三角関数の公式

三角関数の定義

三角関数の相互関係

\( -1 ≦ \sin\theta ≦ 1，-1 ≦ \cos\theta ≦ 1 \)
\[ \tan\theta = \frac{\sin\theta}{\cos\theta}，\sin^2\theta + \cos^2\theta = 1，1 + \tan^2\theta = \frac{1}{\cos^2\theta} \]

負角の公式

• \( \sin(-\theta) = -\sin\theta \)
• \( \cos(-\theta) = \cos\theta \)
• \( \tan(-\theta) = -\tan\theta \)

余角の公式

• \( \sin(\pi \pm \theta) = \mp \sin\theta \)
• \( \cos(\pi \pm \theta) = -\cos\theta \)
• \( \tan(\pi \pm \theta) = \pm \tan\theta \)

補角の公式

• \( \displaystyle\sin\left(\frac{\pi}{2} \pm \theta\right) = \cos\theta \)
• \( \displaystyle\cos\left(\frac{\pi}{2} \pm \theta\right) = \mp \sin\theta \)
• \( \displaystyle\tan\left(\frac{\pi}{2} \pm \theta\right) = \mp \frac{1}{\tan\theta} \)

加法定理

• \( \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta \)
• \( \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \)
• \( \displaystyle\tan(\alpha\pm\beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta} \)

2倍角の公式

• \( \sin 2\theta = 2\sin\theta\cos\theta \)
• \( \cos 2\theta = \cos^2\theta – \sin^2\theta
  = 1 – 2\sin^2\theta
  = 2\cos^2\theta -1 \)
• \( \displaystyle\tan 2\theta = \frac{2\tan\theta}{1 – \tan^2\theta} \)

半角の公式

• \( \displaystyle\sin^2\frac{\theta}{2} = \frac{1 – \cos\theta}{2} \)
• \( \displaystyle\cos^2\frac{\theta}{2} = \frac{1 + \cos\theta}{2} \)
• \( \displaystyle\tan^2\frac{\theta}{2} = \frac{1 – \cos\theta}{1 + \cos\theta} \)

3倍角の公式

• \( \sin 3\theta = 3\sin\theta – 4\sin^3\theta \)
• \( \cos 3\theta = -3\cos\theta + 4\cos^3\theta \)

積 → 和の公式

• \( \displaystyle\sin\alpha\cos\beta = \frac{1}{2}{\sin(\alpha + \beta ) + \sin(\alpha – \beta )} \)
• \( \displaystyle\sin\alpha\sin\beta = -\frac{1}{2}{\cos(\alpha + \beta ) – \sin(\alpha – \beta )} \)
• \( \displaystyle\cos\alpha\cos\beta = \frac{1}{2}{\cos(\alpha + \beta ) + \cos(\alpha – \beta )} \)

和 → 積の公式

• \( \displaystyle\sin A + \sin B = 2\sin\frac{A + B}{2}\cos\frac{A – B}{2} \)
• \( \displaystyle\sin A – \sin B = 2\cos\frac{A + B}{2}\sin\frac{A – B}{2} \)
• \( \displaystyle\cos A + \cos B = 2\cos\frac{A + B}{2}\cos\frac{A – B}{2} \)
• \( \displaystyle\cos A – \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2} \)

三角関数の合成

\[ a\sin\theta + b\cos\theta = \sqrt{a^2+b^2}\sin(\theta +\alpha) \]

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