id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-8600 | De Moivre's Formula/Exponential Form | :$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$ | {{begin-eqn}}
{{eqn | l = \paren {r e^{i \theta} }^\omega
| r = \paren {r \paren {\cos \theta + i \sin \theta} }^\omega
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = r^\omega \paren {\cos \omega \theta + i \sin \omega \theta}
| c = De Moivre's Formula
}}
{{eqn | r = r^\omega e^{i \... | :$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$ | {{begin-eqn}}
{{eqn | l = \paren {r e^{i \theta} }^\omega
| r = \paren {r \paren {\cos \theta + i \sin \theta} }^\omega
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = r^\omega \paren {\cos \omega \theta + i \sin \omega \theta}
| c = [[De Moivre's Formula]]
}}
{{eqn | r = r^\omega e^... | De Moivre's Formula/Exponential Form | https://proofwiki.org/wiki/De_Moivre's_Formula/Exponential_Form | https://proofwiki.org/wiki/De_Moivre's_Formula/Exponential_Form | [
"De Moivre's Formula"
] | [] | [
"De Moivre's Formula"
] |
proofwiki-8601 | Roots of Complex Number/Exponential Form | :$z^{1 / n} = \set {r^{1 / n} e^{i \paren {\theta + 2 \pi k} / n}: k \in \set {0, 1, 2, \ldots, n - 1} }$ | {{begin-eqn}}
{{eqn | l = z^{1 / n}
| r = \paren {r e^{i \theta} }^{1 / n}
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = \paren {r \paren {\cos x + i \sin x} }^{1 / n}
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = \set {r^{1 / n} \paren {\cos \paren {\dfrac {\theta + ... | :$z^{1 / n} = \set {r^{1 / n} e^{i \paren {\theta + 2 \pi k} / n}: k \in \set {0, 1, 2, \ldots, n - 1} }$ | {{begin-eqn}}
{{eqn | l = z^{1 / n}
| r = \paren {r e^{i \theta} }^{1 / n}
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = \paren {r \paren {\cos x + i \sin x} }^{1 / n}
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = \set {r^{1 / n} \paren {\cos \paren {\dfrac {\theta + ... | Roots of Complex Number/Exponential Form | https://proofwiki.org/wiki/Roots_of_Complex_Number/Exponential_Form | https://proofwiki.org/wiki/Roots_of_Complex_Number/Exponential_Form | [
"Complex Roots"
] | [] | [
"Roots of Complex Number"
] |
proofwiki-8602 | Z/(m)-Module Associated with Ring of Characteristic m | Let $\struct {R, +, *}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let the characteristic of $R$ be $m$.
Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$.
Let $\circ$ be the mapping from $\Z_m \times R$ to $R$ defined as:
:$\forall \eqclass a m \in \Z_m: \forall x \in R: \e... | Let us verify that the definition of $\circ$ is well-defined.
Let $\eqclass a m = \eqclass b m$.
Then we need to show that:
:$\forall x \in R: \eqclass a m \circ x = \eqclass b m \circ x$
By the definition of congruence:
:$\eqclass a m = \eqclass b m \iff \exists k \in \Z : a = b + k m$
Then:
{{begin-eqn}}
{{eqn | l = ... | Let $\struct {R, +, *}$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Ring Zero|zero]] is $0_R$ and whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let the [[Definition:Characteristic of Ring|characteristic]] of $R$ be $m$.
Let $\struct {\Z_m, +_m, \times_m}$ be the [[Definition:Ring of In... | Let us verify that the definition of $\circ$ is [[Definition:Well-Defined Operation|well-defined]].
Let $\eqclass a m = \eqclass b m$.
Then we need to show that:
:$\forall x \in R: \eqclass a m \circ x = \eqclass b m \circ x$
By the definition of [[Definition:Congruence Modulo Integer|congruence]]:
:$\eqclass a m... | Z/(m)-Module Associated with Ring of Characteristic m | https://proofwiki.org/wiki/Z/(m)-Module_Associated_with_Ring_of_Characteristic_m | https://proofwiki.org/wiki/Z/(m)-Module_Associated_with_Ring_of_Characteristic_m | [
"Unitary Modules",
"Group Theory"
] | [
"Definition:Ring with Unity",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Characteristic of Ring",
"Definition:Ring of Integers Modulo m",
"Definition:Mapping",
"Definition:Integers Modulo m",
"Definition:Integral Multiple/Rings and Fields",
"Definition:Unitary Mo... | [
"Definition:Well-Defined/Operation",
"Definition:Congruence (Number Theory)/Integers",
"Powers of Group Elements/Sum of Indices",
"Powers of Group Elements/Product of Indices",
"Characteristic times Ring Element is Ring Zero",
"Power of Identity is Identity",
"Definition:Well-Defined/Operation",
"Defi... |
proofwiki-8603 | Sum of Squares of Hyperbolic Secant and Tangent | :$\sech^2 x + \tanh^2 x = 1$
where $\sech$ and $\tanh$ are hyperbolic secant and hyperbolic tangent. | {{begin-eqn}}
{{eqn | l = \sech^2 x + \tanh^2 x
| r = \paren {\frac 2 {e^x + e^{-x} } }^2 + \tanh^2 x
| c = {{Defof|Hyperbolic Secant|index = 1}}
}}
{{eqn | r = \paren {\frac 2 {e^x + e^{-x} } }^2 + \paren {\frac {e^x - e^{-x} } {e^x + e^{-x} } }^2
| c = {{Defof|Hyperbolic Tangent|index = 1}}
}}
{{e... | :$\sech^2 x + \tanh^2 x = 1$
where $\sech$ and $\tanh$ are [[Definition:Hyperbolic Secant|hyperbolic secant]] and [[Definition:Hyperbolic Tangent|hyperbolic tangent]]. | {{begin-eqn}}
{{eqn | l = \sech^2 x + \tanh^2 x
| r = \paren {\frac 2 {e^x + e^{-x} } }^2 + \tanh^2 x
| c = {{Defof|Hyperbolic Secant|index = 1}}
}}
{{eqn | r = \paren {\frac 2 {e^x + e^{-x} } }^2 + \paren {\frac {e^x - e^{-x} } {e^x + e^{-x} } }^2
| c = {{Defof|Hyperbolic Tangent|index = 1}}
}}
{{e... | Sum of Squares of Hyperbolic Secant and Tangent | https://proofwiki.org/wiki/Sum_of_Squares_of_Hyperbolic_Secant_and_Tangent | https://proofwiki.org/wiki/Sum_of_Squares_of_Hyperbolic_Secant_and_Tangent | [
"Sum of Squares of Hyperbolic Secant and Tangent",
"Hyperbolic Secant Function",
"Hyperbolic Tangent Function"
] | [
"Definition:Hyperbolic Secant",
"Definition:Hyperbolic Tangent"
] | [
"Exponent Combination Laws"
] |
proofwiki-8604 | Difference of Squares of Hyperbolic Cotangent and Cosecant | :$\coth^2 x - \csch^2 x = 1$
where $\coth$ and $\csch$ are hyperbolic cotangent and hyperbolic cosecant. | {{begin-eqn}}
{{eqn | l = \coth^2 x - \csch^2 x
| r = \paren {\frac {e^x + e^{-x} } {e^x - e^{-x} } }^2 - \csch^2 x
| c = {{Defof|Hyperbolic Cotangent|index = 1}}
}}
{{eqn | r = \paren {\frac {e^x + e^{-x} } {e^x - e^{-x} } }^2 - \paren {\frac 2 {e^x - e^{-x} } }^2
| c = {{Defof|Hyperbolic Cosecant|in... | :$\coth^2 x - \csch^2 x = 1$
where $\coth$ and $\csch$ are [[Definition:Hyperbolic Cotangent|hyperbolic cotangent]] and [[Definition:Hyperbolic Cosecant|hyperbolic cosecant]]. | {{begin-eqn}}
{{eqn | l = \coth^2 x - \csch^2 x
| r = \paren {\frac {e^x + e^{-x} } {e^x - e^{-x} } }^2 - \csch^2 x
| c = {{Defof|Hyperbolic Cotangent|index = 1}}
}}
{{eqn | r = \paren {\frac {e^x + e^{-x} } {e^x - e^{-x} } }^2 - \paren {\frac 2 {e^x - e^{-x} } }^2
| c = {{Defof|Hyperbolic Cosecant|in... | Difference of Squares of Hyperbolic Cotangent and Cosecant | https://proofwiki.org/wiki/Difference_of_Squares_of_Hyperbolic_Cotangent_and_Cosecant | https://proofwiki.org/wiki/Difference_of_Squares_of_Hyperbolic_Cotangent_and_Cosecant | [
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Hyperbolic Cotangent Function",
"Hyperbolic Cosecant Function"
] | [
"Definition:Hyperbolic Cotangent",
"Definition:Hyperbolic Cosecant"
] | [
"Exponent Combination Laws"
] |
proofwiki-8605 | Hyperbolic Cosecant Function is Odd | Let $\csch: \C \to \C$ be the hyperbolic cosecant function on the set of complex numbers.
Then $\csch$ is odd:
:$\map \csch {-x} = -\csch x$ | {{begin-eqn}}
{{eqn | l = \map \csch {-x}
| r = \frac 1 {\map \sinh {-x} }
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 {-\sinh x}
| c = Hyperbolic Sine Function is Odd
}}
{{eqn | r = -\csch x
}}
{{end-eqn}}
{{qed}} | Let $\csch: \C \to \C$ be the [[Definition:Hyperbolic Cosecant|hyperbolic cosecant function]] on the [[Definition:Complex Number|set of complex numbers]].
Then $\csch$ is [[Definition:Odd Function|odd]]:
:$\map \csch {-x} = -\csch x$ | {{begin-eqn}}
{{eqn | l = \map \csch {-x}
| r = \frac 1 {\map \sinh {-x} }
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 {-\sinh x}
| c = [[Hyperbolic Sine Function is Odd]]
}}
{{eqn | r = -\csch x
}}
{{end-eqn}}
{{qed}} | Hyperbolic Cosecant Function is Odd | https://proofwiki.org/wiki/Hyperbolic_Cosecant_Function_is_Odd | https://proofwiki.org/wiki/Hyperbolic_Cosecant_Function_is_Odd | [
"Hyperbolic Cosecant Function",
"Examples of Odd Functions"
] | [
"Definition:Hyperbolic Cosecant",
"Definition:Complex Number",
"Definition:Odd Function"
] | [
"Hyperbolic Sine Function is Odd"
] |
proofwiki-8606 | Hyperbolic Secant Function is Even | :$\map \sech {-x} = \sech x$ | {{begin-eqn}}
{{eqn | l = \map \sech {-x}
| r = \frac 1 {\map \cosh {-x} }
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac 1 {\cosh x}
| c = Hyperbolic Cosine Function is Even
}}
{{eqn | r = \sech x
}}
{{end-eqn}}
{{qed}} | :$\map \sech {-x} = \sech x$ | {{begin-eqn}}
{{eqn | l = \map \sech {-x}
| r = \frac 1 {\map \cosh {-x} }
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac 1 {\cosh x}
| c = [[Hyperbolic Cosine Function is Even]]
}}
{{eqn | r = \sech x
}}
{{end-eqn}}
{{qed}} | Hyperbolic Secant Function is Even/Proof 1 | https://proofwiki.org/wiki/Hyperbolic_Secant_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Secant_Function_is_Even/Proof_1 | [
"Hyperbolic Secant Function",
"Hyperbolic Secant Function is Even",
"Examples of Even Functions"
] | [] | [
"Hyperbolic Cosine Function is Even"
] |
proofwiki-8607 | Hyperbolic Secant Function is Even | :$\map \sech {-x} = \sech x$ | {{begin-eqn}}
{{eqn | l = \sech \paren {-x}
| r = \frac 1 {\cosh \paren {-x} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {\cos \paren {-i x} }
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \frac 1 {\cos \paren {i x} }
| c = Cosine Function is Even
}}
{{eqn | r = \frac 1 {\cosh x}
| c = H... | :$\map \sech {-x} = \sech x$ | {{begin-eqn}}
{{eqn | l = \sech \paren {-x}
| r = \frac 1 {\cosh \paren {-x} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {\cos \paren {-i x} }
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \frac 1 {\cos \paren {i x} }
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \frac 1 {\cosh x}
... | Hyperbolic Secant Function is Even/Proof 2 | https://proofwiki.org/wiki/Hyperbolic_Secant_Function_is_Even | https://proofwiki.org/wiki/Hyperbolic_Secant_Function_is_Even/Proof_2 | [
"Hyperbolic Secant Function",
"Hyperbolic Secant Function is Even",
"Examples of Even Functions"
] | [] | [
"Hyperbolic Cosine in terms of Cosine",
"Cosine Function is Even",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8608 | Hyperbolic Cotangent Function is Odd | Let $\coth: \C \to \C$ be the hyperbolic cotangent function on the set of complex numbers.
Then $\coth$ is odd:
:$\map \coth {-x} = -\coth x$ | {{begin-eqn}}
{{eqn | l = \map \coth {-x}
| r = \frac {\map \cosh {-x} } {\map \sinh {-x} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\map \cosh {-x} } {-\sinh x}
| c = Hyperbolic Sine Function is Odd
}}
{{eqn | r = \frac {\cosh x} {-\sinh x}
| c = Hyperbolic Cosine F... | Let $\coth: \C \to \C$ be the [[Definition:Hyperbolic Cotangent|hyperbolic cotangent function]] on the [[Definition:Complex Number|set of complex numbers]].
