hcju/proofwikips huggingface dataset:
Particular Values of Tangent Function
The following values of the tangent function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Tangent of Zero === {{:Tangent of Zero}} === Tangent of 15 Degrees === {{:Tangent of 15 Degrees}} === Tangent of 22.5 Degrees === {{:Tangent of 22.5 Degrees}} === Tangent of 30 Degrees === {{:Tangent of 30 Degrees}} === Tangent of 45 Degrees === {{:Tangent of 45 Degrees}} === Tangent of 60 Degrees === {{:Tangent of 60 Degrees}} === Tangent of 67.5 Degrees === {{:Tangent of 67.5 Degrees}} === Tangent of 75 Degrees === {{:Tangent of 75 Degrees}} === Tangent of Right Angle === {{:Tangent of Right Angle}} === Tangent of 105 Degrees === {{:Tangent of 105 Degrees}} === Tangent of 120 Degrees === {{:Tangent of 120 Degrees}} === Tangent of 135 Degrees === {{:Tangent of 135 Degrees}} === Tangent of 150 Degrees === {{:Tangent of 150 Degrees}} === Tangent of 165 Degrees === {{:Tangent of 165 Degrees}} === Tangent of Straight Angle === {{:Tangent of Straight Angle}} === Tangent of 195 Degrees === {{:Tangent of 195 Degrees}} === Tangent of 210 Degrees === {{:Tangent of 210 Degrees}} === Tangent of 225 Degrees === {{:Tangent of 225 Degrees}} === Tangent of 240 Degrees === {{:Tangent of 240 Degrees}} === Tangent of 255 Degrees === {{:Tangent of 255 Degrees}} === Tangent of Three Right Angles === {{:Tangent of Three Right Angles}} === Tangent of 285 Degrees === {{:Tangent of 285 Degrees}} === Tangent of 300 Degrees === {{:Tangent of 300 Degrees}} === Tangent of 315 Degrees === {{:Tangent of 315 Degrees}} === Tangent of 330 Degrees === {{:Tangent of 330 Degrees}} === Tangent of 345 Degrees === {{:Tangent of 345 Degrees}} === Tangent of Full Angle === {{:Tangent of Full Angle}}
Tangent of Zero
:$\tan 0 = 0$
Tangent of 15 Degrees
:$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$
Tangent of 30 Degrees
:$\tan 30^\circ = \tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$
Particular Values of Cotangent Function
The following values of the cotangent function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cotangent of Zero === {{:Cotangent of Zero}} === Cotangent of 15 Degrees === {{:Cotangent of 15 Degrees}} === Cotangent of 30 Degrees === {{:Cotangent of 30 Degrees}} === Cotangent of 45 Degrees === {{:Cotangent of 45 Degrees}} === Cotangent of 60 Degrees === {{:Cotangent of 60 Degrees}} === Cotangent of 75 Degrees === {{:Cotangent of 75 Degrees}} === Cotangent of Right Angle === {{:Cotangent of Right Angle}} === Cotangent of 105 Degrees === {{:Cotangent of 105 Degrees}} === Cotangent of 120 Degrees === {{:Cotangent of 120 Degrees}} === Cotangent of 135 Degrees === {{:Cotangent of 135 Degrees}} === Cotangent of 150 Degrees === {{:Cotangent of 150 Degrees}} === Cotangent of 165 Degrees === {{:Cotangent of 165 Degrees}} === Cotangent of Straight Angle === {{:Cotangent of Straight Angle}} === Cotangent of 195 Degrees === {{:Cotangent of 195 Degrees}} === Cotangent of 210 Degrees === {{:Cotangent of 210 Degrees}} === Cotangent of 225 Degrees === {{:Cotangent of 225 Degrees}} === Cotangent of 240 Degrees === {{:Cotangent of 240 Degrees}} === Cotangent of 255 Degrees === {{:Cotangent of 255 Degrees}} === Cotangent of Three Right Angles === {{:Cotangent of Three Right Angles}} === Cotangent of 285 Degrees === {{:Cotangent of 285 Degrees}} === Cotangent of 300 Degrees === {{:Cotangent of 300 Degrees}} === Cotangent of 315 Degrees === {{:Cotangent of 315 Degrees}} === Cotangent of 330 Degrees === {{:Cotangent of 330 Degrees}} === Cotangent of 345 Degrees === {{:Cotangent of 345 Degrees}} === Cotangent of Full Angle === {{:Cotangent of Full Angle}}
Particular Values of Secant Function
