Triqonometriyanın əsas düsturları

Triqonometriyanın əsas düsturları - triqonometrik funksiyalar üçün istifadə edilən düsturlardır.

Əsas triqonometrik düsturlar

Düstur	Arqumentin mənası
${\displaystyle \operatorname {sin} ^{2}\alpha +\operatorname {cos} ^{2}\alpha =1}$	${\displaystyle \forall \alpha }$
${\displaystyle \operatorname {tg} ^{2}\alpha +1={\frac {1}{\cos ^{2}\alpha }}=\operatorname {sec} ^{2}\alpha }$	${\displaystyle \alpha \neq {\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} }$
${\displaystyle \operatorname {ctg} ^{2}\alpha +1={\frac {1}{\sin ^{2}\alpha }}=\operatorname {cosec} ^{2}\alpha }$	${\displaystyle \alpha \neq \pi n,n\in \mathbb {Z} }$
${\displaystyle ~\operatorname {tg} \alpha \cdot \operatorname {ctg} \alpha =1}$	${\displaystyle \alpha \neq {\frac {\pi n}{2}},n\in \mathbb {Z} }$

Toplama düsturları

Toplama düsturları
${\displaystyle \sin \left(\alpha \pm \beta \right)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$
${\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$
${\displaystyle \operatorname {tg} \left(\alpha \pm \beta \right)={\frac {\operatorname {tg} \alpha \pm \operatorname {tg} \beta }{1\mp \operatorname {tg} \alpha \operatorname {tg} \beta }}}$
${\displaystyle \operatorname {ctg} \left(\alpha \pm \beta \right)={\frac {\operatorname {ctg} \alpha \operatorname {ctg} \beta \mp 1}{\operatorname {ctg} \beta \pm \operatorname {ctg} \alpha }}}$

İkiqat arqument düsturları

İkiqat arqument düsturları
${\displaystyle \operatorname {sin} 2\alpha =2{\sin \alpha }{\cos \alpha }}$
${\displaystyle \operatorname {cos} 2\alpha ={\cos ^{2}\alpha }-{\sin ^{2}\alpha }}$  ${\displaystyle \operatorname {cos} 2\alpha =2{\cos ^{2}\alpha }-1=1-2{\sin ^{2}\alpha }}$
${\displaystyle \operatorname {tg} 2\alpha ={\frac {2\operatorname {tg} \alpha }{1-\operatorname {tg} ^{2}\alpha }}}$
${\displaystyle \operatorname {ctg} 2\alpha ={\frac {\operatorname {ctg} ^{2}\alpha -1}{2\operatorname {ctg} \alpha }}}$

Üçqat arqument düsturları

Üçqat arqument düsturları
${\displaystyle \sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha \,}$
${\displaystyle \cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha \,}$
${\displaystyle \operatorname {tg} 3\alpha ={\frac {3\operatorname {tg} \alpha -\operatorname {tg} ^{3}\alpha }{1-3\operatorname {tg} ^{2}\alpha }}}$
${\displaystyle \operatorname {ctg} 3\alpha ={\frac {3\operatorname {ctg} \alpha -\operatorname {ctg} ^{3}\alpha }{1-3\operatorname {ctg} ^{2}\alpha }}}$

Dərəcənin aşağı salma düsturları

Sinus	Kosinus
${\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\alpha }{2}}}$	${\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}}$
${\displaystyle \sin ^{3}\alpha ={\frac {3\sin \alpha -\sin 3\alpha }{4}}}$	${\displaystyle \cos ^{3}\alpha ={\frac {3\cos \alpha +\cos 3\alpha }{4}}}$
${\displaystyle \sin ^{4}\alpha ={\frac {3-4\cos 2\alpha +\cos 4\alpha }{8}}}$	${\displaystyle \cos ^{4}\alpha ={\frac {3+4\cos 2\alpha +\cos 4\alpha }{8}}}$
${\displaystyle \sin ^{5}\alpha ={\frac {10\sin \alpha -5\sin 3\alpha +\sin 5\alpha }{16}}}$	${\displaystyle \cos ^{5}\alpha ={\frac {10\cos \alpha +5\cos 3\alpha +\cos 5\alpha }{16}}}$

Düstur
${\displaystyle \sin ^{2}\alpha \cos ^{2}\alpha ={\frac {1-\cos 4\alpha }{8}}}$
${\displaystyle \sin ^{3}\alpha \cos ^{3}\alpha ={\frac {3\sin 2\alpha -\sin 6\alpha }{32}}}$
${\displaystyle \sin ^{4}\alpha \cos ^{4}\alpha ={\frac {3-4\cos 4\alpha +\cos 8\alpha }{128}}}$
${\displaystyle \sin ^{5}\alpha \cos ^{5}\alpha ={\frac {10\sin 2\alpha -5\sin 6\alpha +\sin 10\alpha }{512}}}$

Hasilin cəmə çevrilməsi düsturla

Hasilin cəmə çevrilməsi düsturları
${\displaystyle \sin \alpha \sin \beta ={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{2}}}$
${\displaystyle \sin \alpha \cos \beta ={\frac {\sin(\alpha +\beta )+\sin(\alpha -\beta )}{2}}}$
${\displaystyle \cos \alpha \cos \beta ={\frac {\cos(\alpha -\beta )+\cos(\alpha +\beta )}{2}}}$

İstinadlar

1. Otto Forster Analysis 1 Differential- und Integralrechnung einer Veränderlichen vieweg 1983 Seite 87
2. I.N.Bronstein, K.A. Semendjajew: Taschenbuch der Mathematik. 19. Auflage, 1979. B.G. Teubner Verlagsgesellschaft, Leipzig. S 237