Then $\coth$ is [[Definition:Odd Function|odd]]:
:$\map \coth {-x} = -\coth x$ | {{begin-eqn}}
{{eqn | l = \map \coth {-x}
| r = \frac {\map \cosh {-x} } {\map \sinh {-x} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\map \cosh {-x} } {-\sinh x}
| c = [[Hyperbolic Sine Function is Odd]]
}}
{{eqn | r = \frac {\cosh x} {-\sinh x}
| c = [[Hyperbolic Co... | Hyperbolic Cotangent Function is Odd | https://proofwiki.org/wiki/Hyperbolic_Cotangent_Function_is_Odd | https://proofwiki.org/wiki/Hyperbolic_Cotangent_Function_is_Odd | [
"Hyperbolic Cotangent Function",
"Examples of Odd Functions"
] | [
"Definition:Hyperbolic Cotangent",
"Definition:Complex Number",
"Definition:Odd Function"
] | [
"Hyperbolic Sine Function is Odd",
"Hyperbolic Cosine Function is Even"
] |
proofwiki-8609 | Hyperbolic Sine of Sum | :$\map \sinh {a + b} = \sinh a \cosh b + \cosh a \sinh b$ | {{begin-eqn}}
{{eqn | l = \sinh a \cosh b + \cosh a \sinh b
| r = \frac {e^a - e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a + e^{-a} } 2 \frac {e^b - e^{-b} } 2
| c = {{Defof|Hyperbolic Sine}} and {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{a + b} - e^{-a + b} + e^{a - b} - e^{-a - b} } 4
... | :$\map \sinh {a + b} = \sinh a \cosh b + \cosh a \sinh b$ | {{begin-eqn}}
{{eqn | l = \sinh a \cosh b + \cosh a \sinh b
| r = \frac {e^a - e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a + e^{-a} } 2 \frac {e^b - e^{-b} } 2
| c = {{Defof|Hyperbolic Sine}} and {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{a + b} - e^{-a + b} + e^{a - b} - e^{-a - b} } 4
... | Hyperbolic Sine of Sum | https://proofwiki.org/wiki/Hyperbolic_Sine_of_Sum | https://proofwiki.org/wiki/Hyperbolic_Sine_of_Sum | [
"Hyperbolic Sine Function",
"Addition Formulas for Hyperbolic Functions"
] | [] | [
"Exponential of Sum"
] |
proofwiki-8610 | Hyperbolic Cosine of Sum | :$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$ | {{begin-eqn}}
{{eqn | r = \cosh a \cosh b + \sinh a \sinh b
| o =
| c =
}}
{{eqn | r = \frac {e^a + e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a - e^{-a} } 2 \frac {e^b - e^{-b} } 2
| c = {{Defof|Hyperbolic Sine}} and {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{a + b} + e^{-a + b} + e^{a... | :$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$ | {{begin-eqn}}
{{eqn | r = \cosh a \cosh b + \sinh a \sinh b
| o =
| c =
}}
{{eqn | r = \frac {e^a + e^{-a} } 2 \frac {e^b + e^{-b} } 2 + \frac {e^a - e^{-a} } 2 \frac {e^b - e^{-b} } 2
| c = {{Defof|Hyperbolic Sine}} and {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{a + b} + e^{-a + b} + e^{a... | Hyperbolic Cosine of Sum | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Sum | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Sum | [
"Hyperbolic Cosine Function",
"Addition Formulas for Hyperbolic Functions"
] | [] | [
"Exponential of Sum"
] |
proofwiki-8611 | Hyperbolic Tangent of Sum | :$\map \tanh {a + b} = \dfrac {\tanh a + \tanh b} {1 + \tanh a \tanh b}$ | {{begin-eqn}}
{{eqn | l = \map \tanh {a + b}
| r = \frac {\map \sinh {a + b} } {\map \cosh {a + b} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {\sinh a \cosh b + \cosh a \sinh b} {\cosh a \cosh b + \sinh a \sinh b}
| c = Hyperbolic Sine of Sum and Hyperbolic Cosine of Sum
}}
{{... | :$\map \tanh {a + b} = \dfrac {\tanh a + \tanh b} {1 + \tanh a \tanh b}$ | {{begin-eqn}}
{{eqn | l = \map \tanh {a + b}
| r = \frac {\map \sinh {a + b} } {\map \cosh {a + b} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {\sinh a \cosh b + \cosh a \sinh b} {\cosh a \cosh b + \sinh a \sinh b}
| c = [[Hyperbolic Sine of Sum]] and [[Hyperbolic Cosine of Sum... | Hyperbolic Tangent of Sum | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Sum | https://proofwiki.org/wiki/Hyperbolic_Tangent_of_Sum | [
"Hyperbolic Tangent Function",
"Addition Formulas for Hyperbolic Functions"
] | [] | [
"Hyperbolic Sine of Sum",
"Hyperbolic Cosine of Sum",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8612 | Hyperbolic Cotangent of Sum | :$\map \coth {a + b} = \dfrac {\coth a \coth b + 1} {\coth b + \coth a}$ | {{begin-eqn}}
{{eqn | l = \map \coth {a + b}
| r = \frac {\map \cosh {a + b} } {\map \sinh {a + b} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\cosh a \cosh b + \sinh a \sinh b} {\sinh a \cosh b + \cosh a \sinh b}
| c = Hyperbolic Sine of Sum and Hyperbolic Cosine of Sum
}}
... | :$\map \coth {a + b} = \dfrac {\coth a \coth b + 1} {\coth b + \coth a}$ | {{begin-eqn}}
{{eqn | l = \map \coth {a + b}
| r = \frac {\map \cosh {a + b} } {\map \sinh {a + b} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\cosh a \cosh b + \sinh a \sinh b} {\sinh a \cosh b + \cosh a \sinh b}
| c = [[Hyperbolic Sine of Sum]] and [[Hyperbolic Cosine of S... | Hyperbolic Cotangent of Sum | https://proofwiki.org/wiki/Hyperbolic_Cotangent_of_Sum | https://proofwiki.org/wiki/Hyperbolic_Cotangent_of_Sum | [
"Hyperbolic Cotangent Function",
"Addition Formulas for Hyperbolic Functions"
] | [] | [
"Hyperbolic Sine of Sum",
"Hyperbolic Cosine of Sum",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8613 | Double Angle Formulas/Hyperbolic Sine | :$\sinh 2 x = 2 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 2 x
| r = \map \sinh {x + x}
}}
{{eqn | r = \sinh x \cosh x + \cosh x \sinh x
| c = Hyperbolic Sine of Sum
}}
{{eqn | r = 2 \sinh x \cosh x
}}
{{end-eqn}}
{{qed}} | :$\sinh 2 x = 2 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 2 x
| r = \map \sinh {x + x}
}}
{{eqn | r = \sinh x \cosh x + \cosh x \sinh x
| c = [[Hyperbolic Sine of Sum]]
}}
{{eqn | r = 2 \sinh x \cosh x
}}
{{end-eqn}}
{{qed}} | Double Angle Formulas/Hyperbolic Sine/Proof 1 | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Sine | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Sine/Proof_1 | [
"Hyperbolic Sine Function",
"Double Angle Formula for Hyperbolic Sine"
] | [] | [
"Hyperbolic Sine of Sum"
] |
proofwiki-8614 | Double Angle Formulas/Hyperbolic Sine | :$\sinh 2 x = 2 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 2 x
| r = \frac 1 2 \paren {e^{2 x} - e^{-2 x} }
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \paren {e^x + e^{-x} } \paren {e^x - e^{-x} }
| c = Difference of Two Squares
}}
{{eqn | r = 2 \paren {\frac{e^x + e^{-x} } 2 \cdot \frac {e^x - e^{-x} } 2}
}}
{{eqn | r = 2 \sinh... | :$\sinh 2 x = 2 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 2 x
| r = \frac 1 2 \paren {e^{2 x} - e^{-2 x} }
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \paren {e^x + e^{-x} } \paren {e^x - e^{-x} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = 2 \paren {\frac{e^x + e^{-x} } 2 \cdot \frac {e^x - e^{-x} } 2}
}}
{{eqn | r = 2 \... | Double Angle Formulas/Hyperbolic Sine/Proof 2 | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Sine | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Sine/Proof_2 | [
"Hyperbolic Sine Function",
"Double Angle Formula for Hyperbolic Sine"
] | [] | [
"Difference of Two Squares"
] |
proofwiki-8615 | Double Angle Formulas/Hyperbolic Sine | :$\sinh 2 x = 2 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 2 x
| r = -i \sin 2 i x
| c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = -2 i \sin i x \cos i x
| c = Double Angle Formula for Sine
}}
{{eqn | r = 2 \sinh x \cosh x
| c = Hyperbolic Sine in terms of Sine, Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}} | :$\sinh 2 x = 2 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 2 x
| r = -i \sin 2 i x
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{eqn | r = -2 i \sin i x \cos i x
| c = [[Double Angle Formula for Sine]]
}}
{{eqn | r = 2 \sinh x \cosh x
| c = [[Hyperbolic Sine in terms of Sine]], [[Hyperbolic Cosine in terms of Cosine]]
}}
{{end-eqn}}
{{qed}... | Double Angle Formulas/Hyperbolic Sine/Proof 3 | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Sine | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Sine/Proof_3 | [
"Hyperbolic Sine Function",
"Double Angle Formula for Hyperbolic Sine"
] | [] | [
"Hyperbolic Sine in terms of Sine",
"Double Angle Formulas/Sine",
"Hyperbolic Sine in terms of Sine",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8616 | Double Angle Formulas/Hyperbolic Cosine | :$\cosh 2 x = \cosh^2 x + \sinh^2 x$ | {{begin-eqn}}
{{eqn | l = \cosh 2 x
| r = \map \cosh {x + x}
}}
{{eqn | r = \cosh x \cosh x + \sinh x \sinh x
| c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \cosh^2 x + \sinh^2 x
}}
{{end-eqn}}
{{qed}} | :$\cosh 2 x = \cosh^2 x + \sinh^2 x$ | {{begin-eqn}}
{{eqn | l = \cosh 2 x
| r = \map \cosh {x + x}
}}
{{eqn | r = \cosh x \cosh x + \sinh x \sinh x
| c = [[Hyperbolic Cosine of Sum]]
}}
{{eqn | r = \cosh^2 x + \sinh^2 x
}}
{{end-eqn}}
{{qed}} | Double Angle Formulas/Hyperbolic Cosine | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Cosine | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Cosine | [
"Hyperbolic Cosine Function"
] | [] | [
"Hyperbolic Cosine of Sum"
] |
proofwiki-8617 | Double Angle Formulas/Hyperbolic Tangent | : $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$ | {{begin-eqn}}
{{eqn | l = \tanh 2 x
| r = \frac {\sinh 2 x} {\cosh 2 x}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {2 \cosh x \sinh x} {\cosh^2 x + \sinh^2 x}
| c = Double Angle Formula for Hyperbolic Sine and Double Angle Formula for Hyperbolic Cosine
}}
{{eqn | r = \frac {\fra... | : $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$ | {{begin-eqn}}
{{eqn | l = \tanh 2 x
| r = \frac {\sinh 2 x} {\cosh 2 x}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {2 \cosh x \sinh x} {\cosh^2 x + \sinh^2 x}
| c = [[Double Angle Formula for Hyperbolic Sine]] and [[Double Angle Formula for Hyperbolic Cosine]]
}}
{{eqn | r = \fr... | Double Angle Formulas/Hyperbolic Tangent/Proof 1 | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Tangent | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Tangent/Proof_1 | [
"Hyperbolic Tangent Function",
"Double Angle Formula for Hyperbolic Tangent"
] | [] | [
"Double Angle Formulas/Hyperbolic Sine",
"Double Angle Formulas/Hyperbolic Cosine"
] |
proofwiki-8618 | Double Angle Formulas/Hyperbolic Tangent | : $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$ | {{begin-eqn}}
{{eqn | l = \tanh 2 x
| r = \tanh \left({x + x}\right)
}}
{{eqn | r = \frac {\tanh x + \tanh x} {1 + \tanh x \tanh x}
| c = Hyperbolic Tangent of Sum
}}
{{eqn | r = \frac {2 \tanh x} {1 + \tanh^2 x}
}}
{{end-eqn}}
{{qed}} | : $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$ | {{begin-eqn}}
{{eqn | l = \tanh 2 x
| r = \tanh \left({x + x}\right)
}}
{{eqn | r = \frac {\tanh x + \tanh x} {1 + \tanh x \tanh x}
| c = [[Hyperbolic Tangent of Sum]]
}}
{{eqn | r = \frac {2 \tanh x} {1 + \tanh^2 x}
}}
{{end-eqn}}
{{qed}} | Double Angle Formulas/Hyperbolic Tangent/Proof 2 | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Tangent | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Tangent/Proof_2 | [
"Hyperbolic Tangent Function",
"Double Angle Formula for Hyperbolic Tangent"
] | [] | [
"Hyperbolic Tangent of Sum"
] |
proofwiki-8619 | Double Angle Formulas/Hyperbolic Tangent | : $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$ | Starting from the right, we have
{{begin-eqn}}
{{eqn | l = \dfrac {2 \tanh x} {1 + \tanh^2 x}
| r = \dfrac {2 \paren {\dfrac {e^x - e^{-x} } {e^x + e^{-x} } } } {1 + \paren {\dfrac{e^x - e^{-x} } {e^x + e^{-x} } }^2}
| c = {{Defof|Hyperbolic Tangent|index = 1}}
}}
{{eqn | r = \dfrac {2 \paren {e^x + e^{-x} ... | : $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$ | Starting from the right, we have
{{begin-eqn}}
{{eqn | l = \dfrac {2 \tanh x} {1 + \tanh^2 x}
| r = \dfrac {2 \paren {\dfrac {e^x - e^{-x} } {e^x + e^{-x} } } } {1 + \paren {\dfrac{e^x - e^{-x} } {e^x + e^{-x} } }^2}
| c = {{Defof|Hyperbolic Tangent|index = 1}}
}}
{{eqn | r = \dfrac {2 \paren {e^x + e^{-x} ... | Double Angle Formulas/Hyperbolic Tangent/Proof 3 | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Tangent | https://proofwiki.org/wiki/Double_Angle_Formulas/Hyperbolic_Tangent/Proof_3 | [
"Hyperbolic Tangent Function",
"Double Angle Formula for Hyperbolic Tangent"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Square of Sum",
"Square of Difference"
] |
proofwiki-8620 | Half Angle Formulas/Hyperbolic Sine | {{begin-eqn}}
{{eqn | l = \sinh \frac x 2
| r = +\sqrt {\frac {\cosh x - 1} 2}
| c = for $x \ge 0$
}}
{{eqn | l = \sinh \frac x 2
| r = -\sqrt {\dfrac {\cosh x - 1} 2}
| c = for $x \le 0$
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 1 + 2 \ \sinh^2 \frac x 2
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \ \sinh^2 \frac x 2
| r = \cosh x - 1
}}
{{eqn | ll= \leadsto
| l = \sinh \frac x2
| r = \pm \sqrt {\frac {\cosh x - 1} 2... | {{begin-eqn}}
{{eqn | l = \sinh \frac x 2
| r = +\sqrt {\frac {\cosh x - 1} 2}
| c = for $x \ge 0$
}}
{{eqn | l = \sinh \frac x 2
| r = -\sqrt {\dfrac {\cosh x - 1} 2}
| c = for $x \le 0$
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 1 + 2 \ \sinh^2 \frac x 2
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = 2 \ \sinh^2 \frac x 2
| r = \cosh x - 1
}}
{{eqn | ll= \leadsto
| l = \sinh \frac x2
| r = \pm \sqrt {\frac {\cosh x - 1} 2... | Half Angle Formulas/Hyperbolic Sine | https://proofwiki.org/wiki/Half_Angle_Formulas/Hyperbolic_Sine | https://proofwiki.org/wiki/Half_Angle_Formulas/Hyperbolic_Sine | [
"Hyperbolic Sine Function"
] | [] | [] |
proofwiki-8621 | Half Angle Formulas/Hyperbolic Cosine | :$\cosh \dfrac x 2 = +\sqrt {\dfrac {\cosh x + 1} 2}$ | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 2 \cosh^2 \frac x 2 - 1
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|1}}
}}
{{eqn | ll= \leadsto
| l = 2 \cosh^2 \frac x 2
| r = \cosh x + 1
}}
{{eqn | ll= \leadsto
| l = \cosh \frac x 2
| r = \pm \sqrt {\frac {\cosh x + 1} 2}
}... | :$\cosh \dfrac x 2 = +\sqrt {\dfrac {\cosh x + 1} 2}$ | {{begin-eqn}}
{{eqn | l = \cosh x
| r = 2 \cosh^2 \frac x 2 - 1
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|1}}
}}
{{eqn | ll= \leadsto
| l = 2 \cosh^2 \frac x 2
| r = \cosh x + 1
}}
{{eqn | ll= \leadsto
| l = \cosh \frac x 2
| r = \pm \sqrt {\frac {\cosh x + 1} 2}
}... | Half Angle Formulas/Hyperbolic Cosine | https://proofwiki.org/wiki/Half_Angle_Formulas/Hyperbolic_Cosine | https://proofwiki.org/wiki/Half_Angle_Formulas/Hyperbolic_Cosine | [
"Hyperbolic Cosine Function"
] | [] | [] |
proofwiki-8622 | Half Angle Formulas/Hyperbolic Tangent | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = +\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = for $x \ge 1$
}}
{{eqn | l = \tanh \frac x 2
| r = -\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = for $x \le 1$
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = \frac {\sinh \frac x 2} {\cosh \frac x 2}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {\pm \sqrt {\frac {\cosh x - 1} 2} } {\pm \sqrt {\frac {\cosh x + 1} 2} }
| c = Half Angle Formula for Hyperbolic Sine and Half Angle Formula for ... | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = +\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = for $x \ge 1$
}}
{{eqn | l = \tanh \frac x 2
| r = -\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = for $x \le 1$
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = \frac {\sinh \frac x 2} {\cosh \frac x 2}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {\pm \sqrt {\frac {\cosh x - 1} 2} } {\pm \sqrt {\frac {\cosh x + 1} 2} }
| c = [[Half Angle Formula for Hyperbolic Sine]] and [[Half Angle Formul... | Half Angle Formulas/Hyperbolic Tangent | https://proofwiki.org/wiki/Half_Angle_Formulas/Hyperbolic_Tangent | https://proofwiki.org/wiki/Half_Angle_Formulas/Hyperbolic_Tangent | [
"Hyperbolic Tangent Function"
] | [] | [
"Half Angle Formulas/Hyperbolic Sine",
"Half Angle Formulas/Hyperbolic Cosine"
] |
proofwiki-8623 | Half Angle Formula for Hyperbolic Tangent/Corollary 1 | :$\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$ | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = Half Angle Formula for Hyperbolic Tangent
}}
{{eqn | r = \pm \sqrt {\frac {\paren {\cosh x - 1} \paren {\cosh x + 1} } {\paren {\cosh x + 1}^2} }
| c = multiplying top and bottom by $\sqrt {\cosh x + 1... | :$\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$ | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = [[Half Angle Formulas/Hyperbolic Tangent|Half Angle Formula for Hyperbolic Tangent]]
}}
{{eqn | r = \pm \sqrt {\frac {\paren {\cosh x - 1} \paren {\cosh x + 1} } {\paren {\cosh x + 1}^2} }
| c = multip... | Half Angle Formula for Hyperbolic Tangent/Corollary 1 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Hyperbolic_Tangent/Corollary_1 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Hyperbolic_Tangent/Corollary_1 | [
"Hyperbolic Tangent Function"
] | [] | [
"Half Angle Formulas/Hyperbolic Tangent",
"Difference of Two Squares",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-8624 | Half Angle Formula for Hyperbolic Tangent/Corollary 2 | For $x \ne 0$:
:$\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$ | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = Half Angle Formula for Hyperbolic Tangent
}}
{{eqn | r = \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\paren {\cosh x + 1} \paren {\cosh x - 1} } }
| c = multiplying numerator and denominator by $\sqrt {... | For $x \ne 0$:
:$\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$ | {{begin-eqn}}
{{eqn | l = \tanh \frac x 2
| r = \pm \sqrt {\frac {\cosh x - 1} {\cosh x + 1} }
| c = [[Half Angle Formula for Hyperbolic Tangent]]
}}
{{eqn | r = \pm \sqrt {\frac {\paren {\cosh x - 1}^2} {\paren {\cosh x + 1} \paren {\cosh x - 1} } }
| c = multiplying [[Definition:Numerator|numerator]... | Half Angle Formula for Hyperbolic Tangent/Corollary 2 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Hyperbolic_Tangent/Corollary_2 | https://proofwiki.org/wiki/Half_Angle_Formula_for_Hyperbolic_Tangent/Corollary_2 | [
"Hyperbolic Tangent Function"
] | [] | [
"Half Angle Formulas/Hyperbolic Tangent",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-8625 | Triple Angle Formulas/Hyperbolic Sine | :$\sinh 3 x = 3 \sinh x + 4 \sinh^3 x$ | {{begin-eqn}}
{{eqn | l = \sinh {3 x}
| r = \map \sinh {2 x + x}
}}
{{eqn | r = \sinh 2 x \cosh x + \cosh 2 x \sinh x
| c = Hyperbolic Sine of Sum
}}
{{eqn | r = \paren {2 \sinh x \cosh x} \cosh x + \cosh 2 x \sinh x
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \paren {2 \sinh x \cosh... | :$\sinh 3 x = 3 \sinh x + 4 \sinh^3 x$ | {{begin-eqn}}
{{eqn | l = \sinh {3 x}
| r = \map \sinh {2 x + x}
}}
{{eqn | r = \sinh 2 x \cosh x + \cosh 2 x \sinh x
| c = [[Hyperbolic Sine of Sum]]
}}
{{eqn | r = \paren {2 \sinh x \cosh x} \cosh x + \cosh 2 x \sinh x
| c = [[Double Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = \paren {2 \sinh... | Triple Angle Formulas/Hyperbolic Sine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Hyperbolic_Sine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Hyperbolic_Sine | [
"Hyperbolic Sine Function",
"Triple Angle Formulas"
] | [] | [
"Hyperbolic Sine of Sum",
"Double Angle Formulas/Hyperbolic Sine",
"Double Angle Formulas/Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-8626 | Triple Angle Formulas/Hyperbolic Cosine | : $\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$ | {{begin-eqn}}
{{eqn | l = \cosh 3 x
| r = \cosh \paren {2 x + x}
}}
{{eqn | r = \cosh 2 x \cosh x + \sinh 2 x \sinh x
| c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \paren {\cosh^2 x + \sinh^2 x} \cosh x + \sinh 2 x \sinh x
| c = Double Angle Formula for Hyperbolic Cosine
}}
{{eqn | r = \paren {\cosh^2... | : $\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$ | {{begin-eqn}}
{{eqn | l = \cosh 3 x
| r = \cosh \paren {2 x + x}
}}
{{eqn | r = \cosh 2 x \cosh x + \sinh 2 x \sinh x
| c = [[Hyperbolic Cosine of Sum]]
}}
{{eqn | r = \paren {\cosh^2 x + \sinh^2 x} \cosh x + \sinh 2 x \sinh x
| c = [[Double Angle Formula for Hyperbolic Cosine]]
}}
{{eqn | r = \paren ... | Triple Angle Formulas/Hyperbolic Cosine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Hyperbolic_Cosine | https://proofwiki.org/wiki/Triple_Angle_Formulas/Hyperbolic_Cosine | [
"Hyperbolic Cosine Function",
"Triple Angle Formulas"
] | [] | [
"Hyperbolic Cosine of Sum",
"Double Angle Formulas/Hyperbolic Cosine",
"Double Angle Formulas/Hyperbolic Sine",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-8627 | Triple Angle Formulas/Hyperbolic Tangent | :$\tanh {3 x} = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$ | {{begin-eqn}}
{{eqn | l = \tanh {3 x}
| r = \frac {\sinh {3 x} } {\cosh {3 x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {3 \sinh x + 4 \sinh^3 x} {\cosh {3 x} }
| c = Triple Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \frac {3 \sinh x + 4 \sinh^3 x} {4 \cosh^3 x - 3 \cos... | :$\tanh {3 x} = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$ | {{begin-eqn}}
{{eqn | l = \tanh {3 x}
| r = \frac {\sinh {3 x} } {\cosh {3 x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {3 \sinh x + 4 \sinh^3 x} {\cosh {3 x} }
| c = [[Triple Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = \frac {3 \sinh x + 4 \sinh^3 x} {4 \cosh^3 x - 3 ... | Triple Angle Formulas/Hyperbolic Tangent | https://proofwiki.org/wiki/Triple_Angle_Formulas/Hyperbolic_Tangent | https://proofwiki.org/wiki/Triple_Angle_Formulas/Hyperbolic_Tangent | [
"Hyperbolic Tangent Function",
"Triple Angle Formulas"
] | [] | [
"Triple Angle Formulas/Hyperbolic Sine",
"Triple Angle Formulas/Hyperbolic Cosine",
"Sum of Squares of Hyperbolic Secant and Tangent"
] |
proofwiki-8628 | Quadruple Angle Formulas/Hyperbolic Sine | :$\sinh 4 x = 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 4 x
| r = \map \sinh {3 x + x}
}}
{{eqn | r = \sinh 3 x \cosh x + \cosh 3 x \sinh x
| c = Hyperbolic Sine of Sum
}}
{{eqn | r = \paren {3 \sinh x + 4 \sinh^3 x} \cosh x + \cosh 3 x \sinh x
| c = Triple Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \paren {3 \sinh x +... | :$\sinh 4 x = 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x$ | {{begin-eqn}}
{{eqn | l = \sinh 4 x
| r = \map \sinh {3 x + x}
}}
{{eqn | r = \sinh 3 x \cosh x + \cosh 3 x \sinh x
| c = [[Hyperbolic Sine of Sum]]
}}
{{eqn | r = \paren {3 \sinh x + 4 \sinh^3 x} \cosh x + \cosh 3 x \sinh x
| c = [[Triple Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = \paren {3 \... | Quadruple Angle Formulas/Hyperbolic Sine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Hyperbolic_Sine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Hyperbolic_Sine | [
"Hyperbolic Sine Function",
"Quadruple Angle Formulas"
] | [] | [
"Hyperbolic Sine of Sum",
"Triple Angle Formulas/Hyperbolic Sine",
"Triple Angle Formulas/Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-8629 | Quadruple Angle Formulas/Hyperbolic Cosine | :$\cosh 4 x = 8 \cosh^4 x - 8 \cosh^2 x + 1$ | {{begin-eqn}}
{{eqn | l = \cosh 4 x
| r = \map \cosh {2 x + 2 x}
}}
{{eqn | r = \cosh 2 x \cosh 2 x + \sinh 2 x \sinh 2 x
| c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \paren {\cosh^2 x + \sinh^2 x} \paren {\cosh^2 x + \sinh^2 x} + \paren {2 \sinh x \cosh x} \paren {2 \sinh x \cosh x}
| c = Double Ang... | :$\cosh 4 x = 8 \cosh^4 x - 8 \cosh^2 x + 1$ | {{begin-eqn}}
{{eqn | l = \cosh 4 x
| r = \map \cosh {2 x + 2 x}
}}
{{eqn | r = \cosh 2 x \cosh 2 x + \sinh 2 x \sinh 2 x
| c = [[Hyperbolic Cosine of Sum]]
}}
{{eqn | r = \paren {\cosh^2 x + \sinh^2 x} \paren {\cosh^2 x + \sinh^2 x} + \paren {2 \sinh x \cosh x} \paren {2 \sinh x \cosh x}
| c = [[Doub... | Quadruple Angle Formulas/Hyperbolic Cosine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Hyperbolic_Cosine | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Hyperbolic_Cosine | [
"Hyperbolic Cosine Function",
"Quadruple Angle Formulas"
] | [] | [
"Hyperbolic Cosine of Sum",
"Double Angle Formulas",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-8630 | Quadruple Angle Formulas/Hyperbolic Tangent | :$\tanh 4 x = \dfrac {4 \tanh x + 4 \tanh^3 x} {1 + 6 \tanh^2 x + \tanh^4 x}$ | {{begin-eqn}}
{{eqn | l = \tanh 4 x)
| r = \frac {\sinh 4 x} {\cosh 4 x}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {8 \sinh^3 x \cosh x + 4 \sinh x \cosh x} {\cosh 4 x}
| c = Quadruple Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \frac {8 \sinh^3 x \cosh x + 4 \sinh x \cosh... | :$\tanh 4 x = \dfrac {4 \tanh x + 4 \tanh^3 x} {1 + 6 \tanh^2 x + \tanh^4 x}$ | {{begin-eqn}}
{{eqn | l = \tanh 4 x)
| r = \frac {\sinh 4 x} {\cosh 4 x}
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {8 \sinh^3 x \cosh x + 4 \sinh x \cosh x} {\cosh 4 x}
| c = [[Quadruple Angle Formula for Hyperbolic Sine]]
}}
{{eqn | r = \frac {8 \sinh^3 x \cosh x + 4 \sinh x \... | Quadruple Angle Formulas/Hyperbolic Tangent | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Hyperbolic_Tangent | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Hyperbolic_Tangent | [
"Hyperbolic Tangent Function",
"Quadruple Angle Formulas"
] | [] | [
"Quadruple Angle Formulas/Hyperbolic Sine",
"Quadruple Angle Formulas/Hyperbolic Cosine",
"Sum of Squares of Hyperbolic Secant and Tangent"
] |
proofwiki-8631 | Power Reduction Formulas/Hyperbolic Sine Squared | :$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$ | {{begin-eqn}}
{{eqn | l = 2 \sinh^2 x + 1
| r = \cosh 2 x
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = \sinh^2 x
| r = \frac {\cosh 2 x - 1} 2
| c = solving for $\sinh^2 x$
}}
{{end-eqn}}
{{qed}} | :$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$ | {{begin-eqn}}
{{eqn | l = 2 \sinh^2 x + 1
| r = \cosh 2 x
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|2}}
}}
{{eqn | ll= \leadsto
| l = \sinh^2 x
| r = \frac {\cosh 2 x - 1} 2
| c = solving for $\sinh^2 x$
}}
{{end-eqn}}
{{qed}} | Power Reduction Formulas/Hyperbolic Sine Squared/Proof 1 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_Squared/Proof_1 | [
"Hyperbolic Sine Function",
"Square of Hyperbolic Sine"
] | [] | [] |
proofwiki-8632 | Power Reduction Formulas/Hyperbolic Sine Squared | :$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$ | {{begin-eqn}}
{{eqn | l = \sinh^2 x
| r = \paren {\frac {e^x - e^{-x} } 2}^2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 4 \paren {e^{2 x} + e^{-2 x} - 2}
| c = multiplying out
}}
{{eqn | r = \frac 1 2 \paren {\dfrac {e^{2 x} + e^{-2 x} } 2 - 1}
| c = rearranging
}}
{{eqn | r = \frac {\cosh 2 x - 1} 2
|... | :$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$ | {{begin-eqn}}
{{eqn | l = \sinh^2 x
| r = \paren {\frac {e^x - e^{-x} } 2}^2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 4 \paren {e^{2 x} + e^{-2 x} - 2}
| c = multiplying out
}}
{{eqn | r = \frac 1 2 \paren {\dfrac {e^{2 x} + e^{-2 x} } 2 - 1}
| c = rearranging
}}
{{eqn | r = \frac {\cosh 2 x - 1} 2
|... | Power Reduction Formulas/Hyperbolic Sine Squared/Proof 2 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_Squared/Proof_2 | [
"Hyperbolic Sine Function",
"Square of Hyperbolic Sine"
] | [] | [] |
proofwiki-8633 | Power Reduction Formulas/Hyperbolic Cosine Squared | :$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$ | {{begin-eqn}}
{{eqn | l = 2 \cosh^2 x - 1
| r = \cosh 2 x
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|1}}
}}
{{eqn | l = \cosh^2 x
| r = \frac {\cosh 2 x + 1} 2
| c = solving for $\cosh^2 x$
}}
{{end-eqn}}
{{qed}} | :$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$ | {{begin-eqn}}
{{eqn | l = 2 \cosh^2 x - 1
| r = \cosh 2 x
| c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|1}}
}}
{{eqn | l = \cosh^2 x
| r = \frac {\cosh 2 x + 1} 2
| c = solving for $\cosh^2 x$
}}
{{end-eqn}}
{{qed}} | Power Reduction Formulas/Hyperbolic Cosine Squared/Proof 1 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Squared/Proof_1 | [
"Hyperbolic Cosine Function",
"Square of Hyperbolic Cosine"
] | [] | [] |
proofwiki-8634 | Power Reduction Formulas/Hyperbolic Cosine Squared | :$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$ | {{begin-eqn}}
{{eqn | l = \cosh^2 x
| r = \frac 1 4 \paren {e^x + e^{-x} }^2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{2 x} + e^{-2 x} + 2} 4
}}
{{eqn | r = \frac {\cosh 2 x + 1} 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{end-eqn}}
{{qed}} | :$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$ | {{begin-eqn}}
{{eqn | l = \cosh^2 x
| r = \frac 1 4 \paren {e^x + e^{-x} }^2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{2 x} + e^{-2 x} + 2} 4
}}
{{eqn | r = \frac {\cosh 2 x + 1} 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{end-eqn}}
{{qed}} | Power Reduction Formulas/Hyperbolic Cosine Squared/Proof 2 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Squared/Proof_2 | [
"Hyperbolic Cosine Function",
"Square of Hyperbolic Cosine"
] | [] | [] |
proofwiki-8635 | Power Reduction Formulas/Hyperbolic Cosine Squared | :$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$ | {{begin-eqn}}
{{eqn | l = \cosh^2 x
| r = \cos^2 i x
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \frac {\map \cos {2 i x} + 1} 2
| c = Square of Cosine
}}
{{eqn | r = \frac {\cosh 2 x + 1} 2
| c = Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}} | :$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$ | {{begin-eqn}}
{{eqn | l = \cosh^2 x
| r = \cos^2 i x
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \frac {\map \cos {2 i x} + 1} 2
| c = [[Square of Cosine]]
}}
{{eqn | r = \frac {\cosh 2 x + 1} 2
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{end-eqn}}
{{qed}} | Power Reduction Formulas/Hyperbolic Cosine Squared/Proof 3 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Squared | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Squared/Proof_3 | [
"Hyperbolic Cosine Function",
"Square of Hyperbolic Cosine"
] | [] | [
"Hyperbolic Cosine in terms of Cosine",
"Power Reduction Formulas/Cosine Squared",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8636 | Power Reduction Formulas/Hyperbolic Sine Cubed | :$\sinh^3 x = \dfrac {\sinh 3 x - 3 \sinh x} 4$ | {{begin-eqn}}
{{eqn | l = \sinh 3 x
| r = 3 \sinh x + 4 \sinh^3 x
| c = Triple Angle Formula for Hyperbolic Sine
}}
{{eqn | ll= \leadsto
| l = 4 \sinh^3 x
| r = \sinh 3 x - 3 \sinh x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \sinh^3 x
| r = \frac {\sinh 3 x - 3 \sinh x} 4... | :$\sinh^3 x = \dfrac {\sinh 3 x - 3 \sinh x} 4$ | {{begin-eqn}}
{{eqn | l = \sinh 3 x
| r = 3 \sinh x + 4 \sinh^3 x
| c = [[Triple Angle Formula for Hyperbolic Sine]]
}}
{{eqn | ll= \leadsto
| l = 4 \sinh^3 x
| r = \sinh 3 x - 3 \sinh x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \sinh^3 x
| r = \frac {\sinh 3 x - 3 \sinh ... | Power Reduction Formulas/Hyperbolic Sine Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_Cubed | [