The following values of the secant function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Secant of Zero === {{:Secant of Zero}} === Secant of $15 \degrees$ === {{:Secant of 15 Degrees}} === Secant of $30 \degrees$ === {{:Secant of 30 Degrees}} === Secant of $45 \degrees$ === {{:Secant of 45 Degrees}} === Secant of $60 \degrees$ === {{:Secant of 60 Degrees}} === Secant of $75 \degrees$ === {{:Secant of 75 Degrees}} === Secant of Right Angle === {{:Secant of Right Angle}} === Secant of $105 \degrees$ === {{:Secant of 105 Degrees}} === Secant of $120 \degrees$ === {{:Secant of 120 Degrees}} === Secant of $135 \degrees$ === {{:Secant of 135 Degrees}} === Secant of $150 \degrees$ === {{:Secant of 150 Degrees}} === Secant of $165 \degrees$ === {{:Secant of 165 Degrees}} === Secant of Straight Angle === {{:Secant of Straight Angle}} === Secant of $195 \degrees$ === {{:Secant of 195 Degrees}} === Secant of $210 \degrees$ === {{:Secant of 210 Degrees}} === Secant of $225 \degrees$ === {{:Secant of 225 Degrees}} === Secant of $240 \degrees$ === {{:Secant of 240 Degrees}} === Secant of $255 \degrees$ === {{:Secant of 255 Degrees}} === Secant of Three Right Angles === {{:Secant of Three Right Angles}} === Secant of $285 \degrees$ === {{:Secant of 285 Degrees}} === Secant of $300 \degrees$ === {{:Secant of 300 Degrees}} === Secant of $315 \degrees$ === {{:Secant of 315 Degrees}} === Secant of $330 \degrees$ === {{:Secant of 330 Degrees}} === Secant of $345 \degrees$ === {{:Secant of 345 Degrees}} === Secant of Full Angle === {{:Secant of Full Angle}}
Particular Values of Cosecant Function
The following values of the cosecant function can be expressed as exact algebraic numbers. This list is non-exhaustive.   === Cosecant of Zero === {{:Cosecant of Zero}} === Cosecant of 15 Degrees === {{:Cosecant of 15 Degrees}} === Cosecant of 30 Degrees === {{:Cosecant of 30 Degrees}} === Cosecant of 45 Degrees === {{:Cosecant of 45 Degrees}} === Cosecant of 60 Degrees === {{:Cosecant of 60 Degrees}} === Cosecant of 75 Degrees === {{:Cosecant of 75 Degrees}} === Cosecant of Right Angle === {{:Cosecant of Right Angle}} === Cosecant of 105 Degrees === {{:Cosecant of 105 Degrees}} === Cosecant of 120 Degrees === {{:Cosecant of 120 Degrees}} === Cosecant of 135 Degrees === {{:Cosecant of 135 Degrees}} === Cosecant of 150 Degrees === {{:Cosecant of 150 Degrees}} === Cosecant of 165 Degrees === {{:Cosecant of 165 Degrees}} === Cosecant of Straight Angle === {{:Cosecant of Straight Angle}} === Cosecant of 195 Degrees === {{:Cosecant of 195 Degrees}} === Cosecant of 210 Degrees === {{:Cosecant of 210 Degrees}} === Cosecant of 225 Degrees === {{:Cosecant of 225 Degrees}} === Cosecant of 240 Degrees === {{:Cosecant of 240 Degrees}} === Cosecant of 255 Degrees === {{:Cosecant of 255 Degrees}} === Cosecant of Three Right Angles === {{:Cosecant of Three Right Angles}} === Cosecant of 285 Degrees === {{:Cosecant of 285 Degrees}} === Cosecant of 300 Degrees === {{:Cosecant of 300 Degrees}} === Cosecant of 315 Degrees === {{:Cosecant of 315 Degrees}} === Cosecant of 330 Degrees === {{:Cosecant of 330 Degrees}} === Cosecant of 345 Degrees === {{:Cosecant of 345 Degrees}} === Cosecant of Full Angle === {{:Cosecant of Full Angle}}
Tangent of 45 Degrees
:$\tan 45 \degrees = \tan \dfrac \pi 4 = 1$
Tangent of 60 Degrees
:$\tan 60^\circ = \tan \dfrac \pi 3 = \sqrt 3$
Tangent of 75 Degrees
:$\tan 75^\circ = \tan \dfrac {5 \pi} {12} = 2 + \sqrt 3$
Tangent of Complement equals Cotangent
:$\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ for $\theta \ne n \pi$ where $\tan$ and $\cot$ are tangent and cotangent respectively. That is, the cotangent of an angle is the tangent of its complement. This relation is defined wherever $\sin \theta \ne 0$.
