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Useful trigonometry equalities

Posted September 25th, 2007 by Isoscel

$ \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta $
$ \sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta $

$ \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta $
$ \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta $

$$\tan(\alpha+\beta)=\frac{\sin\alpha\cos\beta+\cos\alpha\sin\beta}{\cos\alpha\cos\beta-\sin\alpha\sin\beta}=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$$
$$\tan(\alpha-\beta)=\frac{\sin\alpha\cos\beta-\cos\alpha\sin\beta}{\cos\alpha\cos\beta+\sin\alpha\sin\beta}=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$$
$$ \sin x+\sin y=2\sin \frac{x+y}{2}\cos \frac{x-y}{2}$$
$$ \sin x-\sin y=2\sin \frac{x-y}{2}\cos \frac{x+y}{2}$$
$$\cos x+\cos y =2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$$
$$\cos x-\cos y =-2\sin\frac{x+y}{2}\sin \frac{x-y}{2}$$

$ \sin 2x=2\sin x\cos x $
$ \cos 2x=\cos^2 x-\sin^2 x=2\cos^2 x-1=1-2\sin^2 x $

$ \sin 3x=3\sin x-4\sin^3 x $
$ \cos 3x=4\cos^3 x-3\cos x $

$$\tan 2x=\frac{2\tan x}{1-\tan^2x}$$
$$\tan 3x=\frac{\tan 2x+\tan x}{1-\tan 2x\tan x}=\frac{\frac{2\tan x}{1-\tan^2x}+\tan x}{1-\frac{2\tan x}{1-\tan^2x}\tan x}=\frac{3\tan x-\tan^3 x}{1-3\tan^2x}$$
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