"Hyperbolic Sine Function"
] | [] | [
"Triple Angle Formulas/Hyperbolic Sine"
] |
proofwiki-8637 | Power Reduction Formulas/Hyperbolic Cosine Cubed | :$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$ | {{begin-eqn}}
{{eqn | l = \cosh 3 x
| r = 4 \cosh^3 x - 3 \cosh x
| c = Triple Angle Formula for Hyperbolic Cosine
}}
{{eqn | ll= \leadsto
| l = 4 \cosh^3 x
| r = \cosh 3 x + 3 \cosh x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \cosh^3 x
| r = \dfrac {\cosh 3 x + 3 \cosh x... | :$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$ | {{begin-eqn}}
{{eqn | l = \cosh 3 x
| r = 4 \cosh^3 x - 3 \cosh x
| c = [[Triple Angle Formula for Hyperbolic Cosine]]
}}
{{eqn | ll= \leadsto
| l = 4 \cosh^3 x
| r = \cosh 3 x + 3 \cosh x
| c = rearranging
}}
{{eqn | ll= \leadsto
| l = \cosh^3 x
| r = \dfrac {\cosh 3 x + 3 \co... | Power Reduction Formulas/Hyperbolic Cosine Cubed/Proof 1 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Cubed/Proof_1 | [
"Hyperbolic Cosine Function",
"Cube of Hyperbolic Cosine"
] | [] | [
"Triple Angle Formulas/Hyperbolic Cosine"
] |
proofwiki-8638 | Power Reduction Formulas/Hyperbolic Cosine Cubed | :$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$ | {{begin-eqn}}
{{eqn | l = \cosh^3 x
| r = \frac 1 {2^3} \paren {e^x + e^{-x} }^3
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 8 \paren {e^{3x} + e^{-3x} + 3e^{x} + 3e^{-x} }
}}
{{eqn | r = \frac 1 4 \paren {\frac{ e^{3x} + e^{-3x} } 2} + \frac 3 4 \paren {\frac{e^{x} + e^{-x} } 2}
}}
{{eqn | r = \frac {\c... | :$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$ | {{begin-eqn}}
{{eqn | l = \cosh^3 x
| r = \frac 1 {2^3} \paren {e^x + e^{-x} }^3
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 8 \paren {e^{3x} + e^{-3x} + 3e^{x} + 3e^{-x} }
}}
{{eqn | r = \frac 1 4 \paren {\frac{ e^{3x} + e^{-3x} } 2} + \frac 3 4 \paren {\frac{e^{x} + e^{-x} } 2}
}}
{{eqn | r = \frac {\c... | Power Reduction Formulas/Hyperbolic Cosine Cubed/Proof 2 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Cubed/Proof_2 | [
"Hyperbolic Cosine Function",
"Cube of Hyperbolic Cosine"
] | [] | [] |
proofwiki-8639 | Power Reduction Formulas/Hyperbolic Cosine Cubed | :$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$ | {{begin-eqn}}
{{eqn | l = \cosh^3 x
| r = \cos^3 i x
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \frac {\map \cos {3 i x} + 3 \cos i x} 4
| c = Cube of Cosine
}}
{{eqn | r = \frac {\cosh 3 x + 3 \cosh x} 4
| c = Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}} | :$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$ | {{begin-eqn}}
{{eqn | l = \cosh^3 x
| r = \cos^3 i x
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \frac {\map \cos {3 i x} + 3 \cos i x} 4
| c = [[Cube of Cosine]]
}}
{{eqn | r = \frac {\cosh 3 x + 3 \cosh x} 4
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{end-eqn}}
{{qed}} | Power Reduction Formulas/Hyperbolic Cosine Cubed/Proof 3 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Cubed | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_Cubed/Proof_3 | [
"Hyperbolic Cosine Function",
"Cube of Hyperbolic Cosine"
] | [] | [
"Hyperbolic Cosine in terms of Cosine",
"Power Reduction Formulas/Cosine Cubed",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8640 | Power Reduction Formulas/Hyperbolic Sine to 4th | :$\sinh^4 x = \dfrac {3 - 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn|l = \sinh^4 x
|r = \left({\sinh^2 x}\right)^2
}}
{{eqn|r = \left({\frac {\cosh 2 x - 1} 2}\right)^2
|c = Square of Hyperbolic Sine
}}
{{eqn|r = \frac {\cosh^2 2 x - 2 \cosh 2 x + 1} 4
|c = multiplying out
}}
{{eqn|r = \frac {\frac {\cosh 4 x + 1} 2 - 2 \cosh 2 x + 1} 4
|c = Squar... | :$\sinh^4 x = \dfrac {3 - 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn|l = \sinh^4 x
|r = \left({\sinh^2 x}\right)^2
}}
{{eqn|r = \left({\frac {\cosh 2 x - 1} 2}\right)^2
|c = [[Square of Hyperbolic Sine]]
}}
{{eqn|r = \frac {\cosh^2 2 x - 2 \cosh 2 x + 1} 4
|c = multiplying out
}}
{{eqn|r = \frac {\frac {\cosh 4 x + 1} 2 - 2 \cosh 2 x + 1} 4
|c = [... | Power Reduction Formulas/Hyperbolic Sine to 4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_to_4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Sine_to_4th | [
"Hyperbolic Sine Function"
] | [] | [
"Power Reduction Formulas/Hyperbolic Sine Squared",
"Power Reduction Formulas/Hyperbolic Cosine Squared"
] |
proofwiki-8641 | Power Reduction Formulas/Hyperbolic Cosine to 4th | :$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cosh 4 x
| r = \paren {\cosh^2 x}^2
}}
{{eqn | r = \paren {\frac {\cosh 2 x + 1} 2}^2
| c = Square of Hyperbolic Cosine
}}
{{eqn | r = \frac {\cosh^2 2 x + 2 \cosh 2 x + 1} 4
| c = multiplying out
}}
{{eqn | r = \frac {\frac {\cosh 4 x + 1} 2 + 2 \cosh 2 x + 1} 4
| c =... | :$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cosh 4 x
| r = \paren {\cosh^2 x}^2
}}
{{eqn | r = \paren {\frac {\cosh 2 x + 1} 2}^2
| c = [[Square of Hyperbolic Cosine]]
}}
{{eqn | r = \frac {\cosh^2 2 x + 2 \cosh 2 x + 1} 4
| c = multiplying out
}}
{{eqn | r = \frac {\frac {\cosh 4 x + 1} 2 + 2 \cosh 2 x + 1} 4
|... | Power Reduction Formulas/Hyperbolic Cosine to 4th/Proof 1 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_to_4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_to_4th/Proof_1 | [
"Hyperbolic Cosine Function",
"Fourth Power of Hyperbolic Cosine"
] | [] | [
"Power Reduction Formulas/Hyperbolic Cosine Squared",
"Power Reduction Formulas/Hyperbolic Cosine Squared"
] |
proofwiki-8642 | Power Reduction Formulas/Hyperbolic Cosine to 4th | :$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cosh^4 x
| r = \frac 1 {2^4}\left(e^{x} + e^{-x}\right)^4
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 {16} \left({e^{4 x} + 4 e^{2 x} + 6 e^{0 x} + 4 e^{-2 x} + e^{-4 x} }\right)
| c = Binomial Theorem
}}
{{eqn | r = \frac 1 8 \left({\frac{e^{4 x} + e^{-4 x} } 2}\right) + \fra... | :$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cosh^4 x
| r = \frac 1 {2^4}\left(e^{x} + e^{-x}\right)^4
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac 1 {16} \left({e^{4 x} + 4 e^{2 x} + 6 e^{0 x} + 4 e^{-2 x} + e^{-4 x} }\right)
| c = [[Binomial Theorem]]
}}
{{eqn | r = \frac 1 8 \left({\frac{e^{4 x} + e^{-4 x} } 2}\right) + ... | Power Reduction Formulas/Hyperbolic Cosine to 4th/Proof 2 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_to_4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_to_4th/Proof_2 | [
"Hyperbolic Cosine Function",
"Fourth Power of Hyperbolic Cosine"
] | [] | [
"Binomial Theorem"
] |
proofwiki-8643 | Power Reduction Formulas/Hyperbolic Cosine to 4th | :$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cosh^4 x
| r = \cos^4 i x
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \frac {3 + 4 \cos \paren {2 i x} + \cos \paren {4 i x} } 8
| c = Fourth Power of Cosine
}}
{{eqn | r = \frac {3 + 4 \cosh 2 x + \cosh 4 x} 8
| c = Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}... | :$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$ | {{begin-eqn}}
{{eqn | l = \cosh^4 x
| r = \cos^4 i x
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \frac {3 + 4 \cos \paren {2 i x} + \cos \paren {4 i x} } 8
| c = [[Fourth Power of Cosine]]
}}
{{eqn | r = \frac {3 + 4 \cosh 2 x + \cosh 4 x} 8
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{end-... | Power Reduction Formulas/Hyperbolic Cosine to 4th/Proof 3 | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_to_4th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Hyperbolic_Cosine_to_4th/Proof_3 | [
"Hyperbolic Cosine Function",
"Fourth Power of Hyperbolic Cosine"
] | [] | [
"Hyperbolic Cosine in terms of Cosine",
"Power Reduction Formulas/Cosine to 4th",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8644 | Prosthaphaeresis Formulas/Hyperbolic Sine plus Hyperbolic Sine | :$\sinh x + \sinh y = 2 \map \sinh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \map \sinh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}
| c =
}}
{{eqn | r = 2 \frac {\map \sinh {\dfrac {x + y} 2 + \dfrac {x - y} 2} + \map \sinh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2
| c = Werner Formula for Hyperbolic Sine by Hyperbolic Cosine
}}
{{eqn | r... | :$\sinh x + \sinh y = 2 \map \sinh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \map \sinh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}
| c =
}}
{{eqn | r = 2 \frac {\map \sinh {\dfrac {x + y} 2 + \dfrac {x - y} 2} + \map \sinh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2
| c = [[Werner Formula for Hyperbolic Sine by Hyperbolic Cosine]]
}}
{{eqn... | Prosthaphaeresis Formulas/Hyperbolic Sine plus Hyperbolic Sine | https://proofwiki.org/wiki/Prosthaphaeresis_Formulas/Hyperbolic_Sine_plus_Hyperbolic_Sine | https://proofwiki.org/wiki/Prosthaphaeresis_Formulas/Hyperbolic_Sine_plus_Hyperbolic_Sine | [
"Hyperbolic Sine Function",
"Prosthaphaeresis"
] | [] | [
"Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine"
] |
proofwiki-8645 | Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine | :$\sinh x - \sinh y = 2 \map \cosh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \map \cosh {\frac {x + y} 2} \map \sinh {\frac {x - y} 2}
| c =
}}
{{eqn | r = 2 \frac {\map \sinh {\dfrac {x - y} 2 + \dfrac {x + y} 2} + \map \sinh {\dfrac {x - y} 2 - \dfrac {x + y} 2} } 2
| c = Werner Formula for Hyperbolic Sine by Hyperbolic Cosine
}}
{{eqn | r ... | :$\sinh x - \sinh y = 2 \map \cosh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \map \cosh {\frac {x + y} 2} \map \sinh {\frac {x - y} 2}
| c =
}}
{{eqn | r = 2 \frac {\map \sinh {\dfrac {x - y} 2 + \dfrac {x + y} 2} + \map \sinh {\dfrac {x - y} 2 - \dfrac {x + y} 2} } 2
| c = [[Werner Formula for Hyperbolic Sine by Hyperbolic Cosine]]
}}
{{eqn ... | Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine | https://proofwiki.org/wiki/Prosthaphaeresis_Formulas/Hyperbolic_Sine_minus_Hyperbolic_Sine | https://proofwiki.org/wiki/Prosthaphaeresis_Formulas/Hyperbolic_Sine_minus_Hyperbolic_Sine | [
"Hyperbolic Sine Function",
"Prosthaphaeresis"
] | [] | [
"Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine",
"Hyperbolic Sine Function is Odd"
] |
proofwiki-8646 | Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine | :$\cosh x + \cosh y = 2 \map \cosh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \map \cosh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}
| c =
}}
{{eqn | r = 2 \frac {\map \cosh {\dfrac {x + y} 2 + \dfrac {x - y} 2} + \map \cosh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2
| c = Werner Formula for Hyperbolic Cosine by Hyperbolic Cosine
}}
{{eqn |... | :$\cosh x + \cosh y = 2 \map \cosh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$ | {{begin-eqn}}
{{eqn | o =
| r = 2 \map \cosh {\frac {x + y} 2} \map \cosh {\frac {x - y} 2}
| c =
}}
{{eqn | r = 2 \frac {\map \cosh {\dfrac {x + y} 2 + \dfrac {x - y} 2} + \map \cosh {\dfrac {x + y} 2 - \dfrac {x - y} 2} } 2
| c = [[Werner Formula for Hyperbolic Cosine by Hyperbolic Cosine]]
}}
{{e... | Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine | https://proofwiki.org/wiki/Prosthaphaeresis_Formulas/Hyperbolic_Cosine_plus_Hyperbolic_Cosine | https://proofwiki.org/wiki/Prosthaphaeresis_Formulas/Hyperbolic_Cosine_plus_Hyperbolic_Cosine | [
"Hyperbolic Cosine Function",
"Prosthaphaeresis"
] | [] | [
"Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine"
] |
proofwiki-8647 | Werner Formulas/Hyperbolic Sine by Hyperbolic Sine | :$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \cosh {x + y} - \map \cosh {x - y} } 2
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} - \map \cosh {x - y} } 2
| c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} - \paren {\cosh x \cosh y - \sinh x \s... | :$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\map \cosh {x + y} - \map \cosh {x - y} } 2
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} - \map \cosh {x - y} } 2
| c = [[Hyperbolic Cosine of Sum]]
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} - \paren {\cosh x \cosh y - \sinh ... | Werner Formulas/Hyperbolic Sine by Hyperbolic Sine/Proof 1 | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Sine | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Sine/Proof_1 | [
"Werner Formula for Hyperbolic Sine by Hyperbolic Sine",
"Werner Formulas",
"Hyperbolic Sine Function"
] | [] | [
"Hyperbolic Cosine of Sum",
"Hyperbolic Cosine of Sum/Corollary"
] |
proofwiki-8648 | Werner Formulas/Hyperbolic Sine by Hyperbolic Sine | :$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$ | {{begin-eqn}}
{{eqn | l = \sinh x \sinh y
| r = \frac {e^x - e^{-x} } 2 \frac {e^y - e^{-y} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{x + y} - e^{x - y} - e^{-x + y} + e^{-x - y} } 4
| c = simplifying
}}
{{eqn | r = \frac 1 2 \paren {\dfrac {e^{x + y} + e^{-\paren {x + y} } } 2 - \f... | :$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$ | {{begin-eqn}}
{{eqn | l = \sinh x \sinh y
| r = \frac {e^x - e^{-x} } 2 \frac {e^y - e^{-y} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{x + y} - e^{x - y} - e^{-x + y} + e^{-x - y} } 4
| c = simplifying
}}
{{eqn | r = \frac 1 2 \paren {\dfrac {e^{x + y} + e^{-\paren {x + y} } } 2 - \f... | Werner Formulas/Hyperbolic Sine by Hyperbolic Sine/Proof 2 | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Sine | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Sine/Proof_2 | [
"Werner Formula for Hyperbolic Sine by Hyperbolic Sine",
"Werner Formulas",
"Hyperbolic Sine Function"
] | [] | [] |
proofwiki-8649 | Werner Formulas/Hyperbolic Sine by Hyperbolic Sine | :$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$ | {{begin-eqn}}
{{eqn | l = \sinh x \sinh y
| r = i^2 \map \sin {\frac x i} \map \sin {\frac y i}
| c = Sine in terms of Hyperbolic Sine
}}
{{eqn | r = -\map \sin {\frac x i} \map \sin {\frac y i}
| c = $i^2 = -1$
}}
{{eqn | r = -\frac {\map \cos {\frac x i - \frac y i} - \map \cos {\frac x i + \frac y ... | :$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$ | {{begin-eqn}}
{{eqn | l = \sinh x \sinh y
| r = i^2 \map \sin {\frac x i} \map \sin {\frac y i}
| c = [[Sine in terms of Hyperbolic Sine]]
}}
{{eqn | r = -\map \sin {\frac x i} \map \sin {\frac y i}
| c = $i^2 = -1$
}}
{{eqn | r = -\frac {\map \cos {\frac x i - \frac y i} - \map \cos {\frac x i + \fra... | Werner Formulas/Hyperbolic Sine by Hyperbolic Sine/Proof 3 | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Sine | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Sine/Proof_3 | [
"Werner Formula for Hyperbolic Sine by Hyperbolic Sine",
"Werner Formulas",
"Hyperbolic Sine Function"
] | [] | [
"Sine in terms of Hyperbolic Sine",
"Werner Formulas/Sine by Sine",
"Cosine in terms of Hyperbolic Cosine"
] |
proofwiki-8650 | Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine | :$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \cosh \paren {x - y} } 2
| c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \paren {\cosh x \cosh y - \sin... | :$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \cosh \paren {x - y} } 2
| c = [[Hyperbolic Cosine of Sum]]
}}
{{eqn | r = \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \paren {\cosh x \cosh y - ... | Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Cosine_by_Hyperbolic_Cosine | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Cosine_by_Hyperbolic_Cosine | [