Cotangent of Complement equals Tangent
:$\cot \left({\dfrac \pi 2 - \theta}\right) = \tan \theta$ for $\theta \ne \left({2 n + 1}\right) \dfrac \pi 2$ where $\cot$ and $\tan$ are cotangent and tangent respectively. That is, the tangent of an angle is the cotangent of its complement. This relation is defined wherever $\cos \theta \ne 0$.
Tangent of Angle plus Right Angle
: $\tan \left({x + \dfrac \pi 2}\right) = -\cot x$
Cotangent of Angle plus Right Angle
: $\cot \left({x + \dfrac \pi 2}\right) = -\tan x$
Tangent of Supplementary Angle
:$\map \tan {\pi - \theta} = -\tan \theta$ where $\tan$ denotes tangent. That is, the tangent of an angle is the negative of its supplement.
Cotangent of Supplementary Angle
:$\map \cot {\pi - \theta} = -\cot \theta$ where $\cot$ denotes tangent. That is, the cotangent of an angle is the negative of its supplement.
Tangent of Conjugate Angle
:$\map \tan {2 \pi - \theta} = -\tan \theta$ where $\tan$ denotes tangent. That is, the tangent of an angle is the negative of its conjugate.
Cotangent of Conjugate Angle
:$\map \cot {2 \pi - \theta} = -\cot \theta$ where $\cot$ denotes cotangent. That is, the cotangent of an angle is the negative of its conjugate.
Tangent of Right Angle
:$\tan 90^\circ = \tan \dfrac \pi 2$ is undefined
Tangent of 105 Degrees
:$\tan 105^\circ = \tan \dfrac {7 \pi} {12} = - \left({2 + \sqrt 3}\right)$
Tangent of 120 Degrees
:$\tan 120 \degrees = \tan \dfrac {2 \pi} 3 = -\sqrt 3$
Tangent of 135 Degrees
:$\tan 135^\circ = \tan \dfrac {3 \pi} 4 = - 1$
Tangent of 150 Degrees
:$\tan 150^\circ = \tan \dfrac {5 \pi} 6 = - \dfrac {\sqrt 3} 3$
Tangent of 165 Degrees
:$\tan 165^\circ = \tan \dfrac {11 \pi} {12} = - \left({2 - \sqrt 3}\right)$
Tangent of Straight Angle
:$\tan 180^\circ = \tan \pi = 0$
Tangent of 195 Degrees
:$\tan 195^\circ = \tan \dfrac {13 \pi} {12} = 2 - \sqrt 3$
Tangent of 210 Degrees
:$\tan 210 \degrees = \tan \dfrac {7 \pi} 6 = \dfrac {\sqrt 3} 3$
Tangent of 225 Degrees
:$\tan 225 \degrees = \tan \dfrac {5 \pi} 4 = 1$
Tangent of 240 Degrees
:$\tan 240 \degrees = \tan \dfrac {4 \pi} 3 = \sqrt 3$
Tangent of 255 Degrees
:$\tan 255^\circ = \tan \dfrac {17 \pi} {12} = 2 + \sqrt 3$
Tangent of Three Right Angles
:$\tan 270^\circ = \tan \dfrac {3 \pi} 2$ is undefined
Tangent of 285 Degrees
:$\tan 285^\circ = \tan \dfrac {19 \pi} {12} = -\left({2 + \sqrt 3}\right)$
Tangent of 300 Degrees
:$\tan 300^\circ = \tan \dfrac {5 \pi} 3 = -\sqrt 3$
Tangent of 315 Degrees
:$\tan 315^\circ = \tan \dfrac {7 \pi} 4 = -1$
