These include a 2018 Vallee Research Scholarship, [ 5 ] the 2019 EMBL John Kendrew Award [ 17 ] the 2020 Philip Leverhulme Prize for Biological Sciences, [ 18 ] the 2021 Eppendorf Award for Young European Investigators, [ 19 ] and the 2021 Lister Prize, [ 20 ] the 2022 Colworth Medal from the Biochemical Society [ 6 ] and the 2023 Fleming Prize from the Microbiology Society . [ 4 ] | https://en.wikipedia.org/wiki/Tanmay_A._M._Bharat |
In mathematics , Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations . It is a natural extension of Pontryagin duality , between compact and discrete commutative topological groups, to groups that are compact but noncommutative . The theory is named after Tadao Tannaka and Mark Grigorievich Krein . In contrast to the case of commutative groups considered by Lev Pontryagin , the notion dual to a noncommutative compact group is not a group, but a category of representations Π( G ) with some additional structure, formed by the finite-dimensional representations of G .
Duality theorems of Tannaka and Krein describe the converse passage from the category Π( G ) back to the group G , allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise from a group in this fashion. Alexander Grothendieck later showed that by a similar process, Tannaka duality can be extended to the case of algebraic groups via Tannakian formalism . Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physicists . A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups , and is currently being extended to quantum supergroups , quantum groupoids and their dual Hopf algebroids .
In Pontryagin duality theory for locally compact commutative groups, the dual object to a group G is its character group G ^ , {\displaystyle {\hat {G}},} which consists of its one-dimensional unitary representations . If we allow the group G to be noncommutative, the most direct analogue of the character group is the set of equivalence classes of irreducible unitary representations of G . The analogue of the product of characters is the tensor product of representations . However, irreducible representations of G in general fail to form a group, or even a monoid, because a tensor product of irreducible representations is not necessarily irreducible. It turns out that one needs to consider the set Π ( G ) {\displaystyle \Pi (G)} of all finite-dimensional representations, and treat it as a monoidal category , where the product is the usual tensor product of representations, and the dual object is given by the operation of the contragredient representation .
A representation of the category Π ( G ) {\displaystyle \Pi (G)} is a monoidal natural transformation from the identity functor id Π ( G ) {\displaystyle \operatorname {id} _{\Pi (G)}} to itself. In other words, it is a non-zero function φ {\displaystyle \varphi } that associates with any T ∈ Ob Π ( G ) {\displaystyle T\in \operatorname {Ob} \Pi (G)} an endomorphism of the space of T and satisfies the conditions of compatibility with tensor products, φ ( T ⊗ U ) = φ ( T ) ⊗ φ ( U ) {\displaystyle \varphi (T\otimes U)=\varphi (T)\otimes \varphi (U)} , and with arbitrary intertwining operators f : T → U {\displaystyle f\colon T\to U} , namely, f ∘ φ ( T ) = φ ( U ) ∘ f {\displaystyle f\circ \varphi (T)=\varphi (U)\circ f} . The collection Γ ( Π ( G ) ) {\displaystyle \Gamma (\Pi (G))} of all representations of the category Π ( G ) {\displaystyle \Pi (G)} can be endowed with multiplication φ ψ ( T ) = φ ( T ) ψ ( T ) {\displaystyle \varphi \psi (T)=\varphi (T)\psi (T)} and topology , in which convergence is defined pointwise , i.e., a sequence { φ a } {\displaystyle \{\varphi _{a}\}} converges to some φ {\displaystyle \varphi } if and only if { φ a ( T ) } {\displaystyle \{\varphi _{a}(T)\}} converges to φ ( T ) {\displaystyle \varphi (T)} for all T ∈ Ob Π ( G ) {\displaystyle T\in \operatorname {Ob} \Pi (G)} . It can be shown that the set Γ ( Π ( G ) ) {\displaystyle \Gamma (\Pi (G))} thus becomes a compact (topological) group.
Tannaka's theorem provides a way to reconstruct the compact group G from its category of representations Π( G ).
Let G be a compact group and let F: Π( G ) → Vect C be the forgetful functor from finite-dimensional complex representations of G to complex finite-dimensional vector spaces . One puts a topology on the natural transformations τ: F → F by setting it to be the coarsest topology possible such that each of the projections End( F ) → End( V ) given by τ ↦ τ V {\displaystyle \tau \mapsto \tau _{V}} (taking a natural transformation τ {\displaystyle \tau } to its component τ V {\displaystyle \tau _{V}} at V ∈ Π ( G ) {\displaystyle V\in \Pi (G)} ) is a continuous function . We say that a natural transformation is tensor-preserving if it is the identity map on the trivial representation of G , and if it preserves tensor products in the sense that τ V ⊗ W = τ V ⊗ τ W {\displaystyle \tau _{V\otimes W}=\tau _{V}\otimes \tau _{W}} . We also say that τ is self-conjugate if τ ¯ = τ {\displaystyle {\overline {\tau }}=\tau } where the bar denotes complex conjugation. Then the set T ( G ) {\displaystyle {\mathcal {T}}(G)} of all tensor-preserving, self-conjugate natural transformations of F is a closed subset of End( F ), which is in fact a (compact) group whenever G is a (compact) group. Every element x of G gives rise to a tensor-preserving self-conjugate natural transformation via multiplication by x on each representation, and hence one has a map G → T ( G ) {\displaystyle G\to {\mathcal {T}}(G)} . Tannaka's theorem then says that this map is an isomorphism.
Krein's theorem answers the following question: which categories can arise as a dual object to a compact group?
Let Π be a category of finite-dimensional vector spaces, endowed with operations of tensor product and involution. The following conditions are necessary and sufficient in order for Π to be a dual object to a compact group G .
If all these conditions are satisfied then the category Π = Π( G ), where G is the group of the representations of Π.
Interest in Tannaka–Krein duality theory was reawakened in the 1980s with the discovery of quantum groups in the work of Drinfeld and Jimbo . One of the main approaches to the study of a quantum group proceeds through its finite-dimensional representations, which form a category akin to the symmetric monoidal categories Π( G ), but of more general type, braided monoidal category . It turned out that a good duality theory of Tannaka–Krein type also exists in this case and plays an important role in the theory of quantum groups by providing a natural setting in which both the quantum groups and their representations can be studied. Shortly afterwards different examples of braided monoidal categories were found in rational conformal field theory . Tannaka–Krein philosophy suggests that braided monoidal categories arising from conformal field theory can also be obtained from quantum groups, and in a series of papers, Kazhdan and Lusztig proved that it was indeed so. On the other hand, braided monoidal categories arising from certain quantum groups were applied by Reshetikhin and Turaev to construction of new invariants of knots.
The Doplicher–Roberts theorem (due to Sergio Doplicher and John E. Roberts ) characterises Rep( G ) in terms of category theory , as a type of subcategory of the category of Hilbert spaces . [ 1 ] Such subcategories of compact group unitary representations on Hilbert spaces are: | https://en.wikipedia.org/wiki/Tannaka–Krein_duality |
In mathematics , a Tannakian category is a particular kind of monoidal category C , equipped with some extra structure relative to a given field K . The role of such categories C is to generalise the category of linear representations of an algebraic group G defined over K . A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory .
The name is taken from Tadao Tannaka and Tannaka–Krein duality , a theory about compact groups G and their representation theory . The theory was developed first in the school of Alexander Grothendieck . It was later reconsidered by Pierre Deligne , and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory , which is a theory about finite permutation representations of groups G which are profinite groups .
The gist of the theory is that the fiber functor Φ of the Galois theory is replaced by an exact and faithful tensor functor F from C to the category of finite-dimensional vector spaces over K . The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group G of natural transformations of F into itself, that respect the tensor structure. This is in general not an algebraic group but a more general group scheme that is an inverse limit of algebraic groups ( pro-algebraic group ), and C is then found to be equivalent to the category of finite-dimensional linear representations of G .
More generally, it may be that fiber functors F as above only exists to categories of finite-dimensional vector spaces over non-trivial extension fields L / K . In such cases the group scheme G is replaced by a gerbe G {\displaystyle {\mathcal {G}}} on the fpqc site of Spec( K ), and C is then equivalent to the category of (finite-dimensional) representations of G {\displaystyle {\mathcal {G}}} .