"Werner Formulas",
"Hyperbolic Cosine Function"
] | [] | [
"Hyperbolic Cosine of Sum",
"Hyperbolic Cosine of Sum/Corollary"
] |
proofwiki-8651 | Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine | :$\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2
}}
{{eqn | r = \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \cosh \paren {x - y} } 2
| c = Hyperbolic Sine of Sum
}}
{{eqn | r = \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \paren {\sinh x \cosh y - \cosh ... | :$\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$ | {{begin-eqn}}
{{eqn | o =
| r = \frac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2
}}
{{eqn | r = \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \cosh \paren {x - y} } 2
| c = [[Hyperbolic Sine of Sum]]
}}
{{eqn | r = \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \paren {\sinh x \cosh y - \c... | Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Cosine | https://proofwiki.org/wiki/Werner_Formulas/Hyperbolic_Sine_by_Hyperbolic_Cosine | [
"Hyperbolic Sine Function",
"Hyperbolic Cosine Function",
"Werner Formulas"
] | [] | [
"Hyperbolic Sine of Sum",
"Hyperbolic Sine of Sum/Corollary"
] |
proofwiki-8652 | Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant | Everywhere that the function is defined:
:$\map \arsinh {\dfrac 1 x} = \arcsch x$
where $\arsinh$ and $\arcsch$ denote real area hyperbolic sine and real area hyperbolic cosecant respectively. | {{begin-eqn}}
{{eqn | l = \map \arsinh {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \sinh y
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \csch y
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
... | Everywhere that the function is defined:
:$\map \arsinh {\dfrac 1 x} = \arcsch x$
where $\arsinh$ and $\arcsch$ denote [[Definition:Real Area Hyperbolic Sine|real area hyperbolic sine]] and [[Definition:Real Area Hyperbolic Cosecant|real area hyperbolic cosecant]] respectively. | {{begin-eqn}}
{{eqn | l = \map \arsinh {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \sinh y
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \csch y
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
... | Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Sine_of_Reciprocal_equals_Real_Area_Hyperbolic_Cosecant | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Sine_of_Reciprocal_equals_Real_Area_Hyperbolic_Cosecant | [
"Inverse Hyperbolic Cosecant",
"Inverse Hyperbolic Sine",
"Reciprocals"
] | [
"Definition:Inverse Hyperbolic Sine/Real/Definition 2",
"Definition:Inverse Hyperbolic Cosecant/Real/Definition 2"
] | [] |
proofwiki-8653 | Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant | Everywhere that the function is defined:
:$\map \arcosh {\dfrac 1 x} = \arsech x$
where $\arcosh$ and $\arsech$ denote real area hyperbolic cosine and real area hyperbolic secant respectively. | {{begin-eqn}}
{{eqn | l = \map \arcosh {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \cosh y
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sech y
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
... | Everywhere that the function is defined:
:$\map \arcosh {\dfrac 1 x} = \arsech x$
where $\arcosh$ and $\arsech$ denote [[Definition:Real Area Hyperbolic Cosine|real area hyperbolic cosine]] and [[Definition:Real Area Hyperbolic Secant|real area hyperbolic secant]] respectively. | {{begin-eqn}}
{{eqn | l = \map \arcosh {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \cosh y
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sech y
| c = {{Defof|Hyperbolic Secant|index = 2}}
}}
... | Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosine_of_Reciprocal_equals_Real_Area_Hyperbolic_Secant | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosine_of_Reciprocal_equals_Real_Area_Hyperbolic_Secant | [
"Inverse Hyperbolic Secant",
"Inverse Hyperbolic Cosine",
"Reciprocals"
] | [
"Definition:Inverse Hyperbolic Cosine/Real/Principal Branch",
"Definition:Inverse Hyperbolic Secant/Real/Principal Branch"
] | [] |
proofwiki-8654 | Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent | Everywhere that the function is defined:
:$\map \artanh {\dfrac 1 x} = \arcoth x$
where $\artanh$ and $\arcoth$ denote real area hyperbolic tangent and real area hyperbolic cotangent respectively. | {{begin-eqn}}
{{eqn | l = \map \artanh {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \tanh y
| c = {{Defof|Real Area Hyperbolic Tangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \frac {\cosh y} {\sinh y}
| c = {{Defof|Hyperb... | Everywhere that the function is defined:
:$\map \artanh {\dfrac 1 x} = \arcoth x$
where $\artanh$ and $\arcoth$ denote [[Definition:Real Area Hyperbolic Tangent|real area hyperbolic tangent]] and [[Definition:Real Area Hyperbolic Cotangent|real area hyperbolic cotangent]] respectively. | {{begin-eqn}}
{{eqn | l = \map \artanh {\dfrac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \tanh y
| c = {{Defof|Real Area Hyperbolic Tangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \frac {\cosh y} {\sinh y}
| c = {{Defof|Hyperb... | Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Tangent_of_Reciprocal_equals_Real_Area_Hyperbolic_Cotangent | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Tangent_of_Reciprocal_equals_Real_Area_Hyperbolic_Cotangent | [
"Inverse Hyperbolic Tangent",
"Inverse Hyperbolic Cotangent",
"Reciprocals"
] | [
"Definition:Inverse Hyperbolic Tangent/Real/Definition 2",
"Definition:Inverse Hyperbolic Cotangent/Real/Definition 2"
] | [] |
proofwiki-8655 | Inverse Hyperbolic Sine is Odd Function | Let $x \in \R$.
Then:
:$\map \arsinh {-x} = -\arsinh x$
where $\arsinh$ denotes the inverse hyperbolic sine function. | {{begin-eqn}}
{{eqn | l = \map \arsinh {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \sinh y
| c = {{Defof|Inverse Hyperbolic Sine|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\sinh y
| c =
}}
{{eqn | ll= \leadstoandfrom
... | Let $x \in \R$.
Then:
:$\map \arsinh {-x} = -\arsinh x$
where $\arsinh$ denotes the [[Definition:Real Inverse Hyperbolic Sine|inverse hyperbolic sine function]]. | {{begin-eqn}}
{{eqn | l = \map \arsinh {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \sinh y
| c = {{Defof|Inverse Hyperbolic Sine|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\sinh y
| c =
}}
{{eqn | ll= \leadstoandfrom
... | Inverse Hyperbolic Sine is Odd Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Sine_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Sine_is_Odd_Function | [
"Inverse Hyperbolic Sine",
"Examples of Odd Functions"
] | [
"Definition:Inverse Hyperbolic Sine/Real"
] | [
"Hyperbolic Sine Function is Odd"
] |
proofwiki-8656 | Inverse Hyperbolic Tangent is Odd Function | :$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\tanh^{-1} } {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \tanh y
| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\tanh y
| c =
}}
{{eqn | ll= \leadstoan... | :$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\tanh^{-1} } {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \tanh y
| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\tanh y
| c =
}}
{{eqn | ll= \leadstoan... | Inverse Hyperbolic Tangent is Odd Function/Proof 1 | https://proofwiki.org/wiki/Inverse_Hyperbolic_Tangent_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Tangent_is_Odd_Function/Proof_1 | [
"Inverse Hyperbolic Tangent",
"Inverse Hyperbolic Tangent is Odd Function",
"Examples of Odd Functions"
] | [] | [
"Hyperbolic Tangent Function is Odd"
] |
proofwiki-8657 | Inverse Hyperbolic Tangent is Odd Function | :$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\tanh^{-1} } {-x}
| r = \frac 1 2 \map \ln {\frac {1 + \paren {-x} } {1 - \paren {-x} } }
| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 2}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {1 - x} {1 + x} }
}}
{{eqn | r = \frac 1 2 \paren {\map \ln {1 - x} - \map \ln {1 + x} }... | :$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\tanh^{-1} } {-x}
| r = \frac 1 2 \map \ln {\frac {1 + \paren {-x} } {1 - \paren {-x} } }
| c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 2}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {1 - x} {1 + x} }
}}
{{eqn | r = \frac 1 2 \paren {\map \ln {1 - x} - \map \ln {1 + x} }... | Inverse Hyperbolic Tangent is Odd Function/Proof 2 | https://proofwiki.org/wiki/Inverse_Hyperbolic_Tangent_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Tangent_is_Odd_Function/Proof_2 | [
"Inverse Hyperbolic Tangent",
"Inverse Hyperbolic Tangent is Odd Function",
"Examples of Odd Functions"
] | [] | [
"Difference of Logarithms",
"Difference of Logarithms"
] |
proofwiki-8658 | Inverse Hyperbolic Cotangent is Odd Function | :$\map {\coth^{-1} } {-x} = -\coth^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\coth^{-1} } {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \coth y
| c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\coth y
| c =
}}
{{eqn | ll= \leadsto... | :$\map {\coth^{-1} } {-x} = -\coth^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\coth^{-1} } {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \coth y
| c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\coth y
| c =
}}
{{eqn | ll= \leadsto... | Inverse Hyperbolic Cotangent is Odd Function/Proof 1 | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cotangent_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cotangent_is_Odd_Function/Proof_1 | [
"Inverse Hyperbolic Cotangent",
"Inverse Hyperbolic Cotangent is Odd Function",
"Examples of Odd Functions"
] | [] | [
"Hyperbolic Cotangent Function is Odd"
] |
proofwiki-8659 | Inverse Hyperbolic Cotangent is Odd Function | :$\map {\coth^{-1} } {-x} = -\coth^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\coth^{-1} } {-x}
| r = \frac 1 2 \map \ln {\frac {-z + 1} {-z - 1} }
| c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 2}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {z - 1} {z + 1} }
| c = multiplying the argument by $\dfrac {-1} {-1}$
}}
{{eqn | r = \frac 1 2 \paren {\... | :$\map {\coth^{-1} } {-x} = -\coth^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\coth^{-1} } {-x}
| r = \frac 1 2 \map \ln {\frac {-z + 1} {-z - 1} }
| c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 2}}
}}
{{eqn | r = \frac 1 2 \map \ln {\frac {z - 1} {z + 1} }
| c = multiplying the argument by $\dfrac {-1} {-1}$
}}
{{eqn | r = \frac 1 2 \paren {\... | Inverse Hyperbolic Cotangent is Odd Function/Proof 2 | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cotangent_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cotangent_is_Odd_Function/Proof_2 | [
"Inverse Hyperbolic Cotangent",
"Inverse Hyperbolic Cotangent is Odd Function",
"Examples of Odd Functions"
] | [] | [
"Difference of Logarithms",
"Difference of Logarithms"
] |
proofwiki-8660 | Inverse Hyperbolic Cosecant is Odd Function | Let $x \in \R$.
Then:
:$\map {\csch^{-1} } {-x} = -\csch^{-1} x$
where $\map {\csch^{-1} } {-x}$ denotes the inverse hyperbolic cosecant function. | {{begin-eqn}}
{{eqn | l = \map {\csch^{-1} } {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \csch y
| c = {{Defof|Inverse Hyperbolic Cosecant|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\csch y
| c =
}}
{{eqn | ll= \leadstoa... | Let $x \in \R$.
Then:
:$\map {\csch^{-1} } {-x} = -\csch^{-1} x$
where $\map {\csch^{-1} } {-x}$ denotes the [[Definition:Real Inverse Hyperbolic Cosecant|inverse hyperbolic cosecant function]]. | {{begin-eqn}}
{{eqn | l = \map {\csch^{-1} } {-x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -x
| r = \csch y
| c = {{Defof|Inverse Hyperbolic Cosecant|subdef = Real|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = -\csch y
| c =
}}
{{eqn | ll= \leadstoa... | Inverse Hyperbolic Cosecant is Odd Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cosecant_is_Odd_Function | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cosecant_is_Odd_Function | [
"Inverse Hyperbolic Cosecant",
"Examples of Odd Functions"
] | [
"Definition:Inverse Hyperbolic Cosecant/Real"
] | [
"Hyperbolic Cosecant Function is Odd"
] |
proofwiki-8661 | Hyperbolic Cosecant in terms of Cosecant | Let $z \in \C$ be a complex number.
Then:
:$i \csch z = -\csc \paren {i z}$
where:
: $\csc$ denotes the cosecant function
: $\csch$ denotes the hyperbolic cosecant
: $i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = i \csch z
| r = \frac i {\sinh z}
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-1} {i \sinh z}
| c = as $i^2 = -1$
}}
{{eqn | r = \frac {-1} {\sin \paren {i z} }
| c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = -\csc \paren {i z}
| c ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$i \csch z = -\csc \paren {i z}$
where:
: $\csc$ denotes the [[Definition:Complex Cosecant Function|cosecant function]]
: $\csch$ denotes the [[Definition:Hyperbolic Cosecant|hyperbolic cosecant]]
: $i$ is the [[Definition:Imaginary Unit|imagin... | {{begin-eqn}}
{{eqn | l = i \csch z
| r = \frac i {\sinh z}
| c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-1} {i \sinh z}
| c = as $i^2 = -1$
}}
{{eqn | r = \frac {-1} {\sin \paren {i z} }
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{eqn | r = -\csc \paren {i z}
... | Hyperbolic Cosecant in terms of Cosecant | https://proofwiki.org/wiki/Hyperbolic_Cosecant_in_terms_of_Cosecant | https://proofwiki.org/wiki/Hyperbolic_Cosecant_in_terms_of_Cosecant | [
"Cosecant Function",
"Hyperbolic Cosecant Function"
] | [
"Definition:Complex Number",
"Definition:Cosecant/Complex Function",
"Definition:Hyperbolic Cosecant",
"Definition:Complex Number/Imaginary Unit"
] | [
"Hyperbolic Sine in terms of Sine"
] |
proofwiki-8662 | Hyperbolic Secant in terms of Secant | Let $z \in \C$ be a complex number.
Then:
:$\sech z = \sec \paren {i z}$
where:
: $\sec$ denotes the secant function
: $\sech$ denotes the hyperbolic secant
: $i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \sec \paren {i z}
| r = \frac 1 {\cos \paren {i z} }
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\cosh z}
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \sech z
| c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\sech z = \sec \paren {i z}$
where:
: $\sec$ denotes the [[Definition:Complex Secant Function|secant function]]
: $\sech$ denotes the [[Definition:Hyperbolic Secant|hyperbolic secant]]
: $i$ is the [[Definition:Imaginary Unit|imaginary unit]]:... | {{begin-eqn}}
{{eqn | l = \sec \paren {i z}
| r = \frac 1 {\cos \paren {i z} }
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\cosh z}
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \sech z
| c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}} | Hyperbolic Secant in terms of Secant | https://proofwiki.org/wiki/Hyperbolic_Secant_in_terms_of_Secant | https://proofwiki.org/wiki/Hyperbolic_Secant_in_terms_of_Secant | [
"Secant Function",
"Hyperbolic Secant Function"
] | [
"Definition:Complex Number",
"Definition:Secant Function/Complex",
"Definition:Hyperbolic Secant",
"Definition:Complex Number/Imaginary Unit"
] | [
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8663 | Hyperbolic Cotangent in terms of Cotangent | Let $z \in \C$ be a complex number.
