和積の公式

$3$\displaystyle{\sin {\phi}+\sin {\theta}=2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ⇒公式の導出
$3$\displaystyle{\sin {\phi}- \sin {\theta}=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ⇒公式の導出
$3$\displaystyle{\cos {\phi}+\cos {\theta}=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ⇒公式の導出
$3$\displaystyle{\cos {\phi}- \cos {\theta}=- 2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ⇒公式の導出

■公式の導出

● $3$\displaystyle{\sin {\phi}+\sin {\theta}}$ $3$\displaystyle{=2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出

積和の公式 $3$\displaystyle{\sin {\alpha}\cos {\beta}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ \sin $ {\alpha}+{\beta} $ +\sin $ {\alpha}- {\beta} $ \} }$ に， $3$\displaystyle{{\alpha}=\frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ， $3$\displaystyle{{\beta}=\frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ として代入すると

$3$\displaystyle{\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \left{{ \sin $ \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }$ $3$\displaystyle{ \left.{ +\sin $ \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }\right\} }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ \sin {\phi}+\sin {\theta} $ }$

したがって

$3$\displaystyle{\sin {\phi}+\sin {\theta}}$ $3$\displaystyle{=2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$

となる．

● $3$\displaystyle{\sin {\phi}- \sin {\theta}}$ $3$\displaystyle{=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出

積和の公式 $3$\displaystyle{\sin {\alpha}\cos {\beta}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ \sin $ {\alpha}+{\beta} $ +\sin $ {\alpha}- {\beta} $ \} }$ に， $3$\displaystyle{{\alpha}=\frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ， $3$\displaystyle{{\beta}=\frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ として代入すると

$3$\displaystyle{\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \left{{ \sin $ \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }$ $3$\displaystyle{ \left.{ +\sin $ \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }\right\} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ \sin {\phi}+\sin $ - {\theta} $ \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ \sin {\phi}- \sin {\theta} $ }$

したがって

$3$\displaystyle{\sin {\phi}- \sin {\theta}}$ $3$\displaystyle{=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$

となる．

● $3$\displaystyle{\cos {\phi}+\cos {\theta}}$ $3$\displaystyle{=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出

積和の公式 $3$\displaystyle{\cos {\alpha}\cos {\beta}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ \cos $ {\alpha}+{\beta} $ +\cos $ {\alpha}- {\beta} $ \} }$ に， $3$\displaystyle{{\alpha}=\frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ， $3$\displaystyle{{\beta}=\frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ として代入すると

$3$\displaystyle{\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \left{{ \cos $ \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }$ $3$\displaystyle{ \left.{ +\cos $ \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }\right\} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ \cos {\phi}+\cos {\theta} $ }$

したがって

$3$\displaystyle{\cos {\phi}+\cos {\theta}}$ $3$\displaystyle{=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$

となる．

● $3$\displaystyle{\cos {\phi}- \cos {\theta}}$ $3$\displaystyle{=- 2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出

積和の公式 $3$\displaystyle{\sin {\alpha}\sin {\beta}}$ $3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \{ \cos $ {\alpha}+{\beta} $ - \cos $ {\alpha}- {\beta} $ \} }$ に， $3$\displaystyle{{\alpha}=\frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ ， $3$\displaystyle{{\beta}=\frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ として代入すると

$3$\displaystyle{\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ $3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \left{{ \cos $ \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}+\frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }$ $3$\displaystyle{ \left.{ - \cos $ \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}} $ }\right\} }$

$3$\displaystyle{=- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ \cos {\phi}- \cos {\theta} $ }$

したがって

$3$\displaystyle{\cos {\phi}- \cos {\theta}}$ $3$\displaystyle{=- 2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$

となる．

ホーム>>カテゴリー分類>>三角関数>>和積の公式

最終更新日： 2026年2月10日

和積の公式

■公式の導出

● の公式の導出

● の公式の導出

● の公式の導出

● の公式の導出

● $3$\displaystyle{\sin {\phi}+\sin {\theta}}$ $3$\displaystyle{=2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出

● $3$\displaystyle{\sin {\phi}- \sin {\theta}}$ $3$\displaystyle{=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出

● $3$\displaystyle{\cos {\phi}+\cos {\theta}}$ $3$\displaystyle{=2\cos \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\cos \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出

● $3$\displaystyle{\cos {\phi}- \cos {\theta}}$ $3$\displaystyle{=- 2\sin \frac{\hspace{2} {\phi}+{\theta} \hspace{2}}{\hspace{2}2\hspace{2}}\sin \frac{\hspace{2} {\phi}- {\theta} \hspace{2}}{\hspace{2}2\hspace{2}}}$ の公式の導出