三角関数の公式

基本の公式

$\frac{\sin\theta}{\cos\theta} = \tan\theta$
$\sin^2\theta + \cos^2\theta = 1$
$\tan^2\theta + 1 = \frac{1}{\cos^2\theta}$

角度の変換公式

$\sin(-\theta) = -\sin\theta$	$\cos(-\theta) = \cos\theta$	$\tan(-\theta) = -\tan\theta$
$\sin(\frac{\pi}{2} + \theta) = \cos\theta$	$\cos(\frac{\pi}{2} + \theta) = -\sin\theta$	$\tan(\frac{\pi}{2} + \theta) = -\frac{1}{\tan\theta}$
$\sin(90^\circ + \theta) = \cos\theta$	$\cos(90^\circ + \theta) = -\sin\theta$	$\tan(90^\circ + \theta) = -\frac{1}{\tan\theta}$
$\sin(\frac{\pi}{2} - \theta) = \cos\theta$	$\cos(\frac{\pi}{2} - \theta) = \sin\theta$	$\tan(\frac{\pi}{2} - \theta) = \frac{1}{\tan\theta}$
$\sin(90^\circ - \theta) = \cos\theta$	$\cos(90^\circ - \theta) = \sin\theta$	$\tan(90^\circ - \theta) = \frac{1}{\tan\theta}$
$\sin(\pi + \theta) = -\sin\theta$	$\cos(\pi + \theta) = -\cos\theta$	$\tan(\pi + \theta) = \tan\theta$
$\sin(180^\circ + \theta) = -\sin\theta$	$\cos(180^\circ + \theta) = -\cos\theta$	$\tan(180^\circ + \theta) = \tan\theta$
$\sin(\pi - \theta) = \sin\theta$	$\cos(\pi - \theta) = -\cos\theta$	$\tan(\pi - \theta) = -\tan\theta$
$\sin(180^\circ - \theta) = \sin\theta$	$\cos(180^\circ - \theta) = -\cos\theta$	$\tan(180^\circ - \theta) = -\tan\theta$

正弦定理

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

$\frac{a}{\sin A} = \frac{b}{\sin B}$	$\frac{b}{\sin B} = \frac{c}{\sin C}$	$\frac{c}{\sin C} = \frac{a}{\sin A}$
$\frac{b}{\sin B} = \frac{a}{\sin A}$	$\frac{c}{\sin C} = \frac{b}{\sin B}$	$\frac{a}{\sin A} = \frac{c}{\sin C}$
$\frac{a}{\sin A} = 2R$	$\frac{b}{\sin B} = 2R$	$\frac{c}{\sin C} = 2R$

余弦定理

$a^2 = b^2 + c^2 = 2bc\cos A$	$b^2 = c^2 + a^2 = 2ca\cos B$	$c^2 = a^2 + b^2 = 2ab\cos C$
$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$	$\cos B = \frac{c^2 + a^2 - b^2}{2ca}$	$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

加法定理

$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$	$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$	$\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$
$\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$	$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$	$\tan(\alpha - \beta) = \frac{\tan\alpha + \tan\beta}{1 + \tan\alpha\tan\beta}$
$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$	$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$	$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$

2倍角の公式

$\sin2\theta = 2\sin\theta\cos\theta$

$\cos2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
$\cos2\theta = \cos^2\theta - \sin^2\theta$
$\cos2\theta = 2\cos^2\theta - 1$
$\cos2\theta = 1 - 2\sin^2\theta$

$\tan2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

3倍角の公式

$\sin3\theta = 3\sin\theta - 4\sin3\theta$

$\cos3\theta = 4\cos3\theta - 3\cos\theta$

半角の公式

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

$\cos^2\frac{\theta}{2} = \frac{1 + \cos\theta}{2}$

$\tan^2\frac{\theta}{2} = \frac{1 - \cos\theta}{1 + \cos\theta}$

和積の公式

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

積和の公式

$2\sin\alpha\cos\beta = \sin(\alpha + \beta) + \sin(\alpha - \beta)$
$2\cos\alpha\sin\beta = \sin(\alpha + \beta) - \sin(\alpha - \beta)$
$2\cos\alpha\cos\beta = \cos(\alpha + \beta) + \cos(\alpha - \beta)$
$2\sin\alpha\sin\beta = -\cos(\alpha + \beta) + \cos(\alpha - \beta)$

合成

$a\sin\theta+b\cos\theta=\sqrt{a^2+b^2}\sin(\theta+\alpha)$
ただし
$\sin\alpha=\frac{b}{\sqrt{a^2+b^2}},\cos\alpha=\frac{a}{\sqrt{a^2+b^2}}$

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最終更新：2010年10月29日 13:52

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