Then:
:$\coth z = -\cot \paren {i z}$
where:
: $\cot$ denotes the cotangent function
: $\coth$ denotes the hyperbolic cotangent
: $i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = i \coth z
| r = \frac {i \cosh z} {\sinh z}
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {-\cosh z} {i \sinh z}
| c = $i^2 = -1$
}}
{{eqn | r = \frac {-\cos \paren {i z} } {i \sinh z}
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \fra... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\coth z = -\cot \paren {i z}$
where:
: $\cot$ denotes the [[Definition:Complex Cotangent Function|cotangent function]]
: $\coth$ denotes the [[Definition:Hyperbolic Cotangent|hyperbolic cotangent]]
: $i$ is the [[Definition:Imaginary Unit|imag... | {{begin-eqn}}
{{eqn | l = i \coth z
| r = \frac {i \cosh z} {\sinh z}
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {-\cosh z} {i \sinh z}
| c = $i^2 = -1$
}}
{{eqn | r = \frac {-\cos \paren {i z} } {i \sinh z}
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r =... | Hyperbolic Cotangent in terms of Cotangent | https://proofwiki.org/wiki/Hyperbolic_Cotangent_in_terms_of_Cotangent | https://proofwiki.org/wiki/Hyperbolic_Cotangent_in_terms_of_Cotangent | [
"Cotangent Function",
"Hyperbolic Cotangent Function"
] | [
"Definition:Complex Number",
"Definition:Cotangent/Complex Function",
"Definition:Hyperbolic Cotangent",
"Definition:Complex Number/Imaginary Unit"
] | [
"Hyperbolic Cosine in terms of Cosine",
"Hyperbolic Sine in terms of Sine"
] |
proofwiki-8664 | Sine in terms of Hyperbolic Sine | Let $z \in \C$ be a complex number.
Then:
:$i \sin z = \map \sinh {i z}$
where:
:$\sin$ denotes the complex sine
:$\sinh$ denotes the hyperbolic sine
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \sinh {i z}
| r = \frac {e^{i z} - e^{-i z} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = i \frac {e^{i z} - e^{-i z} } {2 i}
| c = multiplying top and bottom by $i$
}}
{{eqn | r = i \sin z
| c = Euler's Sine Identity
}}
{{end-eqn}}
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$i \sin z = \map \sinh {i z}$
where:
:$\sin$ denotes the [[Definition:Complex Sine Function|complex sine]]
:$\sinh$ denotes the [[Definition:Hyperbolic Sine|hyperbolic sine]]
:$i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \sinh {i z}
| r = \frac {e^{i z} - e^{-i z} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = i \frac {e^{i z} - e^{-i z} } {2 i}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $i$
}}
{{eqn | r = i \sin z
| c = [[Euler's S... | Sine in terms of Hyperbolic Sine | https://proofwiki.org/wiki/Sine_in_terms_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Sine_in_terms_of_Hyperbolic_Sine | [
"Sine Function",
"Hyperbolic Sine Function"
] | [
"Definition:Complex Number",
"Definition:Sine/Complex Function",
"Definition:Hyperbolic Sine",
"Definition:Complex Number/Imaginary Unit"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Euler's Sine Identity"
] |
proofwiki-8665 | Cosine in terms of Hyperbolic Cosine | Let $z \in \C$ be a complex number.
Then:
:$\cos z = \map \cosh {i z}$
where:
:$\cos$ denotes the complex cosine
:$\cosh$ denotes the hyperbolic cosine
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \cosh {i z}
| r = \frac {e^{i z} + e^{-i z} } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \cos z
| c = Euler's Cosine Identity
}}
{{end-eqn}}
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\cos z = \map \cosh {i z}$
where:
:$\cos$ denotes the [[Definition:Complex Cosine Function|complex cosine]]
:$\cosh$ denotes the [[Definition:Hyperbolic Cosine|hyperbolic cosine]]
:$i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i^2 ... | {{begin-eqn}}
{{eqn | l = \map \cosh {i z}
| r = \frac {e^{i z} + e^{-i z} } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \cos z
| c = [[Euler's Cosine Identity]]
}}
{{end-eqn}}
{{qed}} | Cosine in terms of Hyperbolic Cosine | https://proofwiki.org/wiki/Cosine_in_terms_of_Hyperbolic_Cosine | https://proofwiki.org/wiki/Cosine_in_terms_of_Hyperbolic_Cosine | [
"Cosine Function",
"Hyperbolic Cosine Function"
] | [
"Definition:Complex Number",
"Definition:Cosine/Complex Function",
"Definition:Hyperbolic Cosine",
"Definition:Complex Number/Imaginary Unit"
] | [
"Euler's Cosine Identity"
] |
proofwiki-8666 | Tangent in terms of Hyperbolic Tangent | Let $z \in \C$ be a complex number.
Then:
:$i \tan z = \map \tanh {i z}$
where:
:$\tan$ denotes the tangent function
:$\tanh$ denotes the hyperbolic tangent
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \map \tanh {i z}
| r = \frac {\map \sinh {i z} } {\map \cosh {i z} }
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {i \sin z} {\map \cosh {i z} }
| c = Sine in terms of Hyperbolic Sine
}}
{{eqn | r = \frac {i \sin z} {\cos z}
| c = Cosine in terms of Hyperbol... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$i \tan z = \map \tanh {i z}$
where:
:$\tan$ denotes the [[Definition:Complex Tangent Function|tangent function]]
:$\tanh$ denotes the [[Definition:Hyperbolic Tangent|hyperbolic tangent]]
:$i$ is the [[Definition:Imaginary Unit|imaginary unit]]... | {{begin-eqn}}
{{eqn | l = \map \tanh {i z}
| r = \frac {\map \sinh {i z} } {\map \cosh {i z} }
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {i \sin z} {\map \cosh {i z} }
| c = [[Sine in terms of Hyperbolic Sine]]
}}
{{eqn | r = \frac {i \sin z} {\cos z}
| c = [[Cosine in terms of Hy... | Tangent in terms of Hyperbolic Tangent | https://proofwiki.org/wiki/Tangent_in_terms_of_Hyperbolic_Tangent | https://proofwiki.org/wiki/Tangent_in_terms_of_Hyperbolic_Tangent | [
"Tangent Function",
"Hyperbolic Tangent Function"
] | [
"Definition:Complex Number",
"Definition:Tangent Function/Complex",
"Definition:Hyperbolic Tangent",
"Definition:Complex Number/Imaginary Unit"
] | [
"Sine in terms of Hyperbolic Sine",
"Cosine in terms of Hyperbolic Cosine"
] |
proofwiki-8667 | Cosecant in terms of Hyperbolic Cosecant | Let $z \in \C$ be a complex number.
Then:
:$i \csc = -\csch \paren {i z}$
where:
: $\csc$ denotes the cosecant function
: $\csch$ denotes the hyperbolic cosecant
: $i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = i \csc x
| r = \frac i {\sin z}
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = \frac 1 {-i \sin z}
| c = $i^2 = -1$
}}
{{eqn | r = \frac 1 {-\sinh \paren {i z} }
| c = Sine in terms of Hyperbolic Sine
}}
{{eqn | r = -\csch \paren {i z}
| c = {{Defof|Hyp... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$i \csc = -\csch \paren {i z}$
where:
: $\csc$ denotes the [[Definition:Complex Cosecant Function|cosecant function]]
: $\csch$ denotes the [[Definition:Hyperbolic Cosecant|hyperbolic cosecant]]
: $i$ is the [[Definition:Imaginary Unit|imaginar... | {{begin-eqn}}
{{eqn | l = i \csc x
| r = \frac i {\sin z}
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = \frac 1 {-i \sin z}
| c = $i^2 = -1$
}}
{{eqn | r = \frac 1 {-\sinh \paren {i z} }
| c = [[Sine in terms of Hyperbolic Sine]]
}}
{{eqn | r = -\csch \paren {i z}
| c = {{Defo... | Cosecant in terms of Hyperbolic Cosecant | https://proofwiki.org/wiki/Cosecant_in_terms_of_Hyperbolic_Cosecant | https://proofwiki.org/wiki/Cosecant_in_terms_of_Hyperbolic_Cosecant | [
"Cosecant Function",
"Hyperbolic Cosecant Function"
] | [
"Definition:Complex Number",
"Definition:Cosecant/Complex Function",
"Definition:Hyperbolic Cosecant",
"Definition:Complex Number/Imaginary Unit"
] | [
"Sine in terms of Hyperbolic Sine"
] |
proofwiki-8668 | Secant in terms of Hyperbolic Secant | Let $z \in \C$ be a complex number.
Then:
:$\sec z = \map \sech {i z}$
where:
:$\sec$ denotes the secant function
:$\sech$ denotes the hyperbolic secant
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = \sec z
| r = \frac 1 {\cos z}
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\map \cosh {i z} }
| c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = \map \sech {i z}
| c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\sec z = \map \sech {i z}$
where:
:$\sec$ denotes the [[Definition:Complex Secant Function|secant function]]
:$\sech$ denotes the [[Definition:Hyperbolic Secant|hyperbolic secant]]
:$i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i^... | {{begin-eqn}}
{{eqn | l = \sec z
| r = \frac 1 {\cos z}
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\map \cosh {i z} }
| c = [[Cosine in terms of Hyperbolic Cosine]]
}}
{{eqn | r = \map \sech {i z}
| c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}} | Secant in terms of Hyperbolic Secant | https://proofwiki.org/wiki/Secant_in_terms_of_Hyperbolic_Secant | https://proofwiki.org/wiki/Secant_in_terms_of_Hyperbolic_Secant | [
"Secant Function",
"Hyperbolic Secant Function"
] | [
"Definition:Complex Number",
"Definition:Secant Function/Complex",
"Definition:Hyperbolic Secant",
"Definition:Complex Number/Imaginary Unit"
] | [
"Cosine in terms of Hyperbolic Cosine"
] |
proofwiki-8669 | Cotangent in terms of Hyperbolic Cotangent | Let $z \in \C$ be a complex number.
Then:
:$i \cot z = -\map \coth {i z}$
where:
:$\cot$ denotes the cotangent function
:$\coth$ denotes the hyperbolic cotangent
:$i$ is the imaginary unit: $i^2 = -1$. | {{begin-eqn}}
{{eqn | l = i \cot z
| r = i \frac {\cos z} {\sin z}
| c = {{Defof|Complex Cotangent Function}}
}}
{{eqn | r = -\frac {\cos z} {i \sin z}
| c = $i^2 = -1$
}}
{{eqn | r = -\frac {\map \cosh {i z} } {i \sin z}
| c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = -\frac {\map \c... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$i \cot z = -\map \coth {i z}$
where:
:$\cot$ denotes the [[Definition:Complex Cotangent Function|cotangent function]]
:$\coth$ denotes the [[Definition:Hyperbolic Cotangent|hyperbolic cotangent]]
:$i$ is the [[Definition:Imaginary Unit|imagina... | {{begin-eqn}}
{{eqn | l = i \cot z
| r = i \frac {\cos z} {\sin z}
| c = {{Defof|Complex Cotangent Function}}
}}
{{eqn | r = -\frac {\cos z} {i \sin z}
| c = $i^2 = -1$
}}
{{eqn | r = -\frac {\map \cosh {i z} } {i \sin z}
| c = [[Cosine in terms of Hyperbolic Cosine]]
}}
{{eqn | r = -\frac {\ma... | Cotangent in terms of Hyperbolic Cotangent | https://proofwiki.org/wiki/Cotangent_in_terms_of_Hyperbolic_Cotangent | https://proofwiki.org/wiki/Cotangent_in_terms_of_Hyperbolic_Cotangent | [
"Cotangent Function",
"Hyperbolic Cotangent Function"
] | [
"Definition:Complex Number",
"Definition:Cotangent/Complex Function",
"Definition:Hyperbolic Cotangent",
"Definition:Complex Number/Imaginary Unit"
] | [
"Cosine in terms of Hyperbolic Cosine",
"Sine in terms of Hyperbolic Sine"
] |
proofwiki-8670 | General Periodicity Property | Let $f: \R \to \R$ be a periodic real function.
Let $L$ be a periodic element of $f$.
Then:
:$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$
That is, after every distance $L$, the function $f$ repeats itself. | There are two cases to consider: either $n$ is not negative, or it is negative.
Since the Natural Numbers are Non-Negative Integers, the case where $n \ge 0$ will be proved using induction. | Let $f: \R \to \R$ be a [[Definition:Periodic Real Function|periodic real function]].
Let $L$ be a [[Definition:Periodic Element|periodic element]] of $f$.
Then:
:$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$
That is, after every distance $L$, the function $f$ repeats itself. | There are two cases to consider: either $n$ is [[Definition:Non-Negative Integer|not negative]], or it is [[Definition:Negative Integer|negative]].
Since the [[Natural Numbers are Non-Negative Integers]], the case where $n \ge 0$ will be proved using [[Principle of Mathematical Induction|induction]]. | General Periodicity Property | https://proofwiki.org/wiki/General_Periodicity_Property | https://proofwiki.org/wiki/General_Periodicity_Property | [
"Periodic Functions",
"Proofs by Induction"
] | [
"Definition:Periodic Function/Real",
"Definition:Periodic Function/Periodic Element"
] | [
"Definition:Positive/Integer",
"Definition:Negative/Integer",
"Natural Numbers are Non-Negative Integers",
"Principle of Mathematical Induction"
] |
proofwiki-8671 | Periodicity of Hyperbolic Sine | Let $k \in \Z$.
Then:
:$\map \sinh {x + 2 k \pi i} = \sinh x$ | {{begin-eqn}}
{{eqn | l = \map \sinh {x + 2 k \pi i}
| r = \frac {e^{x + 2 k \pi i} - e^{-\paren {x + 2 k \pi i} } } {2 i}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{i \paren {-i x + 2 k \pi} } - e^{i \paren {i x + 2 \paren {-k} \pi} } } {2 i}
| c = $i^2 = -1$ and simplifying
}}
{{eqn | ... | Let $k \in \Z$.
Then:
:$\map \sinh {x + 2 k \pi i} = \sinh x$ | {{begin-eqn}}
{{eqn | l = \map \sinh {x + 2 k \pi i}
| r = \frac {e^{x + 2 k \pi i} - e^{-\paren {x + 2 k \pi i} } } {2 i}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{i \paren {-i x + 2 k \pi} } - e^{i \paren {i x + 2 \paren {-k} \pi} } } {2 i}
| c = $i^2 = -1$ and simplifying
}}
{{eqn | ... | Periodicity of Hyperbolic Sine | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Sine | [
"Hyperbolic Sine Function"
] | [] | [
"Period of Complex Exponential Function"
] |
proofwiki-8672 | Periodicity of Hyperbolic Cosine | Let $k \in \Z$.
Then:
:$\map \cosh {x + 2 k \pi i} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map \cosh {x + 2 k \pi i}
| r = \frac {e^{x + 2 k \pi i} + e^{- \paren {x + 2 k \pi i} } } {2 i}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{i \paren {-i x + 2 k \pi} } + e^{i \paren {i x + 2 \paren {-k} \pi} } } {2 i}
| c = $i^2 = -1$ and simplifying
}}
{{eqn... | Let $k \in \Z$.
Then:
:$\map \cosh {x + 2 k \pi i} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map \cosh {x + 2 k \pi i}
| r = \frac {e^{x + 2 k \pi i} + e^{- \paren {x + 2 k \pi i} } } {2 i}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{i \paren {-i x + 2 k \pi} } + e^{i \paren {i x + 2 \paren {-k} \pi} } } {2 i}
| c = $i^2 = -1$ and simplifying
}}
{{eqn... | Periodicity of Hyperbolic Cosine | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Cosine | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Cosine | [
"Hyperbolic Cosine Function"
] | [] | [
"Period of Complex Exponential Function"
] |
proofwiki-8673 | Periodicity of Hyperbolic Tangent | Let $k \in \Z$.
Then:
:$\map \tanh {x + 2 k \pi i} = \tanh x$ | {{begin-eqn}}
{{eqn | l = \map \tanh {x + 2 k \pi i}
| r = \frac {\map \sinh {x + 2 k \pi i} } {\map \cosh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {\sinh x} {\map \cosh {x + 2 k \pi i} }
| c = Periodicity of Hyperbolic Sine
}}
{{eqn | r = \frac {\sinh x} {\cosh x}
... | Let $k \in \Z$.
