Трыганаметрычныя формулы

З пляцоўкі Вікіпедыя.
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Змест

Трыганаметрычныя тоеснасці: [правіць]

 \sin^2 \alpha + \cos^2 \alpha = 1,\, tg(\alpha) = \frac{sin(\alpha)}{cos(\alpha)}, ctg(\alpha) = \frac{cos(\alpha)}{sin(\alpha)},  1 + \mathop{\mathrm{tg}}\,^2 \alpha = \frac{1}{ \cos^2 \alpha},\,  1 + \mathop{\mathrm{ctg}}\,^2 \alpha = \frac{1}{ \sin^2 \alpha},\,  \mathop{\mathrm{tg}}\,\alpha  \cdot \mathop{\mathrm{ctg}}\,\alpha=1.

Формулы складання: [правіць]

 \sin\left( \alpha \pm \beta \right)= \sin\alpha \, \cos\beta \pm \cos\alpha \, \sin\beta,  \operatorname{tg}\left( \alpha \pm \beta \right) = \frac{\operatorname{tg}\,\alpha \pm \operatorname{tg}\,\beta}{1 \mp \operatorname{tg}\,\alpha \, \operatorname{tg}\,\beta},
 \cos\left( \alpha \pm \beta \right)= \cos\alpha \, \cos\beta \mp \sin\alpha \, \sin\beta,  \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: [правіць]

\sin 2\alpha = 2 \sin \alpha \cos \alpha = \frac{2\,\operatorname{tg}\,\alpha }{1 + \operatorname{tg}^2\alpha},
\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},
\operatorname{tg}\,2 \alpha = \frac{2\,\operatorname{tg}\,\alpha}{1 - \operatorname{tg}^2\alpha},
\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).
\sin\,3\alpha=3\sin\alpha - 4\sin^3\alpha, \cos\,3\alpha=4\cos^3\alpha -3\cos\alpha, \operatorname{tg}\,3\alpha=\frac{3\,\operatorname{tg}\,\alpha - \operatorname{tg}^3\,\alpha}{1 - 3\,\operatorname{tg}^2\,\alpha}, \operatorname{ctg}\,3\alpha=\frac{\operatorname{ctg}^3\,\alpha - 3\,\operatorname{ctg}\,\alpha}{3\,\operatorname{ctg}^2\,\alpha - 1}.

Формулы палавіннага вугла: [правіць]

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

Формулы сумы і рознасці функцый: [правіць]

 \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},
 \operatorname{tg} \alpha \pm \operatorname{tg} \beta = \frac{\sin (\alpha \pm \beta)}{\cos \alpha \cos \beta},
 1 \pm \sin {2 \alpha} = (\sin \alpha \pm \cos \alpha)^2.

Здабыткаў функцый: [правіць]

\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: [правіць]

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