Formelsamling/Matematik/Trigonometri

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

Triangelsatser

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

Areasatsen:

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

Sinussatsen:

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

eller

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

Cosinussatsen:

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

Herons formel:

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

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

för triangelns area gäller

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

Formler

För alla vinklar ${\displaystyle \alpha }$ och ${\displaystyle \beta }$ gäller att

Enkla samband:

${\displaystyle \cos(\alpha )=\sin(\alpha +{\frac {\pi }{2}})\,\!}$ (vinklar i radianer)
${\displaystyle \cos(\alpha )=\sin(\alpha +90^{\circ })\,\!}$ (vinklar i grader)
${\displaystyle \sin(-\alpha )=-\sin \alpha \,\!}$
${\displaystyle \cos(-\alpha )=\cos \alpha \,\!}$
${\displaystyle \tan(-\alpha )=-\tan \alpha \,\!}$
${\displaystyle \sin(90^{\circ }-\alpha )=\cos \alpha \,\!}$
${\displaystyle \cos(90^{\circ }-\alpha )=\sin \alpha \,\!}$
${\displaystyle \tan(90^{\circ }-\alpha )={\frac {1}{\tan \alpha }}\,\!}$
${\displaystyle \sin(180^{\circ }-\alpha )=\sin \alpha \,\!}$
${\displaystyle \cos(180^{\circ }-\alpha )=-\cos \alpha \,\!}$
${\displaystyle \tan(180^{\circ }-\alpha )=-\tan \alpha \,\!}$

Trigonometriska ettan:

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

${\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta \,\!}$
${\displaystyle \sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta \,\!}$
${\displaystyle \cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta \,\!}$
${\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta \,\!}$
${\displaystyle \tan(\alpha +\beta )={\frac {\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }}\,\!}$
${\displaystyle \tan(\alpha -\beta )={\frac {\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }}\,\!}$

Formler för halva vinkeln:

${\displaystyle \sin ^{2}({\frac {\alpha }{2}})={\frac {1-\cos \alpha }{2}}\,\!}$
${\displaystyle \cos ^{2}({\frac {\alpha }{2}})={\frac {1+\cos \alpha }{2}}\,\!}$

Formler för dubbla vinkeln:

${\displaystyle \sin(2\alpha )=2\sin(\alpha )\cos(\alpha )\,\!}$
${\displaystyle \cos(2\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha \,\!}$
${\displaystyle \tan(2\alpha )={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }}\,\!}$

Produktformler:

${\displaystyle 2\cos \alpha \cos \beta =\cos(\alpha -\beta )+\cos(\alpha +\beta )\,\!}$
${\displaystyle 2\sin \alpha \sin \beta =\cos(\alpha -\beta )-\cos(\alpha +\beta )\,\!}$
${\displaystyle 2\sin \alpha \cos \beta =\sin(\alpha -\beta )+\sin(\alpha +\beta )\,\!}$

Linjärkombinationer:

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

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

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

Exakta trigonometriska funktionsvärden

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