Trigonometry identities

Pythagorean identities

$\sin ^{2}A+\cos ^{2}A=1$

$1+\tan ^{2}A=\sec ^{2}A$

$1+\cot ^{2}A=\csc ^{2}A$

Law of sines

The law of sines for an arbitrary triangle states:

${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}},$

or equivalently:

${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R$

Law of cosines

The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem to arbitrary triangles:

$c^{2}=a^{2}+b^{2}-2ab\cos C,$

or equivalently:

$\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}$

Sum and difference identities

$\sin(A+B)=\sin A\cos B+\cos A\sin B\,$

$\cos(A+B)=\cos A\cos B-\sin A\sin B\,$

$\tan(A+B)={\frac {\tan A+\tan B}{1-\tan A\tan B}}\,$

 

$\sin(A-B)=\sin A\cos B-\cos A\sin B\,$

$\cos(A-B)=\cos A\cos B+\sin A\sin B\,$

$\tan(A-B)={\frac {\tan A-\tan B}{1+\tan A\tan B}}={\frac {\cot B-\cot A}{1+\cot A\cot B}}\,$

Double-angle identities

$\sin 2A=2\sin A\cos A\,$

$\cos 2A=\cos ^{2}A-\sin ^{2}A=2\cos ^{2}A-1=1-2\sin ^{2}A={1-\tan ^{2}A \over 1+\tan ^{2}A}\,$

$\tan 2A={2\tan A \over 1-\tan ^{2}A}={2\cot A \over \cot ^{2}A-1}={2 \over \cot A-\tan A}\,$

Half-angle identities

Note that $\pm$ are correct, it means it may be either one, depending on the value of A/2.

$\sin {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{2}}}\,$

$\cos {\frac {A}{2}}=\pm {\sqrt {\frac {1+\cos A}{2}}}\,$

$\tan {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{1+\cos A}}}={\frac {\sin A}{1+\cos A}}={\frac {1-\cos A}{\sin A}}\,$

Law of tangents

The law of tangents:

${\frac {a+b}{a-b}}={\frac {\tan[{\frac {1}{2}}(A+B)]}{\tan[{\frac {1}{2}}(A-B)]}}$