Tangent of 330 Degrees
:$\tan 330^\circ = \tan \dfrac {11 \pi} 6 = -\dfrac {\sqrt 3} 3$
Tangent of 345 Degrees
:$\tan 345^\circ = \tan \dfrac {23 \pi} {12} = -\left({2 - \sqrt 3}\right)$
Tangent of Full Angle
:$\tan 360^\circ = \tan 2 \pi = 0$
Cotangent of Zero
:$\cot 0$ is undefined
Cotangent of 30 Degrees
:$\cot 30^\circ = \cot \dfrac {\pi} 6 = \sqrt 3$
Cotangent of 15 Degrees
:$\cot 15^\circ = \cot \dfrac {\pi} {12} = 2 + \sqrt 3$
Cotangent of 45 Degrees
:$\cot 45 \degrees = \cot \dfrac \pi 4 = 1$
Cotangent of 60 Degrees
:$\cot 60^\circ = \cot \dfrac {\pi} 3 = \dfrac {\sqrt 3} 3$
Cotangent of 75 Degrees
:$\cot 75 \degrees = \cot \dfrac {5 \pi} {12} = 2 - \sqrt 3$
Cotangent of Right Angle
:$\cot 90^\circ = \cot \dfrac \pi 2 = 0$
Cotangent of 105 Degrees
:$\cot 105^\circ = \cot \dfrac {7 \pi} {12} = -\left({2 - \sqrt 3}\right)$
Cotangent of 120 Degrees
:$\cot 120 \degrees = \cot \dfrac {2 \pi} 3 = -\dfrac {\sqrt 3} 3$
Cotangent of 135 Degrees
:$\cot 135 \degrees = \cot \dfrac {3 \pi} 4 = -1$
Cotangent of 150 Degrees
:$\cot 150 \degrees = \cot \dfrac {5 \pi} 6 = -\sqrt 3$
Cotangent of 165 Degrees
:$\cot 165 \degrees = \cot \dfrac {11 \pi} {12} = -\paren {2 + \sqrt 3}$
Cotangent of Straight Angle
:$\cot 180^\circ = \cot \pi$ is undefined
Cotangent of 195 Degrees
:$\cot 195 \degrees = \cot \dfrac {13 \pi} {12} = 2 + \sqrt 3$
Cotangent of 210 Degrees
:$\cot 210^\circ = \cot \dfrac {7 \pi} 6 = \sqrt 3$
Cotangent of 225 Degrees
:$\cot 225^\circ = \cot \dfrac {5 \pi} 4 = 1$
Cotangent of 240 Degrees
:$\cot 240^\circ = \cot \dfrac {4 \pi} 3 = \dfrac {\sqrt 3} 3$
Cotangent of 255 Degrees
:$\cot 255 \degrees = \cot \dfrac {17 \pi} {12} = 2 - \sqrt 3$
Cotangent of Three Right Angles
:$\cot 270^\circ = \cot \dfrac {3 \pi} 2 = 0$
Cotangent of 285 Degrees
:$\cot 285 \degrees = \cot \dfrac {19 \pi} {12} = -\paren {2 - \sqrt 3}$
Cotangent of 300 Degrees
:$\cot 300 \degrees = \cot \dfrac {5 \pi} 3 = - \dfrac {\sqrt 3} 3$
Cotangent of 315 Degrees
:$\cot 315^\circ = \cot \dfrac {7 \pi} 4 = -1$
Cotangent of 330 Degrees
:$\cot 330^\circ = \cot \dfrac {11 \pi} 6 = -\sqrt 3$
Cotangent of 345 Degrees
:$\cot 345 \degrees = \cot \dfrac {23 \pi} {12} = -\paren {2 + \sqrt 3}$
Cotangent of Full Angle
:$\cot 360^\circ = \cot 2 \pi$ is undefined
Secant of Zero
:$\sec 0 = 0$
Secant of 15 Degrees
:$\sec 15 \degrees = \sec \dfrac \pi {12} = \sqrt 6 - \sqrt 2$
Secant of 30 Degrees
:$\sec 30 \degrees = \sec \dfrac \pi 6 = \dfrac {2 \sqrt 3} 3$
Secant of 45 Degrees
:$\sec 45 \degrees = \sec \dfrac \pi 4 = \sqrt 2$
