三角比・三角関数を総まとめ！定義や各種公式【重要記事一覧】

\(\angle \mathrm{C} = 90^\circ\) の直角三角形 \(\triangle{\mathrm{ABC}}\) において、

• \(\sin \theta = \displaystyle \frac{\mathrm{AC}}{\mathrm{AB}}\)
• \(\cos \theta = \displaystyle \frac{\mathrm{BC}}{\mathrm{AB}}\)
• \(\tan \theta = \displaystyle \frac{\mathrm{AC}}{\mathrm{BC}}\)

\(x\) 軸の正の部分を始線とした一般角 \(\theta\) の動径と、単位円との交点を \(\mathrm{P} = (x, y)\) とすると、

• \(\sin \theta = y\)
• \(\cos \theta = x\)
• \(\tan \theta = \displaystyle \frac{y}{x}\)

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

• 度数 → 弧度（ラジアン）
  \begin{align}\theta \ (^\circ) = \theta \times \displaystyle \frac{\pi}{180} \ \text{(rad)}\end{align}
• 弧度（ラジアン） → 度数
  \begin{align}\theta \ \text{(rad)} = \theta \times \displaystyle \frac{180}{\pi} \ (^\circ)\end{align}

• 度数
  \begin{align}\sin (90^\circ − \theta) &= \cos \theta \\ \cos (90^\circ − \theta) &= \sin \theta \\ \tan (90^\circ − \theta) &= \frac{1}{\tan \theta}\end{align}
• 弧度 (ラジアン)
  \begin{align}\sin \left( \frac{\pi}{2} − \theta \right) &= \cos \theta \\ \cos \left( \frac{\pi}{2} − \theta \right) &= \sin \theta \\ \tan \left( \frac{\pi}{2} − \theta \right) &= \frac{1}{\tan \theta}\end{align}

• 度数
  \begin{align}\sin (90^\circ + \theta) &= \cos \theta \\ \cos (90^\circ + \theta) &= −\sin \theta \\ \tan (90^\circ + \theta) &= −\frac{1}{\tan \theta}\end{align}
• 弧度 (ラジアン)
  \begin{align}\sin \left( \frac{\pi}{2} + \theta \right) &= \cos \theta \\ \cos \left( \frac{\pi}{2} + \theta \right) &= −\sin \theta \\ \tan \left( \frac{\pi}{2} + \theta \right) &= −\frac{1}{\tan \theta}\end{align}

• 度数
  \begin{align}\sin (180^\circ − \theta) &= \sin \theta \\ \cos (180^\circ − \theta) &= −\cos \theta \\ \tan (180^\circ − \theta) &= −\tan \theta\end{align}
• 弧度 (ラジアン)
  \begin{align}\sin (\pi − \theta) &= \sin \theta \\ \cos (\pi − \theta) &= −\cos \theta \\ \tan (\pi − \theta) &= −\tan \theta\end{align}

• 度数
  \begin{align}\sin (180^\circ + \theta) &= −\sin \theta \\ \cos (180^\circ + \theta) &= −\cos \theta \\ \tan (180^\circ + \theta) &= \tan \theta\end{align}
• 弧度 (ラジアン)
  \begin{align}\sin (\pi + \theta) &= −\sin \theta \\ \cos (\pi + \theta) &= −\cos \theta \\ \tan (\pi + \theta) &= \tan \theta\end{align}

• 度数
  \begin{align}\sin (− \theta) &= −\sin \theta \\ \cos (− \theta) &= \cos \theta \\ \tan (− \theta) &= −\tan \theta\end{align}
• 弧度 (ラジアン)
  \begin{align}\sin (− \theta) &= −\sin \theta \\ \cos (− \theta) &= \cos \theta \\ \tan (− \theta) &= −\tan \theta\end{align}

\(\triangle \mathrm{ABC}\) と、その外接円（半径 \(r\)）について、

\begin{align}\frac{a}{\sin \mathrm{A}} = \frac{b}{\sin \mathrm{B}} = \frac{c}{\sin \mathrm{C}} = 2R\end{align}

\(\triangle \mathrm{ABC}\) について、

\begin{align}a^2 = b^2 + c^2 − 2bc \cos \mathrm{A}\end{align}

\begin{align}b^2 = c^2 + a^2 − 2ca \cos \mathrm{B}\end{align}

\begin{align}c^2 = a^2 + b^2 − 2ab \cos \mathrm{C}\end{align}

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

（複合同順）

• \(\sin2\theta = 2\sin\theta\cos\theta\)
• \(\begin{align} \cos2\theta &= \cos^2\theta − \sin^2\theta \\ &= 1 − 2\sin^2\theta \\ &= 2\cos^2\theta − 1 \end{align}\)
• \(\displaystyle \tan2\theta = \frac{2\tan\theta}{1 − \tan^2\theta}\)

• \(\sin3\theta = 3\sin\theta − 4\sin^3\theta\)
• \(\cos3\theta = −3\cos\theta + 4\cos^3\theta\)

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

（見切れる場合は横へスクロール）

（見切れる場合は横へスクロール）

\(a \neq 0\), \(b \neq 0\) のとき、

\begin{align}a \sin\theta + b \cos\theta = \sqrt{a^2 + b^2} \sin(\theta + \alpha)\end{align}

ただし、

\begin{align}\displaystyle \cos\alpha = \frac{a}{\sqrt{a^2 + b^2}}, \displaystyle \sin\alpha = \frac{b}{\sqrt{a^2 + b^2}}\end{align}

• \((\sin x)’ = \cos x\)
• \((\cos x)’ = −\sin x\)
• \(\displaystyle (\tan x)’ = \frac{1}{\cos^2 x}\)
• \(\displaystyle \left(\frac{1}{\tan x}\right)’ = −\frac{1}{\sin^2 x}\)

積分定数を \(C\) とする。

• \(\displaystyle \int \sin x \ dx = −\cos x + C\)
• \(\displaystyle \int \cos x \ dx = \sin x + C\)
• \(\displaystyle \int \tan x \ dx = −\log |\cos x| + C\)
• \(\displaystyle \int \frac{1}{\cos^2 x} \ dx = \tan x + C\)
• \(\displaystyle \int \frac{1}{\sin^2 x} \ dx = −\frac{1}{\tan x} + C\)