Трыганаметрычныя формулы

Трыганаметрычныя тоеснасці:

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

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

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

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

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

$\mathrm{tg}\alpha \cdot \mathrm{ctg}\alpha =1.$

Формулы складання:

$\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta ,$	$\mathrm{tg}(\alpha \pm \beta )=\frac{\mathrm{tg}\alpha \pm \mathrm{tg}\beta }{1\mp \mathrm{tg}\alpha \mathrm{tg}\beta },$
$\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta ,$	$\mathrm{ctg}(\alpha \pm \beta )=\frac{\mathrm{ctg}\alpha \mathrm{ctg}\beta \mp 1}{\mathrm{ctg}\beta \pm \mathrm{ctg}\alpha },$

Формулы кратных вуглоў:

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

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

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

$\mathrm{ctg}2\alpha =\frac{{\mathrm{ctg}}^2\alpha -1}{2\mathrm{ctg}\alpha }=\frac{1}{2}(\mathrm{ctg}\alpha -\mathrm{tg}\alpha ).$

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

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

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

$\mathrm{ctg}3\alpha =\frac{{\mathrm{ctg}}^3\alpha -3\mathrm{ctg}\alpha }{3{\mathrm{ctg}}^2\alpha -1}.$

Формулы палавіннага вугла:

$\sin \frac{\alpha }{2}=\sqrt{\frac{1-\cos \alpha }{2}},0⩽\alpha ⩽2\pi ;$	$\cos \frac{\alpha }{2}=\sqrt{\frac{1+\cos \alpha }{2}},-\pi ⩽\alpha ⩽\pi ;$
$\mathrm{tg}\frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }=\frac{\sin \alpha }{1+\cos \alpha };$	$\mathrm{ctg}\frac{\alpha }{2}=\frac{\sin \alpha }{1-\cos \alpha }=\frac{1+\cos \alpha }{\sin \alpha };$
$\mathrm{tg}\frac{\alpha }{2}=\sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }},0⩽\alpha <\pi ;$	$\mathrm{ctg}\frac{\alpha }{2}=\sqrt{\frac{1+\cos \alpha }{1-\cos \alpha }},0<\alpha ⩽\pi .$

Формулы сумы і рознасці функцый:

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

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

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

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

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

Здабыткаў функцый:

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

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

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

Формулы паніжэння цотнай ступені:

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

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

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

${\mathrm{ctg}}^2\alpha =\frac{1+\cos 2\alpha }{1-\cos 2\alpha }.$