Secant of 60 Degrees
:$\sec 60 \degrees = \sec \dfrac \pi 3 = 2$
Secant of 75 Degrees
:$\sec 75 \degrees = \sec \dfrac {5 \pi} {12} = \sqrt 6 + \sqrt 2$
Secant of Right Angle
:$\sec 90 \degrees = \sec \dfrac \pi 2$ is undefined
Secant of Angle plus Right Angle
: $\sec \left({x + \dfrac \pi 2}\right) = -\csc x$
Cosecant of Angle plus Right Angle
: $\csc \left({x + \dfrac \pi 2}\right) = \sec x$
Cosecant of Supplementary Angle
:$\csc \left({\pi - \theta}\right) = \csc \theta$ where $\csc$ denotes cosecant. That is, the cosecant of an angle equals its supplement.
Secant of Conjugate Angle
:$\map \sec {2 \pi - \theta} = \sec \theta$ where $\sec$ denotes secant. That is, the secant of an angle equals its conjugate.
Cosecant of Conjugate Angle
:$\map \csc {2 \pi - \theta} = -\csc \theta$ where $\csc$ denotes cosecant. That is, the cosecant of an angle is the negative of its conjugate.
Secant of 105 Degrees
:$\sec 105 \degrees = \sec \dfrac {7 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$
Secant of 120 Degrees
:$\sec 120 \degrees = \sec \dfrac {2 \pi} 3 = -2$
Secant of 135 Degrees
:$\sec 135 \degrees = \sec \dfrac {3 \pi} 4 = -\sqrt 2$
Secant of 150 Degrees
:$\sec 150 \degrees = \sec \dfrac {5 \pi} 6 = -\dfrac {2 \sqrt 3} 3$
Secant of 165 Degrees
:$\sec 165 \degrees = \sec \dfrac {11 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$
Secant of Straight Angle
:$\sec 180 \degrees = \sec \pi = -1$
Secant of 195 Degrees
:$\sec 195 \degrees = \sec \dfrac {13 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$
Secant of 210 Degrees
:$\sec 210 \degrees = \sec \dfrac {7 \pi} 6 = -2 \dfrac {\sqrt 3} 3$
Secant of 225 Degrees
:$\sec 225 \degrees = \sec \dfrac {5 \pi} 4 = -\sqrt 2$
Secant of 240 Degrees
:$\sec 240 \degrees = \sec \dfrac {4 \pi} 3 = - 2$
Secant of 255 Degrees
:$\sec 255 \degrees = \sec \dfrac {17 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$
Secant of Three Right Angles
:$\sec 270 \degrees = \sec \dfrac {3 \pi} 2$ is undefined
Secant of 285 Degrees
:$\sec 285 \degrees = \sec \dfrac {19 \pi} {12} = \sqrt 6 + \sqrt 2$
Secant of 300 Degrees
:$\sec 300 \degrees = \sec \dfrac {5 \pi} 3 = 2$
Secant of 315 Degrees
:$\sec 315 \degrees = \sec \dfrac {7 \pi} 4 = \sqrt 2$
Secant of 330 Degrees
:$\sec 330 \degrees = \sec \dfrac {11 \pi} 6 = 2 \dfrac {\sqrt 3} 3$
Secant of 345 Degrees
:$\sec 345 \degrees = \sec \dfrac {23 \pi} {12} = \sqrt 6 - \sqrt 2$
Secant of Full Angle
:$\sec 360 \degrees = \sec 2 \pi = 1$
Cosecant of Zero
:$\csc 0$ is undefined
