Основни триґонометрийни формули

Функциуї єдного угла

$\sin ^{2}\alpha +\cos ^{2}\alpha =1,\quad {\frac {\sin \alpha }{\cos \alpha }}=\tan \alpha ,\quad \sin \alpha \cdot \csc \alpha =1$,

$\sec ^{2}\alpha -\tan ^{2}\alpha =1,\qquad \cos \alpha \cdot \sec \alpha =1$,

$\csc ^{2}\alpha -\cot ^{2}\alpha =1,\quad {\frac {\cos \alpha }{\sin \alpha }}=\cot \alpha ,\quad \tan \alpha \cdot \cot \alpha =1$

Медзисобне виражованє функцийох

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

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

$\tan \alpha ={\frac {\sin \alpha }{\sqrt {1-\sin ^{2}\alpha }}}={\frac {1}{\cot \alpha }},$

$\cot \alpha ={\frac {\sqrt {1-\sin ^{2}\alpha }}{\sin \alpha }}={\frac {1}{\tan \alpha }}.$

Функциї суми и розлики углох

$\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta ,\,$

$\cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta ,$

$\tan(\alpha \pm \beta )={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }},\quad \cot(\alpha \pm \beta )={\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}.$

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

Функциї углох

$\sin 2\alpha =2\sin \alpha \cos \alpha ,\quad \sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha ,$

$\cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha ,\quad \cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha ,$

$\tan 2\alpha ={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }},\quad \tan 3\alpha ={\frac {3\tan \alpha -\tan ^{3}\alpha }{1-3\tan ^{2}\alpha }},$

$\cot 2\alpha ={\frac {\cot ^{2}\alpha -1}{2\cot \alpha }},\quad \cot 3\alpha ={\frac {\cot ^{3}\alpha -3\cot \alpha }{3\cot ^{2}\alpha -1}},$

$\tan 4\alpha ={\frac {4\tan \alpha -4\tan ^{3}\alpha }{1-6\tan ^{2}\alpha +\tan ^{4}\alpha }},\quad \cot 4\alpha ={\frac {\cot ^{4}\alpha -6\cot ^{2}\alpha +1}{4\cot ^{3}\alpha -4\cot \alpha }}.$

За векше n хаснує ше Моаврова формула за комплексане число, розвита до биномного шору:

$\cos n\alpha +i\sin n\alpha =(\cos \alpha +i\sin \alpha )^{n}=\,$

$\cos ^{n}\alpha +in\cos ^{n-1}\alpha \sin \alpha -C_{n}^{2}\cos ^{n-2}\alpha \sin ^{2}\alpha --iC_{n}^{3}\cos ^{n-3}\alpha \sin ^{3}\alpha +C_{n}^{4}\cos ^{n-4}\alpha \sin ^{4}\alpha +...,$

дзе $C_{n}^{k}=C(n,k)={n \choose k}={\frac {n\cdot (n-1)\cdots (n-k+1)}{k\cdot (k-1)\cdots 1}}={\frac {n!}{k!(n-k)!}}$ биномни коефициєнт.

Оталь шлїдзи:

$\cos n\alpha =\cos ^{n}\alpha -C_{n}^{2}\cos ^{n-2}\alpha \sin ^{2}\alpha +C_{n}^{4}\cos ^{n-4}\alpha \sin ^{4}\alpha -C_{n}^{6}\cos ^{n-6}\alpha \sin ^{6}\alpha +...,$

$\sin n\alpha =n\cos ^{n-1}\alpha \sin \alpha -C_{n}^{3}\cos ^{n-3}\alpha \sin ^{3}\alpha +C_{n}^{5}\cos ^{n-5}\alpha \sin ^{5}\alpha -....$

Сума и розлика функцийох

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

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

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

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

$\tan \alpha \pm \tan \beta ={\frac {\sin(\alpha \pm \beta )}{\cos \alpha \cos \beta }},\quad \cot \alpha \pm \cot \beta =\pm {\frac {\sin(\alpha \pm \beta )}{\sin \alpha \sin \beta }},$

$\tan \alpha +\cot \beta ={\frac {\cos(\alpha -\beta )}{\cos \alpha \sin \beta }},\quad \cot \alpha -\tan \beta ={\frac {cos(\alpha +\beta )}{\sin \alpha \cos \beta }}.$

Продукт функцийох

$\sin \alpha \sin \beta ={\frac {1}{2}}[\cos(\alpha -\beta )-\cos(\alpha +\beta )],$

$\cos \alpha \cos \beta ={\frac {1}{2}}[\cos(\alpha -\beta )+cos(\alpha +\beta )],$

$\sin \alpha \cos \beta ={\frac {1}{2}}[\sin(\alpha -\beta )+\sin(\alpha +\beta )].$

Функциї половки угла

$\sin {\frac {\alpha }{2}}={\sqrt {\frac {1-\cos \alpha }{2}}},\quad \cos {\frac {\alpha }{2}}={\sqrt {\frac {1+\cos \alpha }{2}}},$

$\tan {\frac {\alpha }{2}}={\sqrt {\frac {1-\cos \alpha }{1+\cos \alpha }}}={\frac {1-\cos \alpha }{\sin \alpha }}={\frac {\sin \alpha }{1+\cos \alpha }},$

$\cot {\frac {\alpha }{2}}={\sqrt {\frac {1+\cos \alpha }{1-\cos \alpha }}}={\frac {1+\cos \alpha }{\sin \alpha }}={\frac {\sin \alpha }{1-\cos \alpha }}.$

Ступньованє функцийох

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

$\sin ^{3}\alpha ={\frac {1}{4}}(3\sin \alpha -\sin 3\alpha ),\quad \cos ^{3}\alpha ={\frac {1}{4}}(\cos 3\alpha +3\cos \alpha ),$

$\sin ^{4}\alpha ={\frac {1}{8}}(\cos 4\alpha -4\cos 2\alpha +3),\quad \cos ^{4}\alpha ={\frac {1}{8}}(\cos 4\alpha +4\cos 2\alpha +3).$

За рахованє $\sin ^{n}\alpha \,$ и $\cos ^{n}\alpha \,$ за векше n хаснує ше Моаврово формули.