Useful trigonometry equalities

Submitted by Isoscel on Tue, 09/25/2007 - 22:21.

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

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

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

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

$\sin x+\sin y=2\sin \frac{x+y}{2}\cos \frac{x-y}{2}$

$\sin x-\sin y=2\sin \frac{x-y}{2}\cos \frac{x+y}{2}$

$\cos x+\cos y =2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$

$\cos x-\cos y =-2\sin\frac{x+y}{2}\sin \frac{x-y}{2}$

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

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

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

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

Navigation

User login

Useful trigonometry equalities