Cosecant of 15 Degrees
:$\csc 15^\circ = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$
Cosecant of 30 Degrees
:$\csc 30^\circ = \csc \dfrac \pi 6 = 2$
Cosecant of 45 Degrees
:$\csc 45^\circ = \csc \dfrac \pi 4 = \sqrt 2$
Cosecant of 60 Degrees
:$\csc 60^\circ = \csc \dfrac \pi 3 = \dfrac {2 \sqrt 3} 3$
Cosecant of 75 Degrees
:$\csc 75 \degrees = \csc \dfrac {5 \pi} {12} = \sqrt 6 - \sqrt 2$
Cosecant of Right Angle
:$\csc 90^\circ = \csc \dfrac \pi 2 = 1$
Cosecant of 105 Degrees
:$\csc 105 \degrees = \csc \dfrac {7 \pi} {12} = \sqrt 6 - \sqrt 2$
Cosecant of 120 Degrees
:$\csc 120 \degrees = \csc \dfrac {2 \pi} 3 = \dfrac {2 \sqrt 3} 3$
Cosecant of 135 Degrees
:$\csc 135 \degrees = \csc \dfrac {3 \pi} 4 = \sqrt 2$
Cosecant of 150 Degrees
:$\csc 150 \degrees = \csc \dfrac {5 \pi} 6 = \sqrt 2$
Cosecant of 165 Degrees
:$\csc 165 \degrees = \csc \dfrac {11 \pi} {12} = \sqrt 6 + \sqrt 2$
Cosecant of Straight Angle
:$\csc 180^\circ = \csc \pi$ is undefined
Cosecant of 195 Degrees
:$\csc 195^\circ = \csc \dfrac {13 \pi} {12} = - \left({\sqrt 6 + \sqrt 2}\right)$
Cosecant of 210 Degrees
:$\csc 210^\circ = \csc \dfrac {7 \pi} 6 = -2$
Cosecant of 225 Degrees
:$\csc 225 \degrees = \csc \dfrac {5 \pi} 4 = -\sqrt 2$
Cosecant of 240 Degrees
:$\csc 240 \degrees = \csc \dfrac {4 \pi} 3 = -\dfrac {2 \sqrt 3} 3$
Cosecant of 255 Degrees
:$\csc 255^\circ = \csc \dfrac {17 \pi} {12} = -\left({\sqrt 6 - \sqrt 2}\right)$
Cosecant of Three Right Angles
:$\csc 270^\circ = \csc \dfrac {3 \pi} 2 = -1$
Cosecant of 285 Degrees
:$\csc 285^\circ = \csc \dfrac {19 \pi} {12} = -\left({\sqrt 6 - \sqrt 2}\right)$
Cosecant of 300 Degrees
:$\csc 300^\circ = \csc \dfrac {5 \pi} 3 = -\dfrac {2 \sqrt 3} 3$
Cosecant of 315 Degrees
:$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$
Cosecant of 330 Degrees
:$\csc 330 \degrees = \csc \dfrac {11 \pi} 6 = -2$
Cosecant of 345 Degrees
:$\csc 345 \degrees = \csc \dfrac {23 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$
Cosecant of Full Angle
:$\csc 360 \degrees = \csc 2 \pi$ is undefined
Shape of Sine Function/Graph
:800px
Shape of Cosine Function/Graph
:800px
Shape of Tangent Function/Graph
:800px
Shape of Cotangent Function/Graph
:800px
Shape of Secant Function/Graph
:800px
Cosecant Function is Odd
:$\map \csc {-x} = -\csc x$ That is, the cosecant function is odd.
Secant Function is Even
:$\map \sec {-x} = \sec x$ That is, the secant function is even.
Cotangent Function is Odd
:$\map \cot {-x} = -\cot x$ That is, the cotangent function is odd.
