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Formulae

General trigonometric identities
Contents

Formulae

A list of important general trigonometric identities,

$\cos^2&space;\theta&space;+&space;\sin^2&space;\theta&space;=&space;1$
$\sec^2&space;\theta&space;=&space;1&space;+&space;\tan^2&space;\theta$
$\csc^2&space;\theta&space;=&space;1&space;+&space;\cot^2&space;\theta$

$\cos(90^{\circ}&space;-&space;\theta)&space;=&space;\sin&space;\theta$
$\sin(90^{\circ}&space;-&space;\theta)&space;=&space;\cos&space;\theta$
$\tan(90^{\circ}&space;-&space;\theta)&space;=&space;\cot&space;\theta$

Some more specific identities that relate to the following general diagram,

$c^2&space;=&space;a^2&space;+&space;b^2&space;-&space;2a&space;b&space;cos&space;C$
$\frac{a}{\sin&space;A}&space;=&space;\frac{b}{\sin&space;B}&space;=&space;\frac{c}{\sin&space;C}&space;=&space;2R$

$\cos(A-b)&space;=&space;\cos&space;A&space;\cos&space;B&space;+&space;\sin&space;A&space;\sin&space;B$
$\cos(A+b)&space;=&space;\cos&space;A&space;\cos&space;B&space;-&space;\sin&space;A&space;\sin&space;B$
$\sin(A-b)&space;=&space;\sin&space;A&space;\cos&space;B&space;-&space;\cos&space;A&space;\sin&space;B$
$\sin(A+b)&space;=&space;\sin&space;A&space;\cos&space;B&space;+&space;\cos&space;A&space;\sin&space;B$

$\cos&space;2A&space;=&space;\cos^2&space;A&space;-&space;\sin^2&space;A$
$\cos&space;2A&space;=&space;2&space;\cos^2&space;A&space;-&space;1$
$\cos&space;2A&space;=&space;1&space;-&space;2&space;\sin^2&space;A$
$\sin&space;2A&space;=&space;2&space;\sin&space;A&space;\cos&space;A$
$\tan&space;(A&space;+&space;B)&space;=&space;\frac{\tan&space;A&space;+&space;\tan&space;B}{1-\tan&space;A&space;\tan&space;B}$
$\tan&space;(A&space;-&space;B)&space;=&space;\frac{\tan&space;A&space;-&space;\tan&space;B}{1+\tan&space;A&space;\tan&space;B}$
$\tan&space;2A&space;&space;=&space;\frac{2&space;\tan&space;A}{1-\tan^2&space;A}$

$\tan&space;(A+B+C)&space;=&space;\frac{\tan&space;A&space;+&space;\tan&space;B&space;+&space;\tan&space;C&space;-&space;\tan&space;A&space;\tan&space;B&space;\tan&space;C}{1-&space;\tan&space;B&space;\tan&space;C&space;-&space;\tan&space;C&space;\tan&space;A&space;-&space;\tan&space;A&space;\tan&space;B}$

If $\inline&space;&space;A+B+C=180^{\circ}$

• $\inline&space;&space;\displaystyle&space;\tan&space;A&space;+&space;\tan&space;B&space;+&space;\tan&space;C&space;=&space;\tan&space;A&space;\tan&space;B&space;\tan&space;C$
• $\inline&space;&space;\displaystyle&space;\cot&space;\tfrac{1}{2}A&space;+&space;\cot&space;\tfrac{1}{2}&space;B&space;+&space;\cot&space;\tfrac{1}{2}&space;C&space;=&space;\cot&space;\tfrac{1}{2}&space;A&space;\cot&space;\tfrac{1}{2}&space;B&space;\cot&space;\tfrac{1}{2}&space;C$
• $\inline&space;&space;\displaystyle&space;\sin&space;A&space;+&space;\sin&space;B&space;+&space;\sin&space;C&space;=&space;4&space;\cos&space;\tfrac{1}{2}&space;A&space;\cos&space;\tfrac{1}{2}&space;B&space;\cos&space;\tfrac{1}{2}&space;C$
• $\inline&space;&space;\displaystyle&space;\cos&space;A&space;+&space;\cos&space;B&space;+&space;\cos&space;C&space;-1&space;=&space;4&space;\sin&space;\tfrac{1}{2}&space;A&space;\sin&space;\tfrac{1}{2}&space;B&space;\sin&space;\tfrac{1}{2}&space;C$

$1+&space;\cos&space;A&space;=&space;2&space;\cos^2(\frac{1}{2}A)$
$1-&space;\cos&space;A&space;=&space;2&space;\sin^2(\frac{1}{2}A)$

$\sin&space;2A&space;=&space;\frac{2&space;\tan&space;A}{1+\tan^2&space;A}$
$\cos&space;2A&space;=&space;\frac{1-&space;\tan^2&space;A}{1+\tan^2&space;A}$

$\sin&space;X&space;+&space;\sin&space;Y&space;=&space;2&space;\sin&space;\frac{X+Y}{2}&space;\cos&space;\frac{X-Y}{2}$
$\sin&space;X&space;-&space;\sin&space;Y&space;=&space;2&space;\cos&space;\frac{X+Y}{2}&space;\sin&space;\frac{X-Y}{2}$
$\cos&space;X&space;+&space;\cos&space;Y&space;=&space;2&space;\cos&space;\frac{X+Y}{2}&space;\cos&space;\frac{X-Y}{2}$
$\cos&space;X&space;-&space;\cos&space;Y&space;=&space;2&space;\sin&space;\frac{X+Y}{2}&space;\sin&space;\frac{Y-X}{2}$

$\frac{a-b}{a+b}&space;\cot&space;\frac{C}{2}&space;=&space;\tan&space;\tfrac{1}{2}(A-B)$
$c&space;=&space;a&space;cos&space;B&space;+&space;b&space;cos&space;A$
$\arctan&space;x&space;+&space;\arctan&space;y&space;\arctan&space;\frac{x+y}{1-xy}$
$\arctan&space;x&space;-&space;\arctan&space;y&space;\arctan&space;\frac{x-y}{1+xy}$

where

• R stands for the curcum-radius of the triangle ABC

To avoid doubt (and for those new to maths):
• $\inline&space;&space;\cot&space;\theta&space;=&space;1&space;/&space;\tan&space;\theta$
• $\inline&space;&space;\csc&space;\theta&space;=&space;1&space;/&space;\sin&space;\theta$
• $\inline&space;&space;\sec&space;\theta&space;=&space;1&space;/&space;\cos&space;\theta$