Let K be a field and C a K -linear abelian rigid tensor (i.e., a symmetric monoidal ) category such that E n d ( 1 ) ≅ K {\displaystyle \mathrm {End} (\mathbf {1} )\cong K} . Then C is a Tannakian category (over K ) if there is an extension field L of K such that there exists a K -linear exact and faithful tensor functor (i.e., a strong monoidal functor ) F from C to the category of finite dimensional L -vector spaces . A Tannakian category over K is neutral if such exact faithful tensor functor F exists with L=K . [ 1 ]
The tannakian construction is used in relations between Hodge structure and l-adic representation . Morally [ clarification needed ] , the philosophy of motives
tells us that the Hodge structure and the Galois representation associated to an algebraic variety are related to each other. The closely-related algebraic groups Mumford–Tate group and motivic Galois group arise from categories of Hodge structures, category of Galois representations and motives through Tannakian categories. Mumford-Tate conjecture proposes that the algebraic groups arising from the Hodge strucuture and the Galois representation by means of Tannakian categories
are isomorphic to one another up to connected components.
Those areas of application are closely connected to the theory of motives . Another place in which Tannakian categories have been used is in connection with the Grothendieck–Katz p-curvature conjecture ; in other words, in bounding monodromy groups .
The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group L G {\displaystyle {}^{L}G} of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G . This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with L G {\displaystyle {}^{L}G} .
Wedhorn (2004) has established partial Tannaka duality results in the situation where the category is R -linear, where R is no longer a field (as in classical Tannakian duality), but certain valuation rings . Iwanari (2018) has initiated and developed Tannaka duality in the context of infinity-categories . | https://en.wikipedia.org/wiki/Tannakian_formalism |
In coding theory , a Tanner graph is a bipartite graph that can be used to express constraints (typically equations) that specify an error correcting code . Tanner graphs play a central role in the design and decoding of LDPC codes . They have also been applied to the construction of longer codes from smaller ones. Both encoders and decoders employ these graphs extensively.
Tanner graphs were proposed by Michael Tanner [ 1 ] as a means to create larger error correcting codes from smaller ones using recursive techniques. He generalized the techniques of Elias for product codes.
Tanner discussed lower bounds on the codes obtained from these graphs irrespective of the specific characteristics of the codes which were being used to construct larger codes.
Tanner graphs are partitioned into subcode nodes and digit nodes. For linear block codes, the subcode nodes denote rows of the parity-check matrix H. The digit nodes represent the columns of the matrix H. An edge connects a subcode node to a digit node if a nonzero entry exists in the intersection of the corresponding row and column.
Tanner proved the following bounds
Let R {\displaystyle R} be the rate of the resulting linear code, let the degree of the digit nodes be m {\displaystyle m} and the degree of the subcode nodes be n {\displaystyle n} . If each subcode node is associated with a linear code (n,k) with rate r = k/n, then the rate of the code is bounded by
The advantage of these recursive techniques is that they are computationally tractable. The coding
algorithm for Tanner graphs is extremely efficient in practice, although it is not
guaranteed to converge except for cycle-free graphs, which are known not to admit asymptotically
good codes. [ 2 ]
Zemor's decoding algorithm , which is a recursive low-complexity approach to code construction, is based on Tanner graphs. | https://en.wikipedia.org/wiki/Tanner_graph |
In mathematical analysis , Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations . It is named after Jules Tannery . [ 1 ]
Let S n = ∑ k = 0 ∞ a k ( n ) {\displaystyle S_{n}=\sum _{k=0}^{\infty }a_{k}(n)} and suppose that lim n → ∞ a k ( n ) = b k {\displaystyle \lim _{n\to \infty }a_{k}(n)=b_{k}} . If | a k ( n ) | ≤ M k {\displaystyle |a_{k}(n)|\leq M_{k}} and ∑ k = 0 ∞ M k < ∞ {\displaystyle \sum _{k=0}^{\infty }M_{k}<\infty } , then lim n → ∞ S n = ∑ k = 0 ∞ b k {\displaystyle \lim _{n\to \infty }S_{n}=\sum _{k=0}^{\infty }b_{k}} . [ 2 ] [ 3 ]
Tannery's theorem follows directly from Lebesgue's dominated convergence theorem applied to the sequence space ℓ 1 {\displaystyle \ell ^{1}} .
An elementary proof can also be given. [ 3 ]
Tannery's theorem can be used to prove that the binomial limit and the infinite series characterizations of the exponential e x {\displaystyle e^{x}} are equivalent. Note that
Define a k ( n ) = ( n k ) x k n k {\displaystyle a_{k}(n)={n \choose k}{\frac {x^{k}}{n^{k}}}} . We have that | a k ( n ) | ≤ | x | k k ! {\displaystyle |a_{k}(n)|\leq {\frac {|x|^{k}}{k!}}} and that ∑ k = 0 ∞ | x | k k ! = e | x | < ∞ {\displaystyle \sum _{k=0}^{\infty }{\frac {|x|^{k}}{k!}}=e^{|x|}<\infty } , so Tannery's theorem can be applied and | https://en.wikipedia.org/wiki/Tannery's_theorem |
The Tanpopo mission is an orbital astrobiology experiment investigating the potential interplanetary transfer of life , organic compounds , and possible terrestrial particles in the low Earth orbit. The purpose is to assess the panspermia hypothesis and the possibility of natural interplanetary transport of microbial life as well as prebiotic organic compounds.
The collection and exposure phase took place from May 2015 through February 2018 utilizing the Exposed Facility located on the exterior of Kibo, the Japanese Experimental Module of the International Space Station . [ 1 ] The mission, designed and performed by Japan, used ultra-low density silica gel ( aerogel ) to collect cosmic dust by, [ 2 ] which is being analyzed for amino acid-related compounds and microorganisms following their return to Earth. [ 3 ] The last samples were retrieved in February 2018 and analyses are ongoing. [ 4 ] The principal investigator is Akihiko Yamagishi, who heads a team of researchers from 26 universities and institutions in Japan, including JAXA .
The capture and exposure experiments in the Tanpopo mission were designed to confirm the hypothesis that extraterrestrial organic compounds played important roles in the generation of the first terrestrial life, as well as examination of the hypothesis of panspermia. If the Tanpopo mission can detect microbes at the higher altitude of low Earth orbit (400 km), it will support the possible interplanetary migration of terrestrial life. [ 5 ] [ 6 ] The mission was named after the plant dandelion (Tanpopo) because the plant's seeds evoke the image of seeds of lifeforms spreading out through space.
The Tanpopo mission exposures took place at the Exposed Facility located on the exterior of the Kibo module of the ISS from May 2015 through February 2018. [ 4 ] It collected cosmic dust and exposed dehydrated microorganisms outside the International Space Station while orbiting 400 km (250 mi) above the Earth. These experiments will test some aspects of panspermia, a hypothesis for an exogenesis origin of life distributed by meteoroids , asteroids , comets and cosmic dust . [ 7 ] This mission will also test if terrestrial microbes (e.g., aerosols embedding microbial colonies) may be present, even temporarily and in freeze-dried form in the low Earth orbit altitudes. [ 7 ]
Three key microorganisms include Deinococcus species: D. radiodurans , D. aerius and D. aetherius . [ 8 ] Containers holding yeast and other microbes were also placed outside the Kibo module to examine whether microbes can survive being exposed to the harsh cold environment of outer space . Also, by evaluating retrieved samples of exposed terrestrial microbes and astronomical organic analogs on the exposure panels, they can investigate their survival and any alterations in the duration of interplanetary transport.