Cotangent of Sum
:$\cot \left({a + b}\right) = \dfrac {\cot a \cot b - 1} {\cot b + \cot a}$ where $\cot $ is cotangent. === Corollary === {{:Cotangent of Sum/Corollary}}
Half-Integer is Half Odd Integer
Let $r$ be a number. Then $r$ is a half-integer  $r = \dfrac n 2$ where $n$ is an odd integer.
Sine of Angle plus Straight Angle
:$\map \sin {x + \pi} = -\sin x$
Cosine of Angle plus Straight Angle
:$\map \cos {x + \pi} = -\cos x$
Sine of Angle plus Full Angle
:$\map \sin {x + 2 \pi} = \sin x$
Cosine of Angle plus Full Angle
: $\cos \paren {x + 2 \pi} = \cos x$
Tangent of Angle plus Full Angle
: $\tan \left({x + 2 \pi}\right) = \tan x$
Tangent of Angle plus Straight Angle
: $\tan \left({x + \pi}\right) = \tan x$
Cotangent of Angle plus Straight Angle
:$\map \cot {x + \pi} = \cot x$
Secant of Angle plus Straight Angle
:$\map \sec {x + \pi} = -\sec x$
Cosecant of Angle plus Straight Angle
:$\map \csc {x + \pi} = -\csc x$
Cotangent of Angle plus Full Angle
: $\cot \left({x + 2 \pi}\right) = \cot x$
Secant of Angle plus Full Angle
: $\sec \left({x + 2 \pi}\right) = \sec x$
Cosecant of Angle plus Full Angle
: $\csc \left({x + 2 \pi}\right) = \csc x$
Sine of Angle plus Three Right Angles
:$\map \sin {x + \dfrac {3 \pi} 2} = -\cos x$
Cosine of Angle plus Three Right Angles
:$\map \cos {x + \dfrac {3 \pi} 2} = \sin x$
Tangent of Angle plus Three Right Angles
:$\map \tan {x + \dfrac {3 \pi} 2} = -\cot x$
Cotangent of Angle plus Three Right Angles
:$\map \cot {x + \dfrac {3 \pi} 2} = -\tan x$
Sine of Three Right Angles less Angle
:$\map \sin {\dfrac {3 \pi} 2 - \theta} = -\cos \theta$ where $\sin$ and $\cos$ are sine and cosine respectively.
Cosine of Three Right Angles less Angle
:$\cos \left({\dfrac {3 \pi} 2 - \theta}\right) = - \sin \theta$ where $\cos$ and $\sin$ are cosine and sine respectively.
Cosecant of Three Right Angles less Angle
:$\map \csc {\dfrac {3 \pi} 2 - \theta} = -\sec \theta$ where $\csc$ and $\sec$ are cosecant and secant respectively.
Double Angle Formulas/Sine
:$\sin 2 \theta = 2 \sin \theta \cos \theta$
Double Angle Formulas/Cosine
:$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$
Double Angle Formulas/Tangent
:$\map \tan {2 \theta} = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$
Triple Angle Formulas
=== Triple Angle Formula for Sine === {{:Triple Angle Formulas/Sine}} === Triple Angle Formula for Cosine === {{:Triple Angle Formulas/Cosine}} === Triple Angle Formula for Tangent === {{:Triple Angle Formulas/Tangent}}
Triple Angle Formulas/Sine
:$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$
Triple Angle Formulas/Cosine
:$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$
Triple Angle Formulas/Tangent
:$\map \tan {3 \theta} = \dfrac {3 \tan \theta - \tan^3 \theta} {1 - 3 \tan^2 \theta}$
Quadruple Angle Formulas
=== Quadruple Angle Formula for Sine === {{:Quadruple Angle Formulas/Sine}} === Quadruple Angle Formula for Cosine === {{:Quadruple Angle Formulas/Cosine}} === Quadruple Angle Formula for Tangent === {{:Quadruple Angle Formulas/Tangent}}
Quadruple Angle Formulas/Sine
:$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$
Quadruple Angle Formulas/Cosine
:$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$
Quadruple Angle Formulas/Tangent
:$\tan 4 \theta = \dfrac {4 \tan \theta - 4 \tan^3 \theta} {1 - 6 \tan^2 \theta + \tan^4 \theta}$
