和積の公式

• ${\displaystyle sin\varphi +sin\theta =2sin\frac{\varphi +\theta }{2}cos\frac{\varphi -\theta }{2}}$
• 　　⇒公式の導出

• ${\displaystyle sin\varphi -sin\theta =2cos\frac{\varphi +\theta }{2}sin\frac{\varphi -\theta }{2}}$
• 　　⇒公式の導出

• ${\displaystyle cos\varphi +cos\theta =2cos\frac{\varphi +\theta }{2}cos\frac{\varphi -\theta }{2}}$
• 　　⇒公式の導出

• ${\displaystyle cos\varphi -cos\theta =-2sin\frac{\varphi +\theta }{2}sin\frac{\varphi -\theta }{2}}$
• 　　⇒公式の導出

■公式の導出

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

積和の公式 ${\displaystyle sin\alpha cos\beta }$${\displaystyle =\frac{1}{2}\left\{sin\left(\alpha +\beta \right)+sin\left(\alpha -\beta \right)\right\}}$ において，${\displaystyle \alpha =\frac{\varphi +\theta }{2}}$，${\displaystyle \beta =\frac{\varphi -\theta }{2}}$  とおいて，代入すると，

• ${\displaystyle sin\frac{\varphi +\theta }{2}cos\frac{\varphi -\theta }{2}}$
• ${\displaystyle =\frac{1}{2}\left\{sin\left(\frac{\varphi +\theta }{2}+\frac{\varphi -\theta }{2}\right)\right.}$
• ${\displaystyle \left.+sin\left(\frac{\varphi +\theta }{2}-\frac{\varphi -\theta }{2}\right)\right\}}$

${\displaystyle =\frac{1}{2}\left(sin\varphi +sin\theta \right)}$

したがって，

${\displaystyle sin\varphi +sin\theta }$${\displaystyle =2sin\frac{\varphi +\theta }{2}cos\frac{\varphi -\theta }{2}}$  

となる．

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

積和の公式 ${\displaystyle sin\alpha cos\beta }$${\displaystyle =\frac{1}{2}\left\{sin\left(\alpha +\beta \right)+sin\left(\alpha -\beta \right)\right\}}$ において，${\displaystyle \alpha =\frac{\varphi -\theta }{2}}$，${\displaystyle \beta =\frac{\varphi +\theta }{2}}$  とおいて，代入すると，

• ${\displaystyle sin\frac{\varphi -\theta }{2}cos\frac{\varphi +\theta }{2}}$
• ${\displaystyle =\frac{1}{2}\left\{sin\left(\frac{\varphi -\theta }{2}+\frac{\varphi +\theta }{2}\right)\right.}$
• ${\displaystyle \left.+sin\left(\frac{\varphi -\theta }{2}-\frac{\varphi +\theta }{2}\right)\right\}}$

${\displaystyle =\frac{1}{2}\left\{sin\varphi +sin\left(-\theta \right)\right\}}$ 

${\displaystyle =\frac{1}{2}\left(sin\varphi -sin\theta \right)}$ 

したがって，

${\displaystyle sin\varphi -sin\theta }$${\displaystyle =2cos\frac{\varphi +\theta }{2}sin\frac{\varphi -\theta }{2}}$  

となる．

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

積和の公式 ${\displaystyle cos\alpha cos\beta }$${\displaystyle =\frac{1}{2}\left\{cos\left(\alpha +\beta \right)+cos\left(\alpha -\beta \right)\right\}}$ において，${\displaystyle \alpha =\frac{\varphi +\theta }{2}}$，${\displaystyle \beta =\frac{\varphi -\theta }{2}}$  とおいて，代入すると，

• ${\displaystyle cos\frac{\varphi +\theta }{2}cos\frac{\varphi -\theta }{2}}$
• ${\displaystyle =\frac{1}{2}\left\{cos\left(\frac{\varphi +\theta }{2}+\frac{\varphi -\theta }{2}\right)\right.}$
• ${\displaystyle \left.+cos\left(\frac{\varphi +\theta }{2}-\frac{\varphi -\theta }{2}\right)\right\}}$

${\displaystyle =\frac{1}{2}\left(cos\varphi +cos\theta \right)}$ 

したがって，

${\displaystyle cos\varphi +cos\theta }$${\displaystyle =2cos\frac{\varphi +\theta }{2}cos\frac{\varphi -\theta }{2}}$  

となる．

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

積和の公式 ${\displaystyle sin\alpha sin\beta }$${\displaystyle =-\frac{1}{2}\left\{cos\left(\alpha +\beta \right)-cos\left(\alpha -\beta \right)\right\}}$ において，${\displaystyle \alpha =\frac{\varphi +\theta }{2}}$，${\displaystyle \beta =\frac{\varphi -\theta }{2}}$  とおいて，代入すると，

• ${\displaystyle sin\frac{\varphi +\theta }{2}sin\frac{\varphi -\theta }{2}}$
• ${\displaystyle =-\frac{1}{2}\left\{cos\left(\frac{\varphi +\theta }{2}+\frac{\varphi -\theta }{2}\right)\right.}$
• ${\displaystyle \left.-cos\left(\frac{\varphi +\theta }{2}-\frac{\varphi -\theta }{2}\right)\right\}}$

${\displaystyle =-\frac{1}{2}\left(cos\varphi -cos\theta \right)}$

したがって，

${\displaystyle cos\varphi -cos\theta }$${\displaystyle =-2sin\frac{\varphi +\theta }{2}sin\frac{\varphi -\theta }{2}}$  

となる．

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最終更新日： 2022年6月30日