Then:
:$\map \tanh {x + 2 k \pi i} = \tanh x$ | {{begin-eqn}}
{{eqn | l = \map \tanh {x + 2 k \pi i}
| r = \frac {\map \sinh {x + 2 k \pi i} } {\map \cosh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \frac {\sinh x} {\map \cosh {x + 2 k \pi i} }
| c = [[Periodicity of Hyperbolic Sine]]
}}
{{eqn | r = \frac {\sinh x} {\cosh x}... | Periodicity of Hyperbolic Tangent | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Tangent | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Tangent | [
"Hyperbolic Tangent Function"
] | [] | [
"Periodicity of Hyperbolic Sine",
"Periodicity of Hyperbolic Cosine"
] |
proofwiki-8674 | Periodicity of Hyperbolic Cosecant | Let $k \in \Z$.
Then:
:$\map \csch {x + 2 k \pi i} = \csch x$ | {{begin-eqn}}
{{eqn | l = \map \csch {x + 2 k \pi i}
| r = \frac 1 {\map \sinh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \frac 1 {\sinh x}
| c = Periodicity of Hyperbolic Sine
}}
{{eqn | r = \csch x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{end-eqn}}
{{qed}} | Let $k \in \Z$.
Then:
:$\map \csch {x + 2 k \pi i} = \csch x$ | {{begin-eqn}}
{{eqn | l = \map \csch {x + 2 k \pi i}
| r = \frac 1 {\map \sinh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \frac 1 {\sinh x}
| c = [[Periodicity of Hyperbolic Sine]]
}}
{{eqn | r = \csch x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{end-eqn}}
{{qed}} | Periodicity of Hyperbolic Cosecant | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Cosecant | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Cosecant | [
"Hyperbolic Cosecant Function"
] | [] | [
"Periodicity of Hyperbolic Sine"
] |
proofwiki-8675 | Periodicity of Hyperbolic Secant | Let $k \in \Z$.
Then:
:$\map \sech {x + 2 k \pi i} = \sech x$ | {{begin-eqn}}
{{eqn | l = \map \sech {x + 2 k \pi i}
| r = \frac 1 {\map \cosh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {\cosh x}
| c = Periodicity of Hyperbolic Cosine
}}
{{eqn | r = \sech
| c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}} | Let $k \in \Z$.
Then:
:$\map \sech {x + 2 k \pi i} = \sech x$ | {{begin-eqn}}
{{eqn | l = \map \sech {x + 2 k \pi i}
| r = \frac 1 {\map \cosh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 1 {\cosh x}
| c = [[Periodicity of Hyperbolic Cosine]]
}}
{{eqn | r = \sech
| c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}} | Periodicity of Hyperbolic Secant | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Secant | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Secant | [
"Hyperbolic Secant Function"
] | [] | [
"Periodicity of Hyperbolic Cosine"
] |
proofwiki-8676 | Periodicity of Hyperbolic Cotangent | Let $k \in \Z$.
Then:
:$\map \coth {x + 2 k \pi i} = \coth x$ | {{begin-eqn}}
{{eqn | l = \map \coth {x + 2 k \pi i}
| r = \frac {\map \cosh {x + 2 k \pi i} } {\map \sinh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\map \cosh {x + 2 k \pi i} } {\sinh x}
| c = Periodicity of Hyperbolic Sine
}}
{{eqn | r = \frac {\cosh x} {... | Let $k \in \Z$.
Then:
:$\map \coth {x + 2 k \pi i} = \coth x$ | {{begin-eqn}}
{{eqn | l = \map \coth {x + 2 k \pi i}
| r = \frac {\map \cosh {x + 2 k \pi i} } {\map \sinh {x + 2 k \pi i} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\map \cosh {x + 2 k \pi i} } {\sinh x}
| c = [[Periodicity of Hyperbolic Sine]]
}}
{{eqn | r = \frac {\cosh ... | Periodicity of Hyperbolic Cotangent | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Cotangent | https://proofwiki.org/wiki/Periodicity_of_Hyperbolic_Cotangent | [
"Hyperbolic Tangent Function"
] | [] | [
"Periodicity of Hyperbolic Sine",
"Periodicity of Hyperbolic Cosine"
] |
proofwiki-8677 | Inverse Sine of Imaginary Number | :$\map {\sin^{-1} } {i x} = i \sinh^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\sin^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \sin y
| r = i x
| c = {{Defof|Complex Inverse Sine}}
}}
{{eqn | ll= \leadsto
| l = i \sin y
| r = -x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map {\sin^{-1} } {i y... | :$\map {\sin^{-1} } {i x} = i \sinh^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\sin^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \sin y
| r = i x
| c = {{Defof|Complex Inverse Sine}}
}}
{{eqn | ll= \leadsto
| l = i \sin y
| r = -x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map {\sin^{-1} } {i y... | Inverse Sine of Imaginary Number | https://proofwiki.org/wiki/Inverse_Sine_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Sine_of_Imaginary_Number | [
"Inverse Sine",
"Inverse Hyperbolic Sine"
] | [] | [
"Sine in terms of Hyperbolic Sine",
"Inverse Hyperbolic Sine is Odd Function",
"Definition:Inverse Sine/Complex"
] |
proofwiki-8678 | Inverse Hyperbolic Sine of Imaginary Number | :$\map {\sinh^{-1} } {i x} = i \sin^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\sinh^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \sinh y
| r = i x
| c = {{Defof|Inverse Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
| l = i \sinh y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \sin {i y... | :$\map {\sinh^{-1} } {i x} = i \sin^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\sinh^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \sinh y
| r = i x
| c = {{Defof|Inverse Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
| l = i \sinh y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \sin {i y... | Inverse Hyperbolic Sine of Imaginary Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Sine_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Sine_of_Imaginary_Number | [
"Inverse Sine",
"Inverse Hyperbolic Sine"
] | [] | [
"Hyperbolic Sine in terms of Sine",
"Inverse Sine is Odd Function",
"Definition:Inverse Sine/Complex"
] |
proofwiki-8679 | Inverse Cosine of Imaginary Number | :$\cos^{-1} x = \pm \, i \cosh^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \cos^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \cos y
| r = x
| c = {{Defof|Inverse Cosine}}
}}
{{eqn | ll= \leadsto
| l = \map \cos {\pm \, y}
| r = x
| c = Cosine Function is Even
}}
{{eqn | ll= \leadsto
| l = \map \cosh {\pm \,... | :$\cos^{-1} x = \pm \, i \cosh^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \cos^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \cos y
| r = x
| c = {{Defof|Inverse Cosine}}
}}
{{eqn | ll= \leadsto
| l = \map \cos {\pm \, y}
| r = x
| c = [[Cosine Function is Even]]
}}
{{eqn | ll= \leadsto
| l = \map \cosh {\p... | Inverse Cosine of Imaginary Number | https://proofwiki.org/wiki/Inverse_Cosine_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Cosine_of_Imaginary_Number | [
"Inverse Cosine",
"Inverse Hyperbolic Cosine"
] | [] | [
"Cosine Function is Even",
"Cosine in terms of Hyperbolic Cosine"
] |
proofwiki-8680 | Inverse Hyperbolic Cosine of Imaginary Number | :$\cosh^{-1} x = \pm \, i \cos^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \cosh^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \cosh y
| r = x
| c = {{Defof|Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \map \cosh {\pm \, y}
| r = x
| c = Hyperbolic Cosine Function is Even
}}
{{eqn | ll= \leadsto
... | :$\cosh^{-1} x = \pm \, i \cos^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \cosh^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \cosh y
| r = x
| c = {{Defof|Inverse Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \map \cosh {\pm \, y}
| r = x
| c = [[Hyperbolic Cosine Function is Even]]
}}
{{eqn | ll= \leadsto
... | Inverse Hyperbolic Cosine of Imaginary Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cosine_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cosine_of_Imaginary_Number | [
"Inverse Cosine",
"Inverse Hyperbolic Cosine"
] | [] | [
"Hyperbolic Cosine Function is Even",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-8681 | Inverse Tangent of Imaginary Number | :$\map {\tan^{-1} } {i x} = i \tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\tan^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \tan y
| r = i x
| c = {{Defof|Inverse Tangent}}
}}
{{eqn | ll= \leadsto
| l = i \tan y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \tanh {i y}
| r... | :$\map {\tan^{-1} } {i x} = i \tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\tan^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \tan y
| r = i x
| c = {{Defof|Inverse Tangent}}
}}
{{eqn | ll= \leadsto
| l = i \tan y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \tanh {i y}
| r... | Inverse Tangent of Imaginary Number/Proof 1 | https://proofwiki.org/wiki/Inverse_Tangent_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Tangent_of_Imaginary_Number/Proof_1 | [
"Inverse Tangent",
"Inverse Hyperbolic Tangent",
"Inverse Tangent of Imaginary Number"
] | [] | [
"Tangent in terms of Hyperbolic Tangent",
"Inverse Hyperbolic Tangent is Odd Function"
] |
proofwiki-8682 | Inverse Tangent of Imaginary Number | :$\map {\tan^{-1} } {i x} = i \tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\tan^{-1} } {i x}
| r = \frac i 2 \map \ln {\frac {1 - i \paren {i x} } {1 + i \paren {i x} } }
| c = Arctangent Logarithmic Formulation
}}
{{eqn | r = \frac i 2 \map \ln {\frac {1 + x} {1 - x} }
}}
{{eqn | r = i \tanh^{-1} x
| c = {{Defof|Inverse Hyperbolic Tangent}}
}}
{{end-eqn}}
... | :$\map {\tan^{-1} } {i x} = i \tanh^{-1} x$ | {{begin-eqn}}
{{eqn | l = \map {\tan^{-1} } {i x}
| r = \frac i 2 \map \ln {\frac {1 - i \paren {i x} } {1 + i \paren {i x} } }
| c = [[Arctangent Logarithmic Formulation]]
}}
{{eqn | r = \frac i 2 \map \ln {\frac {1 + x} {1 - x} }
}}
{{eqn | r = i \tanh^{-1} x
| c = {{Defof|Inverse Hyperbolic Tangent}}
}}
{{end-eq... | Inverse Tangent of Imaginary Number/Proof 2 | https://proofwiki.org/wiki/Inverse_Tangent_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Tangent_of_Imaginary_Number/Proof_2 | [
"Inverse Tangent",
"Inverse Hyperbolic Tangent",
"Inverse Tangent of Imaginary Number"
] | [] | [
"Arctangent Logarithmic Formulation"
] |
proofwiki-8683 | Inverse Hyperbolic Tangent of Imaginary Number | :$\map {\tanh^{-1} } {i x} = i \tan^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\tanh^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \tanh y
| r = i x
| c = {{Defof|Inverse Hyperbolic Tangent}}
}}
{{eqn | ll= \leadsto
| l = i \tanh y
| r = -x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \tan {i ... | :$\map {\tanh^{-1} } {i x} = i \tan^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\tanh^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \tanh y
| r = i x
| c = {{Defof|Inverse Hyperbolic Tangent}}
}}
{{eqn | ll= \leadsto
| l = i \tanh y
| r = -x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \tan {i ... | Inverse Hyperbolic Tangent of Imaginary Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Tangent_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Tangent_of_Imaginary_Number | [
"Inverse Tangent",
"Inverse Hyperbolic Tangent"
] | [] | [
"Hyperbolic Tangent in terms of Tangent",
"Inverse Tangent is Odd Function"
] |
proofwiki-8684 | Inverse Cotangent of Imaginary Number | :$\map {\cot^{-1} } {i x} = - i \coth^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\cot^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \cot y
| r = i x
| c = {{Defof|Inverse Cotangent}}
}}
{{eqn | ll= \leadsto
| l = i \cot y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \coth {i y}
|... | :$\map {\cot^{-1} } {i x} = - i \coth^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\cot^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \cot y
| r = i x
| c = {{Defof|Inverse Cotangent}}
}}
{{eqn | ll= \leadsto
| l = i \cot y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \coth {i y}
|... | Inverse Cotangent of Imaginary Number | https://proofwiki.org/wiki/Inverse_Cotangent_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Cotangent_of_Imaginary_Number | [
"Inverse Cotangent",
"Inverse Hyperbolic Cotangent"
] | [] | [
"Cotangent in terms of Hyperbolic Cotangent"
] |
proofwiki-8685 | Inverse Hyperbolic Cotangent of Imaginary Number | :$\map {\coth^{-1} } {i x} = i \cot^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\coth^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \coth y
| r = i x
| c = {{Defof|Inverse Hyperbolic Cotangent}}
}}
{{eqn | ll= \leadsto
| l = i \coth y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \cot ... | :$\map {\coth^{-1} } {i x} = i \cot^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\coth^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \coth y
| r = i x
| c = {{Defof|Inverse Hyperbolic Cotangent}}
}}
{{eqn | ll= \leadsto
| l = i \coth y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \cot ... | Inverse Hyperbolic Cotangent of Imaginary Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cotangent_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cotangent_of_Imaginary_Number | [
"Inverse Cotangent",
"Inverse Hyperbolic Cotangent"
] | [] | [
"Hyperbolic Cotangent in terms of Cotangent"
] |
proofwiki-8686 | Inverse Secant of Imaginary Number | :$\sec^{-1} x = \pm i \sech^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \sec^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \sec y
| r = x
| c = {{Defof|Inverse Secant}}
}}
{{eqn | ll= \leadsto
| l = \map \sec {\pm \, y}
| r = x
| c = Secant Function is Even
}}
{{eqn | ll= \leadsto
| l = \map \sech {\pm \,... | :$\sec^{-1} x = \pm i \sech^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \sec^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \sec y
| r = x
| c = {{Defof|Inverse Secant}}
}}
{{eqn | ll= \leadsto
| l = \map \sec {\pm \, y}
| r = x
| c = [[Secant Function is Even]]
}}
{{eqn | ll= \leadsto
| l = \map \sech {\p... | Inverse Secant of Imaginary Number | https://proofwiki.org/wiki/Inverse_Secant_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Secant_of_Imaginary_Number | [
"Inverse Secant",
"Inverse Hyperbolic Secant"
] | [] | [
"Secant Function is Even",
"Secant in terms of Hyperbolic Secant"
] |
proofwiki-8687 | Inverse Hyperbolic Secant of Imaginary Number | :$\sech^{-1} x = \pm \, i \sec^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \sech^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \sech y
| r = x
| c = {{Defof|Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \map \sech {\pm \, y}
| r = x
| c = Hyperbolic Secant Function is Even
}}
{{eqn | ll= \leadsto
... | :$\sech^{-1} x = \pm \, i \sec^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \sech^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \sech y
| r = x
| c = {{Defof|Inverse Hyperbolic Secant}}
}}
{{eqn | ll= \leadsto
| l = \map \sech {\pm \, y}
| r = x
| c = [[Hyperbolic Secant Function is Even]]
}}
{{eqn | ll= \leadsto
... | Inverse Hyperbolic Secant of Imaginary Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Secant_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Secant_of_Imaginary_Number | [
"Inverse Secant",
"Inverse Hyperbolic Secant"
] | [] | [
"Hyperbolic Secant Function is Even",
"Hyperbolic Secant in terms of Secant"
] |
proofwiki-8688 | Inverse Cosecant of Imaginary Number | :$\map {\csc^{-1} } {i x} = i \csch^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\csc^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \csc y
| r = i x
| c = {{Defof|Inverse Cosecant}}
}}
{{eqn | ll= \leadsto
| l = i \csc y
| r = -x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \csch {i y}
| r... | :$\map {\csc^{-1} } {i x} = i \csch^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\csc^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \csc y
| r = i x
| c = {{Defof|Inverse Cosecant}}
}}
{{eqn | ll= \leadsto
| l = i \csc y
| r = -x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \csch {i y}
| r... | Inverse Cosecant of Imaginary Number | https://proofwiki.org/wiki/Inverse_Cosecant_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Cosecant_of_Imaginary_Number | [
"Inverse Cosecant",
"Inverse Hyperbolic Cosecant"
] | [] | [
"Cosecant in terms of Hyperbolic Cosecant",
"Inverse Hyperbolic Cosecant is Odd Function"
] |
proofwiki-8689 | Inverse Hyperbolic Cosecant of Imaginary Number | :$\map {\csch^{-1} } {i x} = -i \csc^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\csch^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \csch y
| r = i x
| c = {{Defof|Inverse Hyperbolic Cosecant}}
}}
{{eqn | ll= \leadsto
| l = i \csch y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \csc {... | :$\map {\csch^{-1} } {i x} = -i \csc^{-1} x$ | {{begin-eqn}}
{{eqn | l = y
| r = \map {\csch^{-1} } {i x}
| c =
}}
{{eqn | ll= \leadsto
| l = \csch y
| r = i x
| c = {{Defof|Inverse Hyperbolic Cosecant}}
}}
{{eqn | ll= \leadsto
| l = i \csch y
| r = - x
| c = $i^2 = -1$
}}
{{eqn | ll= \leadsto
| l = \map \csc {... | Inverse Hyperbolic Cosecant of Imaginary Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cosecant_of_Imaginary_Number | https://proofwiki.org/wiki/Inverse_Hyperbolic_Cosecant_of_Imaginary_Number | [
"Inverse Cosecant",
"Inverse Hyperbolic Cosecant"
] | [] | [
"Hyperbolic Cosecant in terms of Cosecant"
] |
proofwiki-8690 | Arccosecant of Reciprocal equals Arcsine | :$\map \arccsc {\dfrac 1 x} = \arcsin x$ | {{begin-eqn}}
{{eqn | l = \map \arccsc {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \csc y
| c = {{Defof|Real Arccosecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sin y
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | ll= \leadstoa... | :$\map \arccsc {\dfrac 1 x} = \arcsin x$ | {{begin-eqn}}
{{eqn | l = \map \arccsc {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \csc y
| c = {{Defof|Real Arccosecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \sin y
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | ll= \lead... | Arccosecant of Reciprocal equals Arcsine | https://proofwiki.org/wiki/Arccosecant_of_Reciprocal_equals_Arcsine | https://proofwiki.org/wiki/Arccosecant_of_Reciprocal_equals_Arcsine | [
"Arcsine Function",
"Arccosecant Function",
"Reciprocals"
] | [] | [
"Cosecant is Reciprocal of Sine",
"Category:Arcsine Function",
"Category:Arccosecant Function",
"Category:Reciprocals"
] |
proofwiki-8691 | Arcsecant of Reciprocal equals Arccosine | :$\map \arcsec {\dfrac 1 x} = \arccos x$ | {{begin-eqn}}
{{eqn | l = \map \arcsec {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \sec y
| c = {{Defof|Real Arcsecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cos y
| c = Secant is Reciprocal of Cosine
}}
{{eqn | ll= \leadstoand... | :$\map \arcsec {\dfrac 1 x} = \arccos x$ | {{begin-eqn}}
{{eqn | l = \map \arcsec {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \sec y
| c = {{Defof|Real Arcsecant}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cos y
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | ll= \leadst... | Arcsecant of Reciprocal equals Arccosine | https://proofwiki.org/wiki/Arcsecant_of_Reciprocal_equals_Arccosine | https://proofwiki.org/wiki/Arcsecant_of_Reciprocal_equals_Arccosine | [
"Arccosine Function",
"Arcsecant Function",
"Reciprocals"
] | [] | [
"Secant is Reciprocal of Cosine",
"Category:Arccosine Function",
"Category:Arcsecant Function",
"Category:Reciprocals"
] |
proofwiki-8692 | Arccotangent of Reciprocal equals Arctangent | :$\map \arccot {\dfrac 1 x} = \arctan x$ | {{begin-eqn}}
{{eqn | l = \map \arccot {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \cot y
| c = {{Defof|Real Arccotangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \tan y
| c = Cotangent is Reciprocal of Tangent
}}
{{eqn | ll= \lea... | :$\map \arccot {\dfrac 1 x} = \arctan x$ | {{begin-eqn}}
{{eqn | l = \map \arccot {\frac 1 x}
| r = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| r = \cot y
| c = {{Defof|Real Arccotangent}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \tan y
| c = [[Cotangent is Reciprocal of Tangent]]
}}
{{eqn | ll= ... | Arccotangent of Reciprocal equals Arctangent | https://proofwiki.org/wiki/Arccotangent_of_Reciprocal_equals_Arctangent | https://proofwiki.org/wiki/Arccotangent_of_Reciprocal_equals_Arctangent | [
"Arctangent Function",
"Arccotangent Function",
"Reciprocals"
] | [] | [
"Cotangent is Reciprocal of Tangent",
"Category:Arctangent Function",
"Category:Arccotangent Function",
"Category:Reciprocals"
] |
proofwiki-8693 | Solution to Quadratic Equation/Real Coefficients | Let $a, b, c \in \R$.