Researchers also aim to capture organic compounds and prebiotic organic compounds — such as aminoacids — drifting in space. [ 9 ] The mission collected cosmic dust and other particles for three years by using a two-layer aerogel ultra-low density silica gel collector with a density of 0.01 g/cc (0.0058 oz/cu in) for the upper layer and ~0.03 g/cc (0.017 oz/cu in) for the base layer. [ 7 ] Some of the aerogel collectors were replaced every one to two years through February 2018. [ 9 ] [ 4 ]
The official ISS experiment code name is "Astrobiology Japan" representing "Astrobiology exposure and micrometeoroid capture experiments". [ 10 ]
The objectives of Tanpopo lie in following 6 topics: [ 11 ]
The aerogels were placed and retrieved by using the robotic arm outside Kibo. The first year samples were returned to Earth in mid-2016, [ 12 ] panels from the second year were brought back in late 2017, and the last set ended exposure in February 2018. [ 4 ] The last aerogels were placed inside the 'landing & return capsule' in early 2018 and ejected toward Earth for retrieval. [ 7 ] After retrieving the aerogels, scientists are investigating the captured microparticles and tracks formed, followed by microbiological, organochemical and mineralogical analyses. Particles potentially containing microbes will be used for PCR amplification of rRNA genes followed by DNA sequencing . [ 13 ]
Early mission results from the first sample show evidence that some clumps of microorganism can survive for at least one year in space. [ 14 ] This may support the idea that clumps greater than 0.5 millimeters of microorganisms could be one way for life to spread from planet to planet. [ 14 ] It was also noted that glycine 's decomposition was less than expected, while hydantoin 's recovery was much lower than glycine. [ 3 ]
In August 2020, scientists reported that bacteria from Earth, particularly Deinococcus radiodurans bacteria, which is highly resistant to environmental hazards , were found to survive for three years in outer space , based on studies conducted on the International Space Station. These findings support the notion of panspermia, the hypothesis that life exists throughout the Universe , distributed in various ways, including space dust , meteoroids , asteroids , comets , planetoids or contaminated spacecraft . [ 15 ] [ 16 ] | https://en.wikipedia.org/wiki/Tanpopo_mission |
Tantalum(III) chloride or tantalum trichloride is non-stoichiometric chemical compound with a range of composition from TaCl 2.9 to TaCl 3.1 [ 2 ] Anionic and neutral clusters containing Ta(III) chloride include [Ta 6 Cl 18 ] 4− and [Ta 6 Cl 14 ](H 2 O) 4 . [ 3 ]
Tantalum(III) chloride is formed by reducing tantalum(V) chloride with tantalum metal. this is done by heating tantalum(III) chloride to 305 °C, passing the vapour over tantalum foil at 600°, and condensing the trichloride at 365 °C. If the condensing region is kept at too high a temperature, then TaCl 2.5 deposits instead. [ 5 ]
The trichloride can also be prepared by thermal decomposition of TaCl 4 , with removal of volatile TaCl 5 . TaCl 5 can be vapourised leaving behind TaCl 3 . [ 6 ]
" Salt-free reduction " of a toluene solution of TaCl 5 with 1,4-disilyl-cyclohexadiene in the presence of ethylene produces a complex of TaCl 3 : [ 7 ]
Above 500 °C, TaCl 3 disproportionates further releasing TaCl 5 . [ 6 ] TaCl 3 is insoluble in room temperature water, or dilute acid, but dissolves in boiling water. A blue-green solution is formed. [ 6 ]
Tantalum(III) chloride can form complexes with some ligands as a monomer or dimer.
Complexes include Ta(=C-CMe 3 )(PMe 3 ) 2 Cl 3 , [TaCl 3 (P(CH 2 C 6 H 5 ) 3 THF] 2 μ-N 2 and [TaCl 3 THF 2 ] 2 μ-N 2 (dinitrogen complexes). [ 8 ]
As a dimer, complexes include Ta 2 Cl 6 (SC 4 H 8 ) 3 (SC 4 H 8 =tetrahydrothiophene). Ta 2 Cl 6 (SMe 2 ) 3 , Ta 2 Cl 6 (thiane) 3 and Ta 2 Cl 6 (thiolane) 3 have a double bond between the two tantalum atoms, and two bridging chlorides, and a bridging ligand. [ 7 ] | https://en.wikipedia.org/wiki/Tantalum(III)_chloride |
Tantalum–tungsten alloys are in the refractory metals group that maintain useful physical and chemical properties even at high temperatures. The tantalum–tungsten alloys are characterized by their high melting point and the tension resistance. The properties of the final alloy are a combination of properties from the two elements: tungsten , the element with the highest melting point in the periodic table , and tantalum which has high corrosion resistance . [ 1 ] [ 2 ]
The tantalum–tungsten alloys typically vary in their percentage of tungsten. Some common variants are:
The alloys of tantalum–tungsten have high corrosion resistance, and refractory properties. The crystalline structure of the material is body-centered cubic with a substitutional solid solution with atoms of tungsten. The alloy also has a high melting point and can reach high elastic modulus and high tensile strength . [ 3 ]
[ 1 ]
The equilibrium phase diagram of the alloy formed between the two components tantalum and tungsten is a binary diagram, where the two components are totally soluble on each other. In this diagram the melting temperature of the two elements are shown. It can be seen that there are two lines, representing the solidus and liquidus . [ 3 ] | https://en.wikipedia.org/wiki/Tantalum–tungsten_alloys |
The tap converter is a variation on the cycloconverter , invented in 1981 by New York City electrical engineer Melvin Sandler and significantly functionally enhanced in 1982 through 1984 by graduate students Mariusz Wrzesniewski, Bruce David Wilner, and Eddie Fung. Whereas the cycloconverter switches among a variety of staggered input phases to piece together an extremely jagged output signal, the tap converter synthesizes a much smoother signal by switching among a variety of (obviously synchronized) transformer output taps.
Both linear spacing and power-of-two-style Vernier spacing can be employed in establishing the tap positions, e.g. , a four-tap transformer can provide taps at 0.25, 0.5, 0.75, 1.0 (linear) or 0.0625, 0.125, 0.25, and 0.5 (Vernier). (The limitations of the Vernier—in this case, that the maximum obtainable amplitude is 0.9375—are less discernible as more taps are added.)
By employing a Scott transformer input connection, in order to provide a quadrature phase , an even smoother output waveform can be obtained.
Prototypes of the device were constructed and field-tested under a variety of conditions—nominally as a variable-speed constant-frequency (VSCF) power source for military aircraft—and ornate computer models were constructed for exploring more ornery considerations, such as flux leakage, hysteresis , and practical thyristor characteristics. All of this work was performed at New York's Cooper Union for the Advancement of Science and Art .
As of 2007, the tap converter remains uncommercialized but is used in several military applications due to the minimal output harmonics . [ citation needed ] | https://en.wikipedia.org/wiki/Tap_converter |
Tap water (also known as running water , piped water or municipal water ) is water supplied through a tap , a water dispenser valve. In many countries, tap water usually has the quality of drinking water . Tap water is commonly used for drinking , cooking , and washing . Indoor tap water is distributed through indoor plumbing , which has been around since antiquity but was available to very few people until the second half of the 19th century when it began to spread in popularity in what are now developed countries . Tap water became common in many regions during the 20th century, and is now lacking mainly among people in poverty , especially in developing countries .
Governmental agencies commonly regulate tap water quality . Calling a water supply "tap water" distinguishes it from the other main types of fresh water which may be available; these include water from rainwater -collecting cisterns , water from village pumps or town pumps, water from wells , or water carried from streams, rivers, or lakes (whose potability may vary).
A synonym for tap water is piped water, a term used by the Joint Monitoring Programme (JMP) for Water Supply and Sanitation by WHO and UNICEF to describe the situation for access to drinking water in developing countries . [ 2 ] Piped water is not necessarily of drinking water quality but does count as an " improved water source " in the logic of Sustainable Development Goal 6 . Other improved water sources include boreholes, protected dug wells or springs, rainwater, and bottled or water delivered by tanker. [ 2 ] : 12
Everything in a building that uses water falls under one of two categories; fixture or appliance. As the consumption points above perform their function, most produce waste/sewage components that will require removal by the waste/sewage side of the system. The minimum is an air gap. See cross connection control & backflow prevention for an overview of backflow prevention methods and devices currently in use, both through the use of mechanical and physical principles. [ citation needed ]
Fixtures are devices that use water without an additional source of power.
Potable water supply systems are composed of pipes , fittings , and valves .
Water flow through a tap can be reduced by inexpensive small plastic flow reducers. These restrict flow between 15 and 50%, aiding water conservation and reducing the burden on both water supply and treatment facilities.
The installation of water pipes can be done using the following plastic [ 3 ] and metal [ 3 ] materials:
Other materials, if the pipes made from them have been let into circulation and the widespread use in the construction of the water supply systems.
For many centuries, water pipes were made of lead, because of its ease of processing and durability. The use of lead pipes was a cause of health problems due to ignorance of the dangers of lead on the human body, which causes miscarriages and high death rates of newborns. Lead pipes, which were installed mostly in the late 1800s in the US, are still common today, much of which are located in the Northeast and the Midwest. [ 4 ] Their impact is relatively small due to the fouling of pipes and stone cessation of the evolution of lead in the water; however, lead pipes are still detrimental. Most of the lead pipes that exist today are being removed and replaced with the more common material, copper or some type of plastic.
Modern plumbing delivers clean, safe, and potable water to each service point in water distribution system , including taps. [ 5 ] It is important that the clean water not be contaminated by the wastewater (disposal) side of the process system. Historically, this contamination of drinking water has been one of the largest killers of humans. [ 6 ]
Most of the mandates for enforcing drinking water quality standards are not for the distribution system, but for the treatment plant. Even though the water distribution system is supposed to deliver the treated water to the consumers' taps without water quality degradation, complicated physical, chemical, and biological factors within the system can cause contamination of tap water. [ 5 ]
Tap water can sometimes appear cloudy and is often mistaken for mineral impurities in the water. It is usually caused by air bubbles coming out of solution due to change in temperature or pressure. Because cold water holds more air than warm water, small bubbles will appear in water. It has a high dissolved gas content that is heated or depressurized, which reduces how much dissolved gas the water can hold. The harmless cloudiness of the water disappears quickly as the gas is released from the water. [ 7 ]
Water supply is the provision of water by public utilities , commercial organisations, community endeavors or by individuals, usually via a system of pumps and pipes . Public water supply systems are crucial to properly functioning societies. These systems are what supply drinking water to populations around the globe. [ 8 ] Aspects of service quality include continuity of supply, water quality and water pressure. The institutional responsibility for water supply is arranged differently in different countries and regions (urban versus rural). It usually includes issues surrounding policy and regulation, service provision and standardization .
