Trigonometric Functions of Supplementary Angles

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Theorem

Sine of Supplementary Angle

$\sin \paren {\pi - \theta} = \sin \theta$

where $\sin$ denotes sine.


That is, the sine of an angle equals its supplement.


Cosine of Supplementary Angle

$\map \cos {\pi - \theta} = -\cos \theta$

where $\cos$ denotes cosine.


That is, the cosine of an angle is the negative of its supplement.


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.


Secant of Supplementary Angle

$\map \sec {\pi - \theta} = -\sec \theta$

where $\sec$ denotes secant.


That is, the secant of an angle is the negative of its supplement.


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.


Sources