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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，${\displaystyle \alpha =\frac{\varphi +\theta }{2}}$，${\displaystyle \beta =\frac{\varphi -\theta }{2}}$  として代入すると

${\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，${\displaystyle \alpha =\frac{\varphi -\theta }{2}}$，${\displaystyle \beta =\frac{\varphi +\theta }{2}}$  として代入すると

${\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，${\displaystyle \alpha =\frac{\varphi +\theta }{2}}$，${\displaystyle \beta =\frac{\varphi -\theta }{2}}$  として代入すると

${\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，${\displaystyle \alpha =\frac{\varphi +\theta }{2}}$，${\displaystyle \beta =\frac{\varphi -\theta }{2}}$  として代入すると

${\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．

 

ホーム>>カテゴリー分類>>三角関数>>