Bottled water may have reduced amounts of copper, lead, and other metal contaminants since it does not run through the plumbing pipes where tap water is exposed to metal corrosion; however, this varies by the household and plumbing system. [ 9 ]
In much of the developed world, chlorine often is added as a disinfectant to tap water. If the water contains organic matter, this may produce other byproducts in the water such as trihalomethanes and haloacetic acids , which has shown to increase the risk of cancer. [ 10 ] The level of residual chlorine found at around 0.0002 g per litre, which is too small to cause any health problems directly. [ 9 ] The chlorine concentration recommended by World Health Organization is between 0.0002 and 0.0005 g/L. [ 11 ]
Contaminant levels found in tap water vary between households and plumbing systems . While the majority of US households have access to high-quality tap water, demand for bottled water increases. [ 15 ] In 2002, the Gallup Public Opinion Poll revealed that the possible health risk associated with tap water consumption is one of the main reasons that cause American consumers to prefer bottled water over tap water. [ 16 ]
The trust level towards tap water depends on various criteria, including the existing governmental regulations towards the water quality and their appliance. In 1993, a cryptosporidiosis outbreak in Milwaukee, Wisconsin led to a massive hospitalization of more than 400,000 residents and was considered the largest in US history. [ 17 ] Severe violations of tap water standards influence the decrease in public trust. [ 18 ]
The difference in water quality between bottled and tap water is debatable. In 1999, the Natural Resources Defense Council (NRDC) released controversial findings from a 4-year study on bottled water. The study claimed that one-third of the tested waters were contaminated with synthetic organic chemicals , bacteria , and arsenic . At least one sample exceeded state guidelines for contamination levels in bottled water. [ 19 ]
In the United States, some municipalities make an effort to use tap water over bottled water on governmental properties and events. Voters in Washington State repealed a bottled water tax via citizen initiative. [ 20 ] [ 21 ] | https://en.wikipedia.org/wiki/Tap_water |
Tape Wrangler is a duct tape dispenser produced by Stexley-Brake, LLC. [ 1 ] [ 2 ] The tape gun is designed to dispense straight and smooth pieces of tape, working in the same way as tape dispensers do for pressure-sensitive tape . [ 3 ] [ 4 ] [ 5 ] The product was launched in 2008. [ citation needed ] | https://en.wikipedia.org/wiki/Tape_Wrangler |
In surveying , tape correction(s) refer(s) to correcting measurements for the effect of slope angle, expansion or contraction due to temperature, and the tape's sag, which varies with the applied tension. Not correcting for these effects gives rise to systematic errors , i.e. effects which act in a predictable manner and therefore can be corrected by mathematical methods.
C v = 2 L ∗ s i n 2 + A 2 {\displaystyle C_{v}=2L*sin^{2}+{\frac {A}{2}}}
Where
When distances are measured along the slope , the equivalent horizontal distance may be determined by applying a slope correction.
The vertical slope angle of the length measured must be measured. (Refer to the figure on the other side) Thus,
Where:
The correction C h {\displaystyle C_{h}} is subtracted from s {\displaystyle s} to obtain the equivalent horizontal distance on the slope line:
When measuring or laying out distances, the standard temperature of the tape and the temperature of the tape at time of measurement are usually different. A difference in temperature will cause the tape to lengthen or shorten, so the measurement taken will not be exactly correct. A correction can be applied to the measured length to obtain the correct length.
The correction of the tape length due to change in temperature is given by:
Where:
The correction C t {\displaystyle C_{t}} is added to L {\displaystyle L} to obtain the corrected distance:
For common tape measurements, the tape used is a steel tape with coefficient of thermal expansion C equal to 0.000,011,6 units per unit length per degree Celsius change. This means that the tape changes length by 1.16 mm per 10 m tape per 10 °C change from the standard temperature of the tape. For a 30 meter long tape with standard temperature of 20 °C used at 40 °C, the change in length is 7 mm over the length of the tape.
A tape not supported along its length will sag and form a catenary between end supports. According to the section of tension correction some tapes are calibrated for sag at standard tension. These tapes will require complex sag and tension corrections if used at non-standard tensions. The correction due to sag must be calculated separately for each unsupported stretch separately and is given by:
Where:
A tape held in catenary will record a value larger than the correct measurement. Thus, the correction C s {\displaystyle C_{s}} is subtracted from L {\displaystyle L} to obtain the corrected distance:
Note that the weight of the tape per unit length is equal to the weight of the tape divided by the length of the tape:
so: W = ω L {\displaystyle W=\omega L}
Therefore, we can rewrite the formula for correction due to sag as:
The general formula for a catenary formed by a tape supported only at its ends is
Here, g {\displaystyle g} is the gravitational acceleration. The arc length between two support points at x = − k / 2 {\displaystyle x=-k/2} and x = + k / 2 {\displaystyle x=+k/2} is found by usual methods via integration:
For convenience set a = P ω g {\displaystyle a={\frac {P}{\omega g}}} . The integrand is simplified as follows using hyperbolic function identities :
The tape length L {\displaystyle L} is then found by integrating:
Now the correction for tape sag is the difference between the actual span between the supports, k {\displaystyle k} , and the arc length of the tape's catenary, L {\displaystyle L} . Call this correction δ = k − L {\displaystyle \delta =k-L} . The absolute value of this δ {\displaystyle \delta } correction is C s {\displaystyle C_{s}} above, the amount you would subtract from the tape measurement to get the true span distance.
A Taylor series expansion of δ {\displaystyle \delta } in terms of the quantity L {\displaystyle L} is desired to give a good first approximation to the correction. In fact, the first nonvanishing term in the Taylor series is cubic in L {\displaystyle L} , and the next nonvanishing term is to the fifth power of L; thus, a series expansion for δ {\displaystyle \delta } is reasonable. To this end, we need to find an expression for δ {\displaystyle \delta } that contains L {\displaystyle L} but not k {\displaystyle k} . We already have an expression for L {\displaystyle L} in terms of k {\displaystyle k} , but now need to find the inverse function (for k {\displaystyle k} in terms of L {\displaystyle L} ):
Evaluating δ {\displaystyle \delta } at L = 0 {\displaystyle L=0} yields zero, so there is no zero-order term in the Taylor series. The first derivative of this function with respect to L is
Evaluated at L=0, it vanishes and so does not contribute a Taylor series term. The second derivative of δ {\displaystyle \delta } is
Again, when evaluated at L=0 it vanishes. When evaluated at L=0, the third derivative survives, however.
Thus, the first surviving term in the Taylor series is:
Notice that the variable P {\displaystyle P} here is the tension on the cable, whereas above, P {\displaystyle P} is the mass whose gravitational force (mass times gravitational acceleration) equals the tension on the cable. The only conversion necessary then is to take P / g {\displaystyle P/g} here and equate it to P {\displaystyle P} above. Also, this formula is the tape sag correction to be added to the measured distance, so the negative sign in front can be removed and the tape sag correction can be made instead by subtracting the absolute value as is done in the preceding section.
Some tapes are already calibrated to account for the sag at a standard tension. [ 1 ] [ 2 ] In this case, errors arise when the tape is pulled at a Tension which differs from the standard tension used at standardization. The tape will pulled less than its standard length when a tension less than the standard tension is applied, making the tape too long.
A tape stretches in an elastic manner until it reaches its elastic limit , when it will deform permanently and ruin the tape.
The correction due to tension is given by:
Where:
The correction C p {\displaystyle C_{p}} is added to L {\displaystyle L} to obtain the corrected distance:
The value for A is given by:
Where:
For steel tapes, the value for U w {\displaystyle U_{w}} is 7.866 × 10 − 3 k g / c m 3 {\displaystyle 7.866\times 10^{-3}kg/cm^{3}} .
Manufacturers of measuring tapes do not usually guarantee the exact length of tapes, and standardization is a process where a standard temperature and tension are determined at which the tape is the exact length. The nominal length of tapes can be affected by physical imperfections, stretching or wear. Constant use of tapes cause wear, tapes can become kinked and may be improperly repaired when breaks occur.
The correction due to tape length is given by:
Where:
In the U.S., some tapes come with United States Bureau of Standards certifications establishing the correction needed per 100' of tape.
