Formelsamling/Matematik/Trigonometri

Förklaring av elementära funktioner som sinus, cosinus och tangens finner du här.

Triangelsatser

Om $T$ är en triangel med sidorna $a$, $b$ och $c$ och motstående vinklarna $A$, $B$ respektive $C$ så gäller

Areasatsen:

$Arean=T=\frac{bc\sin \alpha }{2}$

Sinussatsen:

$\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}$

eller

$\frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }$

Cosinussatsen:

$a^2=b^2+c^2-2bc\cdot \cos \alpha$

Herons formel:

Herons formel säger att givet en godtycklig triangel med sidorna a, b, c, och semiperimetern s där

$s=\frac{1}{2}(a+b+c)$

för triangelns area gäller

$\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{n}=\sqrt{s(s-a)(s-b)(s-c)}=\frac{\sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}}{4}$

Formler

För alla vinklar $\alpha$ och $\beta$ gäller att

Enkla samband:

$\cos (\alpha )=\sin (\alpha +\frac{\pi }{2})$ (vinklar i radianer)

$\cos (\alpha )=\sin (\alpha +90^{\circ })$ (vinklar i grader)

$\sin (-\alpha )=-\sin \alpha$

$\cos (-\alpha )=\cos \alpha$

$\tan (-\alpha )=-\tan \alpha$

$\sin (90^{\circ }-\alpha )=\cos \alpha$

$\cos (90^{\circ }-\alpha )=\sin \alpha$

$\tan (90^{\circ }-\alpha )=\frac{1}{\tan \alpha }$

$\sin (180^{\circ }-\alpha )=\sin \alpha$

$\cos (180^{\circ }-\alpha )=-\cos \alpha$

$\tan (180^{\circ }-\alpha )=-\tan \alpha$

Trigonometriska ettan:

${\sin }^2\alpha +{\cos }^2\alpha =1$

Additionssatser:

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

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

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

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

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

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

Formler för halva vinkeln:

${\sin }^2(\frac{\alpha }{2})=\frac{1-\cos \alpha }{2}$

${\cos }^2(\frac{\alpha }{2})=\frac{1+\cos \alpha }{2}$

Formler för dubbla vinkeln:

$\sin (2\alpha )=2\sin (\alpha )\cos (\alpha )$

$\cos (2\alpha )={\cos }^2\alpha -{\sin }^2\alpha =2{\cos }^2\alpha -1=1-2{\sin }^2\alpha$

$\tan (2\alpha )=\frac{2\tan \alpha }{1-{\tan }^2\alpha }$

Produktformler:

$2\cos \alpha \cos \beta =\cos (\alpha -\beta )+\cos (\alpha +\beta )$

$2\sin \alpha \sin \beta =\cos (\alpha -\beta )-\cos (\alpha +\beta )$

$2\sin \alpha \cos \beta =\sin (\alpha -\beta )+\sin (\alpha +\beta )$

Linjärkombinationer:

Då $a>0,b>0,\tan \beta =\frac{b}{a},0^{\circ }<\beta <90^{\circ }$ gäller

$a\sin \alpha +b\cos \alpha =\sqrt{(}{a}^2+b^2)\sin (\alpha +\beta )$

$a\sin \alpha -b\cos \alpha =\sqrt{(}{a}^2+b^2)\sin (\alpha -\beta )$

Exakta trigonometriska funktionsvärden

Vinkel $\alpha$		$\sin \alpha$	$\cos \alpha$	$\tan \alpha$	$\cot \alpha$	$\sec \alpha$	$\csc \alpha$
i grader	i radianer
$0^{\circ }$	$0$	$0$	$1$	$0$	$\mp \mathrm{\infty }$	$1$	$\mp \mathrm{\infty }$
$15^{\circ }$	$\frac{\pi }{12}$	$\frac{1}{4}(\sqrt{6}-\sqrt{2})$	$\frac{1}{4}(\sqrt{6}+\sqrt{2})$	$2-\sqrt{3}$	$2+\sqrt{3}$	$\sqrt{6}-\sqrt{2}$	$\sqrt{6}+\sqrt{2}$
$18^{\circ }$	$\frac{\pi }{10}$	$\frac{1}{4}(\sqrt{5}-1)$	$\frac{1}{4}\sqrt{10+2\sqrt{5}}$	$\sqrt{1-0.4\sqrt{5}}$	$\sqrt{5+2\sqrt{5}}$	$\sqrt{2-\frac{2}{\sqrt{5}}}$	$\sqrt{5}+1$
$30^{\circ }$	$\frac{\pi }{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$	$\sqrt{3}$	$\frac{2}{\sqrt{3}}$	$2$
$36^{\circ }$	$\frac{\pi }{5}$	$\frac{1}{4}\sqrt{10-2\sqrt{5}}$	$\frac{1}{4}(\sqrt{5}+1)$	$\sqrt{5-2\sqrt{5}}$	$\sqrt{1+0.4\sqrt{5}}$	$\sqrt{5}-1$	$\sqrt{2+\frac{2}{\sqrt{5}}}$
$45^{\circ }$	$\frac{\pi }{4}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$1$	$1$	$\sqrt{2}$	$\sqrt{2}$
$60^{\circ }$	$\frac{\pi }{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{1}{\sqrt{3}}$	$2$	$\frac{2}{\sqrt{3}}$
$90^{\circ }$	$\frac{\pi }{2}$	$1$	$0$	$\pm \mathrm{\infty }$	$0$	$\pm \mathrm{\infty }$	$1$
$120^{\circ }$	$\frac{2\pi }{3}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$	$-\frac{1}{\sqrt{3}}$	$-2$	$\frac{2}{\sqrt{3}}$
$180^{\circ }$	$\pi$	$0$	$-1$	$0$	$\mp \mathrm{\infty }$	$-1$	$\pm \mathrm{\infty }$
$270^{\circ }$	$\frac{3\pi }{2}$	$-1$	$0$	$\pm \mathrm{\infty }$	$0$	$\mp \mathrm{\infty }$	$-1$
$360^{\circ }$	$2\pi$	$0$	$1$	$0$	$\mp \mathrm{\infty }$	$1$	$\mp \mathrm{\infty }$