Trigonometry/Identities/Table

Reciprocal

$\csc \theta =\frac{1}{\sin \theta }$

$\sec \theta =\frac{1}{\cos \theta }$

$\cot \theta =\frac{1}{\tan \theta }$

Quotient

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

$\cot \theta =\frac{\cos \theta }{\sin \theta }$

Pythagorean

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

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

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

Negative-angle

$\sin (-\theta )=-\sin \theta$

$\cos (-\theta )=\cos \theta$

$\tan (-\theta )=-\tan \theta$

Sum and difference

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

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

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

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

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

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

Cofunction

$\cos (90^o-\theta )=\sin \theta$

$\sec (90^o-\theta )=\csc \theta$

$\cot (90^o-\theta )=\tan \theta$

$\sin (90^o-\theta )=\cos \theta$

$\csc (90^o-\theta )=\sec \theta$

$\tan (90^o-\theta )=\cot \theta$

Double-angle

$\cos 2A={\cos }^{2}A-{\sin }^{2}A$

$\cos 2A=2{\cos }^{2}A-1$

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

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

$\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A}$

Half-angle

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

$\tan \frac{A}{2}=\frac{\sin A}{1+\cos A}$

$\tan \frac{A}{2}=\frac{1-\cos A}{\sin A}$

Template:Multicol-break

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

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

See also

• Trigonometry