Note that incorrect tape length introduces a systematic error that must be calibrated periodically.
Mostly in pdf : | https://en.wikipedia.org/wiki/Tape_correction_(surveying) |
A tape dispenser is an object that holds a roll of tape and has a mechanism at one end to shear the tape. Dispensers vary widely based on the tape they dispense. Abundant and most common, clear tape dispensers (like those used in an office or at home) are commonly made of plastic, and may be disposable . Other dispensers are stationary and may have sophisticated features to control tape usage and improve ergonomics .
Prior to the development of the tape dispenser, 3M's standard clear scotch tape was sold as a roll, and had to be carefully peeled from the end and cut with scissors. To make the product more useful, the scotch tape sales manager at 3M, John Borden, designed the first tape dispenser in 1932, which had a built-in cutting mechanism and would hold the cut end of the tape until its next use. [ 1 ] [ 2 ]
A handheld dispenser is a variation of handheld tape dispenser used to apply tape to close boxes, etc. Some refer to it as a "tape gun".
Some dispensers are small enough so that the dispenser, with the tape in it, can be taken to the point of application for operator ease. The dispenser allows for a convenient cut-off and helps the operator apply (and sometimes helps rub down) the tape.
Tabletop or desk dispensers are frequently used to hold the tape and allow the operator to pull off the desired amount, tear the tape off, and take the tape to the job.
Tabletop dispensers are available with electrical assists to dispense and cut pressure-sensitive tape to a predetermined length. They are often used in an industrial setting to increase productivity along manufacturing or assembly lines. They eliminate the need to manually measure and cut each individual piece of tape on high volumes of product or packaging. By automating this process, automatic tape dispensers reduce material waste caused by human error . They also reduce the time needed to cut each piece of tape, therefore reducing labor costs and increasing productivity.
Some taping machinery is semi-automatic: the operator takes an object and puts it in or through a machine which automatically applies the tape. This helps save time and controls the consumption of tape.
Fully automatic equipment is available which does not require an operator. All functions can be automated.
High speed packaging machinery is an example of highly automated equipment.
Gummed (water activated) tape dispensers measure, dispense, moisten, and cut gummed or water-activated adhesive tape . This tape is often composed of a paper backing and adhesive glue that is unable to adhere until it is "activated" by contact with water. To perform this step, gummed dispensers often employ a water bottle and wetting brush to moisten each piece of tape as it is dispensed. Many gummed dispensers feature a heater, which is mounted over the feed area to maintain the dispenser's water temperature. These heaters ensure maximum wetting, and are ideal in cold climates. Gummed tape dispensers are often used in packaging or shipping departments for closing corrugated boxes . | https://en.wikipedia.org/wiki/Tape_dispenser |
A tape operator or tape op , also known as a second engineer, is a person who performs menial operations in a recording studio in a similar manner to a tea boy or gopher. [ 1 ] They may act as an apprentice or an assistant to a recording engineer and duties can consist of threading audio tape , setting up microphones and stands, configuring MIDI equipment and cables, and sometimes pressing the relevant transport controls on the recorder or digital audio workstation . [ 2 ] Abbey Road Studios always assigned at least one tape op to each recording session. [ 3 ]
The role of tape op was a useful entry into a professional recording environment, and several went on to successful careers as engineers and record producers . The music and film soundtrack producer John Kurlander started his production career at Abbey Road Studios in 1967 as a tea boy, progressing to principal tape op (or assistant engineer) by 1969. [ 4 ] He was partially responsible for including " Her Majesty " on the Beatles ' Abbey Road after carefully splicing a discarded take of the song onto the master tape. [ 5 ] Alan Parsons also began his production career as an Abbey Road tape op, which led to him to assisting with the mixing of Pink Floyd 's Atom Heart Mother and engineering on The Dark Side of the Moon . [ 6 ]
Due to the increasing ability to produce professional quality recordings at home studios , the experience that can be gained by working as a tape op is being lost, resulting in people having a harder learning curve with music engineering and production. [ 7 ] | https://en.wikipedia.org/wiki/Tape_op |
In computing, tapered floating point ( TFP ) is a format similar to floating point , but with variable-sized entries for the significand and exponent instead of the fixed-length entries found in normal floating-point formats. In addition to this, tapered floating-point formats provide a fixed-size pointer entry indicating the number of digits in the exponent entry. The number of digits of the significand entry (including the sign) results from the difference of the fixed total length minus the length of the exponent and pointer entries. [ 1 ]
Thus numbers with a small exponent, i.e. whose order of magnitude is close to the one of 1, have a higher relative precision than those with a large exponent.
The tapered floating-point scheme was first proposed by Robert Morris of Bell Laboratories in 1971, [ 2 ] and refined with leveling by Masao Iri and Shouichi Matsui of University of Tokyo in 1981, [ 3 ] [ 4 ] [ 1 ] and by Hozumi Hamada of Hitachi, Ltd. [ 5 ] [ 6 ] [ 7 ]
Alan Feldstein of Arizona State University and Peter Turner [ 8 ] of Clarkson University described a tapered scheme resembling a conventional floating-point system except for the overflow or underflow conditions. [ 7 ]
In 2013, John Gustafson proposed the Unum number system, a variant of tapered floating-point arithmetic with an exact bit added to the representation and some interval interpretation to the non-exact values. [ 9 ] [ 10 ] | https://en.wikipedia.org/wiki/Tapered_floating_point |
In mathematics , physics , and theoretical computer graphics , tapering is a kind of shape deformation. [ 1 ] [ 2 ] Just as an affine transformation , such as scaling or shearing , is a first-order model of shape deformation, tapering is a higher order deformation just as twisting and bending. Tapering can be thought of as non-constant scaling by a given tapering function. The resultant deformations can be linear or nonlinear.
To create a nonlinear taper, instead of scaling in x and y for all z with constants as in:
let a and b be functions of z so that:
An example of a linear taper is a ( z ) = α 0 + α 1 z {\displaystyle a(z)=\alpha _{0}+\alpha _{1}z} , and a quadratic taper a ( z ) = α 0 + α 1 z + α 2 z 2 {\displaystyle a(z)={\alpha }_{0}+{\alpha }_{1}z+{\alpha }_{2}z^{2}} .
As another example, if the parametric equation of a cube were given by ƒ ( t ) = ( x ( t ), y ( t ), z ( t )), a nonlinear taper could be applied so that the cube's volume slowly decreases (or tapers) as the function moves in the positive z direction. For the given cube, an example of a nonlinear taper along z would be if, for instance, the function T ( z ) = 1/( a + bt ) were applied to the cube's equation such that ƒ ( t ) = ( T ( z ) x ( t ), T ( z ) y ( t ), T ( z ) z ( t )), for some real constants a and b . | https://en.wikipedia.org/wiki/Tapering_(mathematics) |
A taphotaxon (from the Greek ταφος, taphos meaning burial and ταξις, taxis meaning ordering ) is an invalid taxon based on fossils remains that have been altered in a characteristic way during burial and diagenesis . The fossils so altered have distinctive characteristics that make them appear to be a new taxon, but these characteristics are spurious and do not reflect any significant taxonomic distinction from an existing fossil taxon. The term was first proposed by Spencer G. Lucas in 2001, who particularly applied it to spurious ichnotaxons , [ 1 ] [ 2 ] but it has since been applied to body fossils such as Nuia (interpreted as cylindrical oncolites formed around filamentous cyanobacteria ) [ 3 ] or Ivanovia (thought to be a taphotaxon of Anchicondium or Eugonophyllum ); [ 4 ] conulariids , [ 5 ] and crustaceans . [ 6 ]
In his original definition of the term, Lucas emphasized that he was not seeking to create a new field of taphotaxonomy. The term is intended simply as a useful description of a particular type of invalid taxon. [ 1 ] It should not be used indiscriminately, particularly with ichnotaxons, where the fact that an ichnotaxon derives part of its morphology from taphonomic processes may not always render it an invalid ichnotaxon. [ 7 ]
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Taphotaxon |
Taplitumomab paptox is a mouse monoclonal antibody . The antibody itself, taplitumomab, is linked to the protein PAP, an antiviral from Phytolacca americana , a species of pokeweed . [ 1 ] This is reflected by the 'paptox' in the drug's name.
This monoclonal antibody –related article is a stub . You can help Wikipedia by expanding it .
This antineoplastic or immunomodulatory drug article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Taplitumomab_paptox |
Taptu was a social media and technology company that built platforms, tools and applications that enabled content on touch screen mobile devices, including phones running iOS and Android . Taptu was a privately held company that was founded in Cambridge , England, in 2007 and was funded by DFJ Esprit and Sofinnova. The company was based in Cambridge and Denver , Colorado, United States.
