【数学II】三角関数〜基本公式集・例題一覧〜

半径１の円の弧の長さに対する角度　→　弧度法

\(180^\circ = \pi\)ラジアン

1. \(\tan\theta=\displaystyle\frac{\sin\theta}{\cos\theta}\)
2. \(\sin^2{\theta}+\cos^2{\theta}=1\)
3. \(1+\tan^2{\theta}=\displaystyle\frac{1}{\cos^2{\theta}}\)

• \(\sin(\theta + 2n\pi) = \sin\theta\)
• \(\cos(\theta + 2n\pi) = \cos\theta\)
• \(\tan(\theta + n\pi) = \tan\theta\)  

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

※\(\displaystyle\frac{\pi}{2}\)関連の加減　→　変わる

\(\sin ⇒ \cos\)
\(\cos ⇒ \sin\)
\(\tan ⇒ \displaystyle\frac{1}{\tan}\)

※\(\pi\)関連の加減　→　変わらない

\(\sin ⇒ \sin\)
\(\cos ⇒ \cos\)
\(\tan ⇒ \tan \)

と覚える

周期：\(2\pi\), 原点対称

周期：\(2\pi\), \(y\)軸対称

周期：\(\pi\), 原点対称

1. \(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\)
2. \(\sin(\alpha – \beta) = \sin\alpha\cos\beta – \cos\alpha\sin\beta\)
3. \(\cos(\alpha + \beta) = \cos\alpha\cos\beta – \sin\alpha\sin\beta\)
4. \(\cos(\alpha – \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\)
5. \(\tan(\alpha+\beta) = \displaystyle\frac{\tan\alpha + \tan\beta}{1 – \tan\alpha \tan\beta}\)
6. \(\tan(\alpha – \beta) = \displaystyle\frac{\tan\alpha – \tan\beta}{1 + \tan\alpha \tan\beta}\)

①\(\sin2\theta = 2\sin\theta\cos\theta\)

②\(\cos2\theta = \cos^2\theta – \sin^2\theta\)
\(= 1 – 2\sin^2\theta\)
\(=2\cos^2\theta – 1\)

③\(\tan2\theta = \displaystyle\frac{2\tan\theta}{1 – \tan^2\theta}\)

①\(\sin^2\theta = \displaystyle\frac{1 – \cos2\theta}{2}\)

②\(\cos^2\theta = \displaystyle\frac{1 + \cos2\theta}{2}\)

③\(\tan^2\theta = \displaystyle\frac{1 – \cos2\theta}{1 + \cos2\theta }\)

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

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

③\(\tan^2\displaystyle\frac{\theta}{2} = \displaystyle\frac{1 – \cos\theta}{1 + \cos\theta }\)

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

※ただし、

\(\sin\alpha = \displaystyle\frac{b}{\sqrt{a^2 + b^2}}\)

\(\cos\alpha = \displaystyle\frac{a}{\sqrt{a^2 + b^2}}\)