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:

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

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

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

för triangelns area gäller

$\mathrm {Arean} ={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}\ =\ {\frac {\sqrt {(a+b+c)(a+b-c)(b+c-a)(c+a-b)}}{4}}$

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

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

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

Här saknas information! Du kan hjälpa Wikibooks genom att fylla i mer!

Exakta trigonometriska funktionsvärden

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