Sivu:Lukion taulukot.pdf/30

MERKINTÖJÄ, MÄÄRITELMIÄ JA KAAVOJA 29

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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.	$\tan {\frac {1}{2}}x={\frac {\sin x}{1+\cos x}}={\frac {1-\cos x}{\sin x}}$

Trigonometriset yhtälöt

1.	$\sin x=\sin \alpha$
	$x=\alpha +n\cdot 2\pi$ tai $x=(\pi -\alpha )+n\cdot 2\pi$
2.	$\cos x=\cos \alpha$
	$x=\pm \alpha +n\cdot 2\pi$
3.	$\tan x=\tan \alpha$
	$x=\alpha +n\cdot \pi \quad x,\alpha \neq {\frac {\pi }{2}}+n\cdot \pi$