In September 2012, Taptu was acquired by Mediafed Ltd. The Taptu service shut down March 31, 2015, [ 1 ] a year before Mediafed went into administration. [ 2 ]
Taptu's first product was a mobile search engine, provided as a website using html optimised for phones, and dedicated apps for iPhone and Android. It was closed in early 2011.
Its other product was My Taptu, a social news aggregator that drew heavily on the company’s mobile search heritage but attempts to move them beyond search. [ 3 ] According to Taptu, the app, which was available on the iPhone and Android devices, was "aimed at solving information overload," or what Taptu calls "app hopping." [ 4 ] It presented all the information that a person "is into" in a "one- stop app" through "streams," or what CNET described as “content playlists.” [ 5 ] My Taptu tried to separate itself in the social news space by offering ways of personalization. | https://en.wikipedia.org/wiki/Taptu |
TaqMan probes are hydrolysis probes that are designed to increase the specificity of quantitative PCR . The method was first reported in 1991 by researcher David Gefland at Cetus Corporation, [ 1 ] and the technology was subsequently developed by Hoffmann-La Roche for diagnostic assays and by Applied Biosystems (now part of Thermo Fisher Scientific ) for research applications.
The TaqMan probe principle relies on the 5´–3´ exonuclease activity of Taq polymerase to cleave a dual-labeled probe during hybridization to the complementary target sequence and fluorophore -based detection. [ 2 ] As in other quantitative PCR methods, the resulting fluorescence signal permits quantitative measurements of the accumulation of the product during the exponential stages of the PCR; however, the TaqMan probe significantly increases the specificity of the detection. TaqMan probes were named after the videogame Pac-Man ( Taq Polymerase + PacMan = TaqMan) as its mechanism is based on the Pac-Man principle. [ 3 ]
TaqMan probes consist of a fluorophore covalently attached to the 5’-end of the oligonucleotide probe and a quencher at the 3’-end. [ 4 ] Several different fluorophores (e.g. 6-carboxyfluorescein , acronym: FAM , or tetrachlorofluorescein, acronym: TET) and quenchers (e.g. tetramethyl rhodamine , acronym: TAMRA) are available. [ 5 ] The quencher molecule quenches the fluorescence emitted by the fluorophore when excited by the cycler’s light source via Förster resonance energy transfer (FRET). [ 6 ] As long as the fluorophore and the quencher are in proximity, quenching inhibits any fluorescence signals.
TaqMan probes are designed such that they anneal within a DNA region amplified by a specific set of primers. (Unlike the diagram, the probe binds to single stranded DNA.) TaqMan probes can be conjugated to a minor groove binder (MGB) moiety, dihydrocyclopyrroloindole tripeptide (DPI 3 ), in order to increase its binding affinity to the target sequence; MGB-conjugated probes have a higher melting temperature (T m ) due to increased stabilization of van der Waals forces. As the Taq polymerase extends the primer and synthesizes the nascent strand (from the single-stranded template), the 5' to 3' exonuclease activity of the Taq polymerase degrades the probe that has annealed to the template. Degradation of the probe releases the fluorophore from it and breaks the proximity to the quencher, thus relieving the quenching effect and allowing fluorescence of the fluorophore. Hence, fluorescence detected in the quantitative PCR thermal cycler is directly proportional to the fluorophore released and the amount of DNA template present in the PCR.
TaqMan probe-based assays are widely used in quantitative PCR in research and medical laboratories :
1. TaqMan RT-PCR resources - primer databases, software, protocols Archived 2010-11-25 at the Wayback Machine 2. Beacon Designer - Software to design real time PCR primers and probes including SYBR Green primers, TaqMan Probes, Molecular Beacons. | https://en.wikipedia.org/wiki/TaqMan |
Taq polymerase is a thermostable DNA polymerase I named after the thermophilic eubacterial microorganism Thermus aquaticus , from which it was originally isolated by master's student Alice Chien et al. in 1976. [ 1 ] Its name is often abbreviated to Taq or Taq pol . It is frequently used in the polymerase chain reaction (PCR), a method for greatly amplifying the quantity of short segments of DNA .
T. aquaticus is a bacterium that lives in hot springs and hydrothermal vents , and Taq polymerase was identified [ 1 ] as an enzyme able to withstand the protein-denaturing conditions (high temperature) required during PCR. [ 2 ] Therefore, it replaced the DNA polymerase from E. coli originally used in PCR. [ 3 ]
Taq' s optimum temperature for activity is 75–80 °C, with a half-life of greater than 2 hours at 92.5 °C, 40 minutes at 95 °C and 9 minutes at 97.5 °C, and can replicate a 1000 base pair strand of DNA in less than 10 seconds at 72 °C. [ 4 ] At 75–80 °C, Taq reaches its optimal polymerization rate of about 150 nucleotides per second per enzyme molecule, and any deviations from the optimal temperature range inhibit the extension rate of the enzyme. A single Taq synthesizes about 60 nucleotides per second at 70 °C, 24 nucleotides/sec at 55 °C, 1.5 nucleotides/sec at 37 °C, and 0.25 nucleotides/sec at 22 °C. At temperatures above 90 °C, Taq demonstrates very little or no activity at all, but the enzyme itself does not denature and remains intact. [ 5 ] Presence of certain ions in the reaction vessel also affects specific activity of the enzyme. Small amounts of potassium chloride (KCl) and magnesium ion (Mg 2+ ) promote Taq 's enzymatic activity. Taq polymerase is maximally activated at 50mM KCl, while optimal Mg 2+ concentration is determined by the concentration of nucleoside triphosphates (dNTPs). High concentrations of KCl and Mg 2+ inhibit Taq 's activity. [ 6 ] The common metal ion chelator EDTA directly binds to Taq in the absence of these metal ions. [ 7 ]
One of Taq' s drawbacks is its lack of 3' to 5' exonuclease proofreading activity [ 4 ] resulting in relatively low replication fidelity. Originally its error rate was measured at about 1 in 9,000 nucleotides. [ 8 ] Some thermostable DNA polymerases have been isolated from other thermophilic bacteria and archaea, such as Pfu DNA polymerase , possessing a proofreading activity, and are being used instead of (or in combination with) Taq for high-fidelity amplification. [ 9 ] Fidelity can vary widely between Taqs, which has profound effects in downstream sequencing applications. [ 10 ]
Taq makes DNA products that have A ( adenine ) overhangs at their 3' ends. This may be useful in TA cloning , whereby a cloning vector (such as a plasmid ) that has a T ( thymine ) 3' overhang is used, which complements with the A overhang of the PCR product, thus enabling ligation of the PCR product into the plasmid vector.
In the early 1980s, Kary Mullis was working at Cetus Corporation on the application of synthetic DNAs to biotechnology . He was familiar with the use of DNA oligonucleotides as probes for binding to target DNA strands, as well as their use as primers for DNA sequencing and cDNA synthesis. In 1983, he began using two primers, one to hybridize to each strand of a target DNA, and adding DNA polymerase to the reaction. This led to exponential DNA replication , [ 11 ] greatly amplifying discrete segments of DNA between the primers. [ 3 ]
However, after each round of replication the mixture needs to be heated above 90 °C to denature the newly formed DNA, allowing the strands to separate and act as templates in the next round of amplification. This heating step also inactivates the DNA polymerase that was in use before the discovery of Taq polymerase, the Klenow fragment (sourced from E. coli ). Taq polymerase is well-suited for this application because it is able to withstand the temperature of 95 °C which is required for DNA strand separation without denaturing.