The quadratic equation $a x^2 + b x + c = 0$ has:
:Two real solutions if $b^2 - 4 a c > 0$
:One real solution if $b^2 - 4 a c = 0$
:Two complex solutions if $b^2 - 4 a c < 0$, and those two solutions are complex conjugates. | From Solution to Quadratic Equation:
:$x = \dfrac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}$
If the discriminant $b^2 - 4 a c > 0$ then $\sqrt {b^2 - 4 a c}$ has two values and the result follows.
If the discriminant $b^2 - 4 a c = 0$ then $\sqrt {b^2 - 4 a c} = 0$ and $x = \dfrac {-b} {2 a}$.
If the discriminant $b^2 - 4 a ... | Let $a, b, c \in \R$.
The [[Definition:Quadratic Equation|quadratic equation]] $a x^2 + b x + c = 0$ has:
:Two [[Definition:Real Number|real]] [[Definition:Solution to Equation|solutions]] if $b^2 - 4 a c > 0$
:One [[Definition:Real Number|real]] [[Definition:Solution to Equation|solution]] if $b^2 - 4 a c = 0$
:Two [... | From [[Solution to Quadratic Equation]]:
:$x = \dfrac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}$
If the [[Definition:Discriminant of Quadratic Equation|discriminant]] $b^2 - 4 a c > 0$ then $\sqrt {b^2 - 4 a c}$ has two values and the result follows.
If the [[Definition:Discriminant of Quadratic Equation|discriminant]] ... | Solution to Quadratic Equation/Real Coefficients | https://proofwiki.org/wiki/Solution_to_Quadratic_Equation/Real_Coefficients | https://proofwiki.org/wiki/Solution_to_Quadratic_Equation/Real_Coefficients | [
"Quadratic Equations"
] | [
"Definition:Quadratic Equation",
"Definition:Real Number",
"Definition:Fiber of Truth/Solution",
"Definition:Real Number",
"Definition:Fiber of Truth/Solution",
"Definition:Complex Number",
"Definition:Fiber of Truth/Solution",
"Definition:Fiber of Truth/Solution",
"Definition:Complex Conjugate"
] | [
"Solution to Quadratic Equation",
"Definition:Discriminant of Polynomial/Quadratic Equation",
"Definition:Discriminant of Polynomial/Quadratic Equation",
"Definition:Discriminant of Polynomial/Quadratic Equation",
"Definition:Fiber of Truth/Solution"
] |
proofwiki-8694 | Sum of Roots of Quadratic Equation | Let $P$ be the quadratic equation $a x^2 + b x + c = 0$.
Let $\alpha$ and $\beta$ be the roots of $P$.
Then:
:$\alpha + \beta = -\dfrac b a$ | {{begin-eqn}}
{{eqn | l = \alpha
| r = \frac {-b + \sqrt {b^2 - 4 a c} } {2 a}
| c = Solution to Quadratic Equation
}}
{{eqn | l = \beta
| r = \frac {-b - \sqrt {b^2 - 4 a c} } {2 a}
| c = {{WLOG}}, selecting $\alpha$ and $\beta$ as such
}}
{{eqn | ll= \leadsto
| l = \alpha + \beta
|... | Let $P$ be the [[Definition:Quadratic Equation|quadratic equation]] $a x^2 + b x + c = 0$.
Let $\alpha$ and $\beta$ be the [[Definition:Root of Polynomial|roots]] of $P$.
Then:
:$\alpha + \beta = -\dfrac b a$ | {{begin-eqn}}
{{eqn | l = \alpha
| r = \frac {-b + \sqrt {b^2 - 4 a c} } {2 a}
| c = [[Solution to Quadratic Equation]]
}}
{{eqn | l = \beta
| r = \frac {-b - \sqrt {b^2 - 4 a c} } {2 a}
| c = {{WLOG}}, selecting $\alpha$ and $\beta$ as such
}}
{{eqn | ll= \leadsto
| l = \alpha + \beta
... | Sum of Roots of Quadratic Equation | https://proofwiki.org/wiki/Sum_of_Roots_of_Quadratic_Equation | https://proofwiki.org/wiki/Sum_of_Roots_of_Quadratic_Equation | [
"Quadratic Equations"
] | [
"Definition:Quadratic Equation",
"Definition:Root of Polynomial"
] | [
"Solution to Quadratic Equation"
] |
proofwiki-8695 | Product of Roots of Quadratic Equation | Let $P$ be the quadratic equation $a x^2 + b x + c = 0$.
Let $\alpha$ and $\beta$ be the roots of $P$.
Then:
:$\alpha \beta = \dfrac c a$ | {{begin-eqn}}
{{eqn | l = \alpha
| r = \frac {-b + \sqrt {b^2 - 4 a c} } {2 a}
| c = Solution to Quadratic Equation
}}
{{eqn | l = \beta
| r = \frac {-b - \sqrt {b^2 - 4 a c} } {2 a}
| c = {{WLOG}}, selecting $\alpha$ and $\beta$ as such
}}
{{eqn | ll= \leadsto
| l = \alpha \beta
| r... | Let $P$ be the [[Definition:Quadratic Equation|quadratic equation]] $a x^2 + b x + c = 0$.
Let $\alpha$ and $\beta$ be the [[Definition:Root of Polynomial|roots]] of $P$.
Then:
:$\alpha \beta = \dfrac c a$ | {{begin-eqn}}
{{eqn | l = \alpha
| r = \frac {-b + \sqrt {b^2 - 4 a c} } {2 a}
| c = [[Solution to Quadratic Equation]]
}}
{{eqn | l = \beta
| r = \frac {-b - \sqrt {b^2 - 4 a c} } {2 a}
| c = {{WLOG}}, selecting $\alpha$ and $\beta$ as such
}}
{{eqn | ll= \leadsto
| l = \alpha \beta
... | Product of Roots of Quadratic Equation | https://proofwiki.org/wiki/Product_of_Roots_of_Quadratic_Equation | https://proofwiki.org/wiki/Product_of_Roots_of_Quadratic_Equation | [
"Quadratic Equations"
] | [
"Definition:Quadratic Equation",
"Definition:Root of Polynomial"
] | [
"Solution to Quadratic Equation",
"Difference of Two Squares"
] |
proofwiki-8696 | Cardano's Formula/Real Coefficients | :$(1): \quad$ If $D > 0$, then one root is real and two are complex conjugates.
:$(2): \quad$ If $D = 0$, then all roots are real, and at least two are equal.
:$(3): \quad$ If $D < 0$, then all roots are real and unequal. | From Cardano's Formula, the roots of $P$ are:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1
| r = S + T - \dfrac b {3 a}
}}
{{eqn | n = 2
| l = x_2
| r = -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}
}}
{{eqn | n = 3
| l = x_3
| r = -\dfrac {S + T} 2 - \dfrac b {3 ... | :$(1): \quad$ If $D > 0$, then one [[Definition:Root of Polynomial|root]] is [[Definition:Real Number|real]] and two are [[Definition:Complex Conjugate|complex conjugates]].
:$(2): \quad$ If $D = 0$, then all [[Definition:Root of Polynomial|roots]] are [[Definition:Real Number|real]], and at least two are equal.
:$(3):... | From [[Cardano's Formula]], the [[Definition:Root of Polynomial|roots]] of $P$ are:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1
| r = S + T - \dfrac b {3 a}
}}
{{eqn | n = 2
| l = x_2
| r = -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}
}}
{{eqn | n = 3
| l = x_3
... | Cardano's Formula/Real Coefficients | https://proofwiki.org/wiki/Cardano's_Formula/Real_Coefficients | https://proofwiki.org/wiki/Cardano's_Formula/Real_Coefficients | [
"Cardano's Formula"
] | [
"Definition:Root of Polynomial",
"Definition:Real Number",
"Definition:Complex Conjugate",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Definition:Root of Polynomial",
"Definition:Real Number"
] | [
"Cardano's Formula",
"Definition:Root of Polynomial",
"Definition:Root of Polynomial"
] |
proofwiki-8697 | Cardano's Formula/Trigonometric Form | Let $a, b, c, d \in \R$.
Let the discriminant $D < 0$, where $D := Q^3 + R^2$.
Then the solutions of $P$ can be expressed as:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1
| r = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3} - \dfrac b {3 a}
}}
{{eqn | n = 2
| l = x_2
| r = 2 \sqrt {-Q} \map \cos {\dfrac \the... | From Cardano's Formula, the roots of $P$ are:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1
| r = S + T - \dfrac b {3 a}
}}
{{eqn | n = 2
| l = x_2
| r = -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}
}}
{{eqn | n = 3
| l = x_3
| r = -\dfrac {S + T} 2 - \dfrac b {3 ... | Let $a, b, c, d \in \R$.
Let the [[Definition:Discriminant of Cubic Equation|discriminant]] $D < 0$, where $D := Q^3 + R^2$.
Then the solutions of $P$ can be expressed as:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1
| r = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3} - \dfrac b {3 a}
}}
{{eqn | n = 2
| l = x... | From [[Cardano's Formula]], the [[Definition:Root of Polynomial|roots]] of $P$ are:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1
| r = S + T - \dfrac b {3 a}
}}
{{eqn | n = 2
| l = x_2
| r = -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}
}}
{{eqn | n = 3
| l = x_3
... | Cardano's Formula/Trigonometric Form | https://proofwiki.org/wiki/Cardano's_Formula/Trigonometric_Form | https://proofwiki.org/wiki/Cardano's_Formula/Trigonometric_Form | [
"Cardano's Formula"
] | [
"Definition:Discriminant of Polynomial/Cubic Equation"
] | [
"Cardano's Formula",
"Definition:Root of Polynomial",
"Definition:Complex Number/Polar Form"
] |
proofwiki-8698 | Ring is Module over Itself | Let $\struct {R, +, \circ}$ be a ring.
Then $\struct {R, +, \circ}_R$ is an $R$-module. | Note that:
$\struct {R, +, \circ}$ is a ring by assumption.
$\struct {R, +}$ is an abelian group by the definition of a ring.
Let us verify the module axioms:
{{begin-axiom}}
{{axiom | n = 1
| q = \forall x, y, z \in R
| m = x \circ \paren {y + z} = \paren {x \circ y} + \paren {x \circ z}
}}
{{axiom | n... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Then $\struct {R, +, \circ}_R$ is an [[Definition:Module over Ring|$R$-module]]. | Note that:
$\struct {R, +, \circ}$ is a [[Definition:Ring (Abstract Algebra)|ring]] by assumption.
$\struct {R, +}$ is an [[Definition:Abelian Group|abelian group]] by the definition of a [[Definition:Ring (Abstract Algebra)|ring]].
Let us verify the [[Axiom:Module Axioms|module axioms]]:
{{begin-axiom}}
{{axiom |... | Ring is Module over Itself/Proof 1 | https://proofwiki.org/wiki/Ring_is_Module_over_Itself | https://proofwiki.org/wiki/Ring_is_Module_over_Itself/Proof_1 | [
"Module Theory",
"Ring is Module over Itself"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Module over Ring"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Definition:Ring (Abstract Algebra)",
"Axiom:Left Module Axioms",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-8699 | Ring is Module over Itself | Let $\struct {R, +, \circ}$ be a ring.
Then $\struct {R, +, \circ}_R$ is an $R$-module. | This is a special case of Module on Cartesian Product is Module:
:$\struct {R^n, +, \circ}_R$ is an $R$-module
where $n = 1$.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Then $\struct {R, +, \circ}_R$ is an [[Definition:Module over Ring|$R$-module]]. | This is a special case of [[Module on Cartesian Product is Module]]:
:$\struct {R^n, +, \circ}_R$ is an [[Definition:Module over Ring|$R$-module]]
where $n = 1$.
{{qed}} | Ring is Module over Itself/Proof 2 | https://proofwiki.org/wiki/Ring_is_Module_over_Itself | https://proofwiki.org/wiki/Ring_is_Module_over_Itself/Proof_2 | [
"Module Theory",
"Ring is Module over Itself"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Module over Ring"
] | [
"Module on Cartesian Product is Module",
"Definition:Module over Ring"
] |
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