Tables of trigonometric functions

Properties of trigonometric functions

Representation by other trigonometric functions

The trigonometric functions sine, cosine and tangent can be expressed by other trigonometric functions. Here are some important identities:

Function values of trigonometric functions

Reduction formulas

Context of the trigonometric functions for the same argument

${\sin \left(x\right)}^2+{\cos \left(x\right)}^2$$=1$

$\tan \left(x\right)$$=\frac{\sin \left(x\right)}{\cos \left(x\right)}$

$\cot \left(x\right)$$=\frac{\cos \left(x\right)}{\sin \left(x\right)}$$=\frac{1}{\tan \left(x\right)}$

$\cot \left(x\right)\cdot \tan \left(x\right)$$=1$

Addition theorems

Half of the argument value

$\sin \left(\frac{x}{2}\right)$$=\pm \sqrt{\frac{1-\cos \left(x\right)}{2}}$

$\cos \left(\frac{x}{2}\right)$$=\pm \sqrt{\frac{1+\cos \left(x\right)}{2}}$

$\tan \left(\frac{x}{2}\right)$$=\pm \sqrt{\frac{1-\cos \left(x\right)}{1+\cos \left(x\right)}}$

Sum and difference of trigonometric functions

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

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

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

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

$\cos \left(x\right)\pm \sin \left(x\right)$$=\sqrt{2}\sin \left(\frac{\pi }{4}\pm x\right)$$=\sqrt{2}\cos \left(\frac{\pi }{4}\mp x\right)$

$\tan \left(x\right)\pm \tan \left(y\right)$$=\frac{\sin \left(x\pm y\right)}{\cos \left(x\right)\cos (y)}$

$\cot \left(x\right)\pm \cot \left(y\right)$$=\pm \frac{\sin \left(x\pm y\right)}{\sin \left(x\right)\sin (y)}$

$\tan \left(x\right)+\cot \left(y\right)$$=\frac{\cos \left(x-y\right)}{\cos \left(x\right)\sin (y)}$

$\cot \left(x\right)-\tan \left(y\right)$$=\frac{\cos \left(x+y\right)}{\sin \left(x\right)\cos (y)}$

Products of trigonometric functions

$\sin \left(x\right)\sin \left(y\right)$$=\frac{1}{2}\left(\cos \left(x-y\right)-\cos \left(x+y\right)\right)$

$\cos \left(x\right)\cos \left(y\right)$$=\frac{1}{2}\left(\cos \left(x-y\right)+\cos \left(x+y\right)\right)$

$\sin \left(x\right)\cos \left(y\right)$$=\frac{1}{2}\left(\sin \left(x-y\right)+\sin \left(x+y\right)\right)$

$\cos \left(x\right)\sin \left(y\right)$$=\frac{1}{2}\left(\sin \left(x+y\right)-\sin \left(x-y\right)\right)$

Potencies of trigonometric functions

${\sin }^2\left(x\right)$$=\frac{1}{2}\left(1-\cos \left(2x\right)\right)$

${\sin }^3\left(x\right)$$=\frac{1}{4}\left(3\sin \left(x\right)-\sin \left(3x\right)\right)$

${\sin }^n\left(x\right)$$=\frac{1}{2^n}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)\cos \left(\left(n-2k\right)\left(x-\frac{\pi }{2}\right)\right)$

${\cos }^2\left(x\right)$$=\frac{1}{2}\left(1+\cos \left(2x\right)\right)$

${\cos }^3\left(x\right)$$=\frac{1}{4}\left(3\cos \left(x\right)+\cos \left(3x\right)\right)$

${\cos }^n\left(x\right)$$=\frac{1}{2^n}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)\cos \left(\left(n-2k\right)x\right)$

Releated sites

Here is a list of of further useful sites: