# Eikonal Blog

## 2010.04.09

### Simple expression for trigonometric functions of mutiple of an angle

Filed under: mathematics — Tags: , , , , , , — sandokan65 @ 15:16
• $\sin(n x) = \frac{i}2 [(\cos(x)-i \sin(x))^n - (\cos(x)+i \sin(x))^n]$,
• $\cos(n x) = \frac12 [(\cos(x)-i \sin(x))^n + (\cos(x)+i \sin(x))^n]$,
• $\tan(n x) = i \frac{(1-i \tan(x))^n - (1+i \tan(x))^n}{(1-i \tan(x))^n + (1+i \tan(x))^n}$

Examples:

• $\sin(2x) = 2 \sin(x) \cos(x)$,
• $\sin(3x) = \sin(x) [3\cos(x)^2-\sin(x)^2]$,
• $\sin(4x) = 4 \sin(x) \cos(x) [\cos(x)^2-\sin(x)^2]$,
• $\sin(5x) = \sin(x) [5 \cos(x)^4 - 10 \cos(x)^2 \sin(x)^2 + \sin(x)^4]$,
• $\cos(2x) = [\cos(x)^2-\sin(x)^2]$,
• $\cos(3x) = \cos(x) [\cos(x)^2- 3 \sin(x)^2]$,
• $\cos(4x) = [\cos(x)^4 - 6 \cos(x)^2 \sin(x)^2 + \sin(x)^4]$,
• $\cos(5x) = \cos(x) [\cos(x)^4 - 10 \cos(x)^2 \sin(x)^2 + 5 \sin(x)^4]$,
• $\tan(2x) = \frac{2\tan(x)}{1-\tan(x)^2}$,
• $\tan(3x) = \frac{\tan(x)[3-\tan(x)^2]}{1-3\tan(x)^2}$,
• $\tan(4x) = \frac{4\tan(x)[1-\tan(x)^2]}{1-6\tan(x)^2+\tan(x)^4}$,
• $\tan(5x) = \frac{\tan(x)[5- 10 \tan(x)^2 + \tan(x)^4]}{1-10\tan(x)^2+5\tan(x)^4}$, …

Much more on the web: