# Power and n-th arguments of trigonometric functions

Calculator for powers and n-fold arguments of sine and cosine

Calculator for calculating powers and n-fold arguments of the trigonometric functions sine and cosine.

n =

## Addition theorem for multiples of the argument of sine and cosine

$\mathrm{sin}\left(2x\right)$$=2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$

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

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

$\mathrm{cos}\left(3x\right)$$=4{\mathrm{cos}}^{3}\left(x\right)-3\mathrm{cos}\left(x\right)$

## Powers of sine and cosine

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

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

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

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

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

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

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