Quintuple Angle Formulas/Sine
:$\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$
Quintuple Angle Formulas/Cosine
:$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$
Quintuple Angle Formulas/Tangent
:$\tan 5 \theta = \dfrac {\tan^5 \theta - 10 \tan^3 \theta + 5 \tan \theta} {1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$
Power Reduction Formulas/Sine Squared
:$\sin^2 x = \dfrac {1 - \cos 2 x} 2$
Power Reduction Formulas/Cosine Squared
:$\cos^2 x = \dfrac {1 + \cos 2 x} 2$
Power Reduction Formulas/Tangent Squared
:$\tan^2x = \dfrac {1 - \cos2x} {1 + \cos2x}$
Power Reduction Formulas/Sine Cubed
:$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$
Power Reduction Formulas/Cosine Cubed
:$\cos^3 x = \dfrac {3 \cos x + \cos 3 x} 4$
Power Reduction Formulas/Sine to 4th
:$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$
Power Reduction Formulas/Cosine to 4th
:$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$
Power Reduction Formulas/Sine to 5th
:$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$
Power Reduction Formulas/Cosine to 5th
:$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$
Orthocomplement of Subset of Orthocomplement is Superset
Let $H$ be a Hilbert space. Let $A, B \subseteq H$ be subsets of $H$ such that $B \subseteq A^\perp$, where $A^\perp$ is the orthocomplement of $A$. Then $A \subseteq B^\perp$.
Simpson's Formulas/Cosine by Cosine
:$\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$
Simpson's Formulas/Sine by Sine
:$\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$
Simpson's Formulas/Cosine by Sine
:$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$
Simpson's Formulas/Sine by Cosine
:$\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$
Prosthaphaeresis Formulas/Sine plus Sine
:$\sin \alpha + \sin \beta = 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$
Prosthaphaeresis Formulas/Sine minus Sine
:$\sin \alpha - \sin \beta = 2 \map \cos {\dfrac {\alpha + \beta} 2} \map \sin {\dfrac {\alpha - \beta} 2}$
Prosthaphaeresis Formulas/Cosine plus Cosine
:$\cos \alpha + \cos \beta = 2 \, \map \cos {\dfrac {\alpha + \beta} 2} \, \map \cos {\dfrac {\alpha - \beta} 2}$
Prosthaphaeresis Formulas/Cosine minus Cosine
:$\cos \alpha - \cos \beta = -2 \, \map \sin {\dfrac {\alpha + \beta} 2} \, \map \sin {\dfrac {\alpha - \beta} 2}$
Sine of Integer Multiple of Argument/Formulation 1
{{begin-eqn}} {{eqn | l = \sin n \theta       | r = \sin \theta \paren {\paren {2 \cos \theta}^{n - 1} - \dbinom {n - 2} 1 \paren {2 \cos \theta}^{n - 3} + \dbinom {n - 3} 2 \paren {2 \cos \theta}^{n - 5} - \cdots}       | c =  }} {{eqn | r = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }       | c =  }} {{end-eqn}}
Cosine to Power of Odd Integer
:$\displaystyle \cos^{2n+1} \theta = \frac 1 {2^{2n}} \sum_{k \mathop = 0}^n \binom {2n+1} k \cos \left({2n - 2k + 1}\right) \theta$
Sum of Arctangent and Arccotangent
Let $x \in \R$ be a real number. Then: : $\arctan x + \operatorname{arccot} x = \dfrac \pi 2$ where $\arctan$ and $\operatorname{arccot}$ denote arctangent and arccotangent respectively.
Arcsine of Reciprocal equals Arccosecant
:$\map \arcsin {\dfrac 1 x} = \arccsc x$
Arccosine of Reciprocal equals Arcsecant
:$\map \arccos {\dfrac 1 x} = \arcsec x$
Arctangent of Reciprocal equals Arccotangent
:$\map \arctan {\dfrac 1 x} = \arccot x$