【数学Ⅱ】三角関数の公式まとめ（加法定理・変換・合成）

・\( \displaystyle \color{red}{ \tan \theta = \frac{\sin \theta}{\cos \theta} } \)

・\( \displaystyle \color{red}{ \sin^2 \theta + \cos^2 \theta = 1 } \)

・\( \displaystyle \color{red}{ 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta} } \)

【正弦の加法定理】

・\( \color{red}{ \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta } \)

・\( \color{red}{ \sin (\alpha – \beta) = \sin \alpha \cos \beta – \cos \alpha \sin \beta } \)

【余弦の加法定理】

・\( \color{red}{ \cos (\alpha + \beta) = \cos \alpha \cos \beta – \sin \alpha \sin \beta } \)

・\( \color{red}{ \cos (\alpha – \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta  } \)

【正接の加法定理】

・\( \color{red}{ \displaystyle \tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta }{1 – \tan \alpha \tan \beta}  } \)

・\( \color{red}{ \displaystyle \tan (\alpha – \beta) = \frac{\tan \alpha – \tan \beta }{1 + \tan \alpha \tan \beta}  } \)

• \( \color{red}{ \sin 2 \alpha = 2 \sin \alpha \cos \alpha } \)
• \( \color{red}{ \begin{align}
  \cos 2 \alpha & = \cos^2 \alpha – \sin^2 \alpha \\
  & = 2 \cos^2 \alpha – 1 \\
  & = 1 – 2 \sin^2 \alpha
  \end{align} } \)
• \( \color{red}{ \displaystyle \tan 2 \alpha = \frac{2 \tan \alpha}{1 – \tan^2 \alpha} } \)

・\( \displaystyle \color{red}{ \sin^2 \frac{\alpha}{2} = \frac{1 – \cos \alpha}{2} } \)

・\( \displaystyle \color{red}{ \cos^2 \frac{\alpha}{2} = \frac{1 + \cos \alpha}{2} } \)

・\( \displaystyle \color{red}{ \tan^2 \frac{\alpha}{2} = \frac{1 – \cos \alpha}{1 + \cos \alpha } } \)

・\( \color{red}{ \sin 3 \alpha = 3 \sin \alpha – 4 \sin^3 \alpha } \)

・\( \color{red}{ \cos 3 \alpha = -3 \cos \alpha + 4 \cos^3 \alpha } \)

\( \displaystyle ・ \color{red}{ \sin \alpha \cos \beta= \frac{1}{2} \left\{ \sin (\alpha + \beta) + \sin (\alpha – \beta) \right\} } \)

\( \displaystyle ・ \color{red}{ \cos \alpha \sin \beta= \frac{1}{2} \left\{ \sin (\alpha + \beta) – \sin (\alpha – \beta) \right\} } \)

\( \displaystyle ・ \color{red}{ \cos \alpha \cos \beta= \frac{1}{2} \left\{ \cos (\alpha + \beta) + \cos (\alpha – \beta) \right\} } \)

\( \displaystyle ・ \color{red}{ \sin \alpha \sin \beta= – \frac{1}{2} \left\{ \cos (\alpha + \beta) – \cos (\alpha – \beta) \right\} } \)

\( \displaystyle ・ \color{red}{ \sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2} } \)

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

\( \displaystyle ・ \color{red}{ \cos A + \cos B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2} } \)

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

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

ただし，\( \alpha \) は \( \displaystyle \cos \alpha = \frac{a}{\sqrt{a^2 + b^2}}, \ \sin \alpha = \frac{b}{\sqrt{a^2 + b^2}} \) を満たす角度。

\( \displaystyle \color{red}{ a \sin \theta + b \cos \theta = \sqrt{a^2 + b^2} \cos ( \theta \ – \beta ) } \)

ただし，\( \beta \) は \( \displaystyle \sin \beta = \frac{a}{\sqrt{a^2 + b^2}}, \ \cos \beta = \frac{b}{\sqrt{a^2 + b^2}} \) を満たす角度。