Calculus/Table of Trigonometry

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Definitions

$\tan(x)={\frac {\sin(x)}{\cos(x)}}$
$\sec(x)={\frac {1}{\cos(x)}}$
$\cot(x)={\frac {\cos(x)}{\sin(x)}}={\frac {1}{\tan(x)}}$
$\csc(x)={\frac {1}{\sin(x)}}$

Pythagorean Identities

$\sin ^{2}(x)+\cos ^{2}(x)=1$
$1+\tan ^{2}(x)=\sec ^{2}(x)$
$1+\cot ^{2}(x)=\csc ^{2}(x)$

Double Angle Identities

$\sin(2x)=2\sin(x)\cos(x)$
$\cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)$
$\tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}$
$\cos ^{2}(x)={\frac {1+\cos(2x)}{2}}$
$\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}$

Angle Sum Identities

\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)

\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)

\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)

\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)

\sin(x)+\sin(y)=2\sin \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)

\sin(x)-\sin(y)=2\cos \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)

\cos(x)+\cos(y)=2\cos \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)

\cos(x)-\cos(y)=-2\sin \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)

\tan(x)+\tan(y)={\frac {\sin(x+y)}{\cos(x)\cos(y)}}

\tan(x)-\tan(y)={\frac {\sin(x-y)}{\cos(x)\cos(y)}}

\cot(x)+\cot(y)={\frac {\sin(x+y)}{\sin(x)\sin(y)}}

\cot(x)-\cot(y)={\frac {-\sin(x-y)}{\sin(x)\sin(y)}}

Product-to-sum identities

\cos(x)\cos(y)={\frac {\cos(x+y)+\cos(x-y)}{2}}

\sin(x)\sin(y)={\frac {\cos(x-y)-\cos(x+y)}{2}}

\sin(x)\cos(y)={\frac {\sin(x+y)+\sin(x-y)}{2}}

\cos(x)\sin(y)={\frac {\sin(x+y)-\sin(x-y)}{2}}

See also

Trigonometry

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