Трыганаметрычныя формулы

Трыганаметрычныя тоеснасці:

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

${\displaystyle \operatorname {tg} (\alpha )={\frac {\sin(\alpha )}{\cos(\alpha )}},}$

${\displaystyle \operatorname {ctg} (\alpha )={\frac {\cos(\alpha )}{\sin(\alpha )}},}$

${\displaystyle 1+\operatorname {tg} ^{2}\alpha ={\frac {1}{\cos ^{2}\alpha }},\,}$

${\displaystyle 1+\operatorname {ctg} ^{2}\alpha ={\frac {1}{\sin ^{2}\alpha }},\,}$

${\displaystyle \operatorname {tg} \alpha \cdot \operatorname {ctg} \alpha =1.}$

Формулы складання:

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

Формулы кратных вуглоў:

${\displaystyle \sin 2\alpha =2\sin \alpha \cos \alpha ={\frac {2\,\operatorname {tg} \,\alpha }{1+\operatorname {tg} ^{2}\alpha }},}$

${\displaystyle \cos 2\alpha =\cos ^{2}\alpha \,-\,\sin ^{2}\alpha =2\cos ^{2}\alpha \,-\,1=1\,-\,2\sin ^{2}\alpha ={\frac {1-\operatorname {tg} ^{2}\alpha }{1+\operatorname {tg} ^{2}\alpha }}={\frac {\operatorname {ctg} \,\alpha -\operatorname {tg} \,\alpha }{\operatorname {ctg} \,\alpha +\operatorname {tg} \,\alpha }},}$

${\displaystyle \operatorname {tg} \,2\alpha ={\frac {2\,\operatorname {tg} \,\alpha }{1-\operatorname {tg} ^{2}\alpha }},}$

${\displaystyle \operatorname {ctg} \,2\alpha ={\frac {\operatorname {ctg} ^{2}\alpha -1}{2\,\operatorname {ctg} \,\alpha }}={\frac {1}{2}}\left(\operatorname {ctg} \,\alpha -\operatorname {tg} \,\alpha \right).}$

${\displaystyle \sin \,3\alpha =3\sin \alpha -4\sin ^{3}\alpha ,}$

${\displaystyle \cos \,3\alpha =4\cos ^{3}\alpha -3\cos \alpha ,}$

${\displaystyle \operatorname {tg} \,3\alpha ={\frac {3\,\operatorname {tg} \,\alpha -\operatorname {tg} ^{3}\,\alpha }{1-3\,\operatorname {tg} ^{2}\,\alpha }},}$

${\displaystyle \operatorname {ctg} \,3\alpha ={\frac {\operatorname {ctg} ^{3}\,\alpha -3\,\operatorname {ctg} \,\alpha }{3\,\operatorname {ctg} ^{2}\,\alpha -1}}.}$

Формулы палавіннага вугла:

${\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {1-\cos \alpha }{2}}},\quad 0\leqslant \alpha \leqslant 2\pi$ ;}	${\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {1+\cos \alpha }{2}}},\quad -\pi \leqslant \alpha \leqslant \pi$ ;}
${\displaystyle \operatorname {tg} \,{\frac {\alpha }{2}}={\frac {1-\cos \alpha }{\sin \alpha }}={\frac {\sin \alpha }{1+\cos \alpha }};}$	${\displaystyle \operatorname {ctg} \,{\frac {\alpha }{2}}={\frac {\sin \alpha }{1-\cos \alpha }}={\frac {1+\cos \alpha }{\sin \alpha }};}$
${\displaystyle \operatorname {tg} \,{\frac {\alpha }{2}}={\sqrt {\frac {1-\cos \alpha }{1+\cos \alpha }}},\quad 0\leqslant \alpha <\pi$ ;}	${\displaystyle \operatorname {ctg} \,{\frac {\alpha }{2}}={\sqrt {\frac {1+\cos \alpha }{1-\cos \alpha }}},\quad 0<\alpha \leqslant \pi .}$

Формулы сумы і рознасці функцый:

${\displaystyle \sin \alpha \pm \sin \beta =2\sin {\frac {\alpha \pm \beta }{2}}\cos {\frac {\alpha \mp \beta }{2}},}$

${\displaystyle \cos \alpha +\cos \beta =2\cos {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}},}$

${\displaystyle \cos \alpha -\cos \beta =-2\sin {\frac {\alpha +\beta }{2}}\sin {\frac {\alpha -\beta }{2}},}$

${\displaystyle \operatorname {tg} \alpha \pm \operatorname {tg} \beta ={\frac {\sin(\alpha \pm \beta )}{\cos \alpha \cos \beta }},}$

${\displaystyle 1\pm \sin {2\alpha }=(\sin \alpha \pm \cos \alpha )^{2}.}$

Здабыткаў функцый:

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

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

${\displaystyle \cos \alpha \cos \beta ={\frac {\cos(\alpha -\beta )+\cos(\alpha +\beta )}{2}}.}$

Формулы паніжэння цотнай ступені:

${\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\,\alpha }{2}},}$

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

${\displaystyle \operatorname {tg} ^{2}\,\alpha ={\frac {1-\cos 2\,\alpha }{1+\cos 2\,\alpha }},}$

${\displaystyle \operatorname {ctg} ^{2}\,\alpha ={\frac {1+\cos 2\,\alpha }{1-\cos 2\,\alpha }}.}$