Use of the thermostable Taq enables running the PCR at high temperature (~60 °C and above), which facilitates high specificity of the primers and reduces the production of nonspecific products, such as primer dimer . Also, use of a thermostable polymerase eliminates the need to add new enzyme to each round of thermocycling. A single closed tube in a relatively simple machine can be used to carry out the entire process. Thus, the use of Taq polymerase was the key idea that made PCR applicable to a large variety of molecular biology problems concerning DNA analysis. [ 2 ]
Hoffmann-La Roche eventually bought the PCR and Taq patents from Cetus for $330 million, from which it may have received up to $2 billion in royalties. [ 12 ] In 1989, Science Magazine named Taq polymerase its first " Molecule of the Year ". Kary Mullis received the Nobel Prize in Chemistry in 1993, the only one awarded for research performed at a biotechnology company. By the early 1990s, the PCR technique with Taq polymerase was being used in many areas, including basic molecular biology research, clinical testing , and forensics . It also began to find a pressing application in direct detection of the HIV in AIDS . [ 13 ]
In December 1999, U.S. District Judge Vaughn Walker ruled that the 1990 patent involving Taq polymerase was issued, in part, on misleading information and false claims by scientists with Cetus Corporation . The ruling supported a challenge by Promega Corporation against Hoffman-La Roche , which purchased the Taq patents in 1991. Judge Walker cited previous discoveries by other laboratories, including the laboratory of John Trela at the University of Cincinnati department of biological sciences, as the basis for the ruling. [ 14 ]
Taq Pol A has an overall structure similar to that of E. coli PolA. The middle 3'–5' exonuclease domain responsible for proofreading has been dramatically changed and is not functional. [ 15 ] It has a functional 5'-3' exonuclease domain at the amino terminal, described below. The remaining two domains act in coordination, via coupled domain motion. [ 16 ]
Taq polymerase exonuclease is a domain found in the amino-terminal of Taq DNA polymerase I (thermostable). It assumes a ribonuclease H-like motif . The domain confers 5' -3' exonuclease activity to the polymerase. [ 17 ]
Unlike the same domain in E. coli , which would degrade primers and must be removed by digestion for PCR use, [ 9 ] this domain is not said to degrade the primer. [ 18 ] This activity is used in the TaqMan probe: as the daughter strands are formed, the probes complementary to the template come in contact with the polymerase and are cleaved into fluorescent pieces. [ 19 ]
Taq polymerase is bound at its polymerase active-site cleft with the blunt end of duplex DNA. As the Taq polymerase is in contact with the bound DNA, its side chains form hydrogen bonds with the purines and pyrimidines of the DNA. The same region of Taq polymerase that has bonded to DNA also binds with exonuclease. These structures bound to the Taq polymerase have different interactions.
A site-directed mutagenesis experiment that improves the vestigial 3'-5' exonuclease activity by a factor of 2 has been reported, but it was never reported whether doing so decreases the error rate. [ 20 ] Following a similar line of thought, chimera proteins have been made by cherry-picking domains from E. coli , Taq , and T. neapolitana polymerase I. Swapping out the vestigial domain for a functional one from E. coli created a protein with proof-reading ability but a lower optimal temperature and low thermostability. [ 21 ]
Versions of the polymerase without the 5'-3' exonuclease domain has been produced, among which Klentaq or the Stoffel fragment are best known. The complete lack of exonuclease activity make these variants suitable for primers that exhibit secondary structure as well as for copying circular molecules. [ 9 ] Other variations include using Klentaq with a high-fidelity polymerase, a Thermosequenase that recognizes substrates like T7 DNA polymerase does, mutants with higher tolerances to inhibitors, or "domain-tagged" versions that have an extra helix-hairpin-helix motif around the catalytic site to hold the DNA more tightly despite adverse conditions. [ 22 ]
Because of the improvements Taq polymerase provided in PCR DNA replication: higher specificity, fewer nonspecific products, and simpler processes and equipment, it has been instrumental in the efforts made to detect diseases. "The use of Polymerase Chain Reaction (PCR) in infectious disease diagnosis, has resulted in an ability to diagnose early and treat appropriately diseases due to fastidious pathogens, determine the antimicrobial susceptibility of slow growing organisms, and ascertain the quantum of infection." [ 23 ] The implementation of Taq polymerase has saved countless lives. It has served an essential role in the detection of many of the world's worst diseases, including: tuberculosis, streptococcal pharyngitis, atypical pneumonia, AIDS, measles, hepatitis, and ulcerative urogenital infections. PCR, the method used to recreate copies of specific DNA samples, makes disease detection possible by targeting a specific DNA sequence of a targeted pathogen from a patient's sample and amplifying trace amounts of the indicative sequences by copying them up to billions of times. Although this is the most accurate method of disease detection, especially for HIV, it is not performed as often as alternative, inferior tests because of the relatively high cost, labor, and time required. [ 24 ]
The reliance upon Taq polymerase as a catalyst for the PCR replication process has been highlighted during the COVID-19 Pandemic of 2020. Shortages of the necessary enzyme have impaired the ability of countries worldwide to produce test kits for the virus. Without Taq polymerase, the disease detection process is much slower and tedious. [ 25 ]
Despite the advantages of using Taq polymerase in PCR disease detection, the enzyme is not without its shortcomings. Retroviral diseases (HIV, HTLV-1, and HTLV-II) often include mutations from guanine to adenine in their genome. Mutations such as these are what allow PCR tests to detect the diseases but Taq polymerase’s relatively low fidelity rate makes the same G-to-A mutation occur and possibly yield a false positive test result. [ 26 ] | https://en.wikipedia.org/wiki/Taq_polymerase |
Tar is a dark brown or black viscous liquid of hydrocarbons and free carbon , obtained from a wide variety of organic materials through destructive distillation . Tar can be produced from coal , wood , petroleum , or peat . [ 1 ]
Mineral products resembling tar can be produced from fossil hydrocarbons , such as petroleum . Coal tar is produced from coal as a byproduct of coke production.
"Tar" and " pitch " can be used interchangeably. Asphalt (naturally occurring pitch) may also be called either "mineral tar" or "mineral pitch". There is a tendency to use "tar" for more liquid substances and "pitch" for more solid ( viscoelastic ) substances. [ 2 ] Both "tar" and "pitch" are applied to viscous forms of asphalt, such as the asphalt found in naturally occurring tar pits (e.g., the La Brea Tar Pits in Los Angeles). "Rangoon tar", also known as "Burmese oil" or "Burmese naphtha", is also a form of petroleum. [ 3 ] Oil sands , found extensively in Alberta, Canada , and composed of asphalt, are colloquially referred to as "tar sands". [ 4 ] [ 5 ]
Since prehistoric times wood tar has been used as a water repellent coating for boats , ships , sails , and roofs . In Scandinavia , it was produced as a cash crop. "Peasant Tar" might be named for the district of its production. [ 6 ]
Wood tar is still used as an additive in the flavoring of candy , alcohol , and other foods. Wood tar is microbicidal . Producing tar from wood was known in ancient Greece and has probably been used in Scandinavia since the Iron Age . Production and trade in pine-derived tar was a major contributor in the economies of Northern Europe [ 7 ] and Colonial America . Its main use was in preserving wooden sailing vessels against rot. For centuries, dating back at least to the 14th century, tar was among Sweden 's most important exports. Sweden exported 13,000 barrels of tar in 1615 and 227,000 barrels in the peak year of 1863. The largest user was the Royal Navy of the United Kingdom . Demand for tar declined with the advent of iron and steel ships. Production nearly stopped in the early 20th century. Traditional wooden boats are still sometimes tarred.
The heating ( dry distilling ) of pine wood causes tar and pitch to drip away from the wood [ citation needed ] and leave behind charcoal. Birch bark is used to make particularly fine tar, known as " Russian oil", used in Russian leather protection. The by-products of wood tar are turpentine and charcoal . When deciduous tree woods are subjected to destructive distillation , the products are methanol (wood alcohol) and charcoal .
Tar kilns ( Swedish : tjärmila , Danish : tjæremile , Norwegian : tjæremile , Finnish : tervahauta ) are dry distillation ovens, historically used in Scandinavia for producing tar from wood. They were built close to the forest, from limestone or from more primitive holes in the ground. The bottom is sloped into an outlet hole to allow the tar to pour out. The wood is split into dimensions of a finger, stacked densely, and finally covered tight with earth and moss. If oxygen can enter, the wood might catch fire, and the production would be ruined. On top of this, a fire is stacked and lit. After a few hours, the tar starts to pour out and continues to do so for a few days.
Tar was used as seal for roofing shingles and tar paper and to seal the hulls of ships and boats. For millennia, wood tar was used to waterproof sails and boats, but today, sails made from inherently waterproof synthetic substances have reduced the demand for tar. Wood tar is still used to seal traditional wooden boats and the roofs of historic, shingle-roofed churches, as well as painting exterior walls of log buildings. Tar is also a general disinfectant. Pine tar oil, or wood tar oil, is used for the surface treatment of wooden shingle roofs, boats, buckets, and tubs and in the medicine, soap, and rubber industries. Pine tar has good penetration on the rough wood. An old wood tar oil recipe for the treatment of wood is one-third each genuine wood tar, balsam turpentine, and boiled or raw linseed oil or Chinese tung oil. [ citation needed ]
In Finland , wood tar was once considered a panacea reputed to heal "even those cut in twain through their midriff". A Finnish proverb states that "if sauna , vodka and tar won't help, the disease is fatal." [ 8 ] Wood tar is used in traditional Finnish medicine because of its microbicidal properties.
Wood tar is also available diluted as tar water , which has numerous uses:
Mixing tar with linseed oil varnish produces tar paint. Tar paint has a translucent brownish hue and can be used to saturate and tone wood and protect it from weather. Tar paint can also be toned with various pigments, producing translucent colors and preserving the wood texture.
