Lukion taulukot/Trigonometriset funktiot (Trigonometria)

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 Katso myös Trigonometriset funktiot (Merkintöjä)

Trigonometriset funktiot

Asteet ja radiaanit

Asteet	0	30	45	60	90	180	270	360	180 : π
Radiaanit	0	${\frac {\pi }{6}}$	${\frac {\pi }{4}}$	${\frac {\pi }{3}}$	${\frac {\pi }{2}}$	$\pi$	${\frac {3\pi }{2}}$	$2\pi$	1

Funktioiden määritelmät

$\sin \alpha =y$

$\cos \alpha =x$

$\tan \alpha ={\frac {y}{x}}={\frac {\sin \alpha }{\cos \alpha }},\;\alpha \neq {\frac {\pi }{2}}+n\cdot \pi$

$\cot \alpha ={\frac {x}{y}}={\frac {\cos \alpha }{\sin \alpha }}={\frac {1}{\tan \alpha }},\;\alpha \neq n\cdot \pi$

Peruskaavat

1.	$\sin ^{2}x+\cos ^{2}x=1$
	$\sin x=\pm {\sqrt {1-\cos ^{2}x}}$
	$\cos x=\pm {\sqrt {1-\sin ^{2}x}}$
	$\sin ^{2}x={\frac {1-\cos 2x}{2}}$
	$\cos ^{2}x={\frac {1+\cos 2x}{2}}$
2.	$\sin \left({\frac {\pi }{2}}-x\right)=\cos x$
	$\cos \left({\frac {\pi }{2}}-x\right)=\sin x$
3.	$\tan \left({\frac {\pi }{2}}-x\right)=\cot x$
	$\cot \left({\frac {\pi }{2}}-x\right)=\tan x$
4.	$\sin x=\pm {\frac {\tan x}{\sqrt {1+\tan ^{2}x}}};\;\cos x=\pm {\frac {1}{\sqrt {1+\tan ^{2}x}}}$

Jaksollisuus

$\sin(x+n\cdot 2\pi )=\sin x$	perusjakso $2\pi$
$\cos(x+n\cdot 2\pi )=\cos x$
$\tan(x+n\cdot \pi )=\tan x$	perusjakso $\pi$
$\cot(x+n\cdot \pi )=\cot x$

Palautuskaavat

1.	$\sin(-x)=-\sin x$	vastakulmat
	$\cos(-x)=\cos x$
	$\tan(-x)=-\tan x$
2.	$\sin(\pi -x)=\sin x$	suplementtikulmat
	$\cos(\pi -x)=-\cos x$
	$\tan(\pi -x)=-\tan x$
3.	$\sin(\pi +x)=-\sin x$
	$\cos(\pi +x)=-\cos x$

Summakaavoja

1.	$\sin(x+y)=\sin x\cos y+\cos x\sin y$
	$\sin(x-y)=\sin x\cos y-\cos x\sin y$
2.	$\cos(x+y)=\cos x\cos y-\sin x\sin y$
	$\cos(x-y)=\cos x\cos y+\sin x\sin y$

3.	$\tan(x+y)={\frac {\tan x+\tan y}{1-\tan x\tan y}}$
	$\tan(x-y)={\frac {\tan x-\tan y}{1+\tan x\tan y}}$
4.	$\cot(x+y)={\frac {\cot x\cot y-1}{\cot y+\cot x}}$
	$\cot(x-y)={\frac {\cot x\cot y+1}{\cot y-\cot x}}$
5.	$\sin x+\sin y=2\sin {\frac {x+y}{2}}\cos {\frac {x-y}{2}}$
	$\sin x-\sin y=2\cos {\frac {x+y}{2}}\sin {\frac {x-y}{2}}$
6.	$\cos x+\cos y=2\cos {\frac {x+y}{2}}\cos {\frac {x-y}{2}}$
	$\cos x-\cos y=-2\sin {\frac {x+y}{2}}\sin {\frac {x-y}{2}}$

Kaksinkertaiset ja puolet kulmat

1.	$\sin 2x=2\sin x\cos x$
2.	$\cos 2x=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x$
3.	$\tan 2x={\frac {2\tan x}{1-\tan ^{2}x}};\;\cot 2x={\frac {\cot ^{2}x-1}{2\cot x}}$
4.	$1+\cos x=2\cos ^{2}{\frac {1}{2}}x;\;1-\cos x=2\sin ^{2}{\frac {1}{2}}x$
5.	$1+\sin x=2\sin ^{2}\left({\frac {\pi }{4}}+{\frac {1}{2}}x\right)=2\cos ^{2}\left({\frac {\pi }{4}}-{\frac {1}{2}}x\right)$
6.	$1-\sin x=2\sin ^{2}\left({\frac {\pi }{4}}-{\frac {1}{2}}x\right)=2\cos ^{2}\left({\frac {\pi }{4}}+{\frac {1}{2}}x\right)$
7.	$\tan {\frac {1}{2}}x={\frac {\sin x}{1+\cos x}}={\frac {1-\cos x}{\sin x}}$