Tar was once used for public humiliation , known as tarring and feathering . By pouring hot wood tar onto somebody's bare skin and waiting for it to cool, they would remain stuck in one position. From there, people would attach feathers to the tar, which would remain stuck on the tarred person for the duration of the punishment. That person would then become a public example for the rest of the day. [ 9 ]
Pitch was familiar in 9th-century Iraq , derived from petroleum that became accessible from natural fields in the region. It was sometimes used in the construction of baths or in shipbuilding. [ 10 ]
Coal tar was formerly one of the products of gasworks . Tar made from coal or petroleum is considered toxic and carcinogenic because of its high benzene content, [ citation needed ] though coal tar in low concentrations is used as a topical medicine for conditions such as psoriasis . [ 11 ] [ 12 ] Coal and petroleum tar has a pungent odor.
Coal tar is listed at number 1999 in the United Nations list of dangerous goods . | https://en.wikipedia.org/wiki/Tar |
Tar is a dark brown or black viscous liquid of hydrocarbons and free carbon , obtained from a wide variety of organic materials through destructive distillation . Tar can be produced from coal , wood , petroleum , or peat . [ 1 ]
Mineral products resembling tar can be produced from fossil hydrocarbons , such as petroleum . Coal tar is produced from coal as a byproduct of coke production.
"Tar" and " pitch " can be used interchangeably. Asphalt (naturally occurring pitch) may also be called either "mineral tar" or "mineral pitch". There is a tendency to use "tar" for more liquid substances and "pitch" for more solid ( viscoelastic ) substances. [ 2 ] Both "tar" and "pitch" are applied to viscous forms of asphalt, such as the asphalt found in naturally occurring tar pits (e.g., the La Brea Tar Pits in Los Angeles). "Rangoon tar", also known as "Burmese oil" or "Burmese naphtha", is also a form of petroleum. [ 3 ] Oil sands , found extensively in Alberta, Canada , and composed of asphalt, are colloquially referred to as "tar sands". [ 4 ] [ 5 ]
Since prehistoric times wood tar has been used as a water repellent coating for boats , ships , sails , and roofs . In Scandinavia , it was produced as a cash crop. "Peasant Tar" might be named for the district of its production. [ 6 ]
Wood tar is still used as an additive in the flavoring of candy , alcohol , and other foods. Wood tar is microbicidal . Producing tar from wood was known in ancient Greece and has probably been used in Scandinavia since the Iron Age . Production and trade in pine-derived tar was a major contributor in the economies of Northern Europe [ 7 ] and Colonial America . Its main use was in preserving wooden sailing vessels against rot. For centuries, dating back at least to the 14th century, tar was among Sweden 's most important exports. Sweden exported 13,000 barrels of tar in 1615 and 227,000 barrels in the peak year of 1863. The largest user was the Royal Navy of the United Kingdom . Demand for tar declined with the advent of iron and steel ships. Production nearly stopped in the early 20th century. Traditional wooden boats are still sometimes tarred.
The heating ( dry distilling ) of pine wood causes tar and pitch to drip away from the wood [ citation needed ] and leave behind charcoal. Birch bark is used to make particularly fine tar, known as " Russian oil", used in Russian leather protection. The by-products of wood tar are turpentine and charcoal . When deciduous tree woods are subjected to destructive distillation , the products are methanol (wood alcohol) and charcoal .
Tar kilns ( Swedish : tjärmila , Danish : tjæremile , Norwegian : tjæremile , Finnish : tervahauta ) are dry distillation ovens, historically used in Scandinavia for producing tar from wood. They were built close to the forest, from limestone or from more primitive holes in the ground. The bottom is sloped into an outlet hole to allow the tar to pour out. The wood is split into dimensions of a finger, stacked densely, and finally covered tight with earth and moss. If oxygen can enter, the wood might catch fire, and the production would be ruined. On top of this, a fire is stacked and lit. After a few hours, the tar starts to pour out and continues to do so for a few days.
Tar was used as seal for roofing shingles and tar paper and to seal the hulls of ships and boats. For millennia, wood tar was used to waterproof sails and boats, but today, sails made from inherently waterproof synthetic substances have reduced the demand for tar. Wood tar is still used to seal traditional wooden boats and the roofs of historic, shingle-roofed churches, as well as painting exterior walls of log buildings. Tar is also a general disinfectant. Pine tar oil, or wood tar oil, is used for the surface treatment of wooden shingle roofs, boats, buckets, and tubs and in the medicine, soap, and rubber industries. Pine tar has good penetration on the rough wood. An old wood tar oil recipe for the treatment of wood is one-third each genuine wood tar, balsam turpentine, and boiled or raw linseed oil or Chinese tung oil. [ citation needed ]
In Finland , wood tar was once considered a panacea reputed to heal "even those cut in twain through their midriff". A Finnish proverb states that "if sauna , vodka and tar won't help, the disease is fatal." [ 8 ] Wood tar is used in traditional Finnish medicine because of its microbicidal properties.
Wood tar is also available diluted as tar water , which has numerous uses:
Mixing tar with linseed oil varnish produces tar paint. Tar paint has a translucent brownish hue and can be used to saturate and tone wood and protect it from weather. Tar paint can also be toned with various pigments, producing translucent colors and preserving the wood texture.
Tar was once used for public humiliation , known as tarring and feathering . By pouring hot wood tar onto somebody's bare skin and waiting for it to cool, they would remain stuck in one position. From there, people would attach feathers to the tar, which would remain stuck on the tarred person for the duration of the punishment. That person would then become a public example for the rest of the day. [ 9 ]
Pitch was familiar in 9th-century Iraq , derived from petroleum that became accessible from natural fields in the region. It was sometimes used in the construction of baths or in shipbuilding. [ 10 ]
Coal tar was formerly one of the products of gasworks . Tar made from coal or petroleum is considered toxic and carcinogenic because of its high benzene content, [ citation needed ] though coal tar in low concentrations is used as a topical medicine for conditions such as psoriasis . [ 11 ] [ 12 ] Coal and petroleum tar has a pungent odor.
Coal tar is listed at number 1999 in the United Nations list of dangerous goods . | https://en.wikipedia.org/wiki/Tar_kiln |
Tara Matise is an American geneticist at Rutgers University . Since 2018, she has served as chair of the Department of Genetics. Her research interests span computational genetics , data science , and human genetics . She is co-director of the Rutgers University Genetics Coordinating Center.
A native of Buffalo, New York, Matise attended high school at The Buffalo Seminary , a private secular school for girls, graduating in 1982, after which she attended Cornell University , where she earned a bachelor’s degree in biology with a concentration in genetics. [ 1 ] She went on to earn a master’s degree in Genetic Counseling in 1988 from the University of Pittsburgh , and a doctorate in 1992 in human genetics from the University of Pittsburgh School of Public Health under the direction of Aravinda Chakravarti . Matise then worked as a postdoctoral fellow at the University of Pittsburgh , Columbia University and Rockefeller University . [ 1 ] At Rockefeller, Matise worked under the supervision of Jürg Ott . [ 1 ]
Matise moved to Rutgers University in 2000. She was appointed Head of the Computational Genetics Program in the Human Genetics Institute of New Jersey in 2014. [ 2 ] [ 3 ] In 2021, Matise was elected as a fellow in the American Association for the Advancement of Science . [ 4 ] Matise began her career using genetic linkage to identify genes for genetic diseases. Her contribution to the identification of the cystic fibrosis transmembrane conductance regulator (CFTR) gene was honored by the Cystic Fibrosis Foundation in 1990. [ 1 ] Matise was the creator of MultiMap, a computer program that automated the construction of genetic linkage maps of the human genome. [ 5 ] Her work facilitated the development of several genome-wide gene maps in humans and other organisms, and led to the development of the Rutgers Maps, which contain over 28,000 markers and provide an interpolated position for all human markers, the largest linkage map of human polymorphic markers. [ 5 ] [ 6 ]
In 2008, Matise, as co-director of the Rutgers University Genetics Coordinating Center (RUGCC), was funded to lead a coordinating center for the multi-center Population Architecture using Genomics and Epidemiology (PAGE) study. [ 7 ] [ 8 ] [ 9 ] Funded by the National Human Genome Research Institute (NHGRI) at the National Institutes of Health (NIH), the PAGE consortium was a pioneer in its approach to performing globally representative epidemiological genomics. [ 9 ] The RUGCC was responsible for data quality control and dissemination, and study logistics. [ 10 ] [ 11 ] [ 12 ]
Matise, as co-lead of the RUGCC, was funded to direct the coordinating center for the National Human Genome Research Institute (NHGRI) Genome Sequencing Program in 2015. The program made use of genome sequencing to understand the genes that underpin inherited disease. [ 13 ] | https://en.wikipedia.org/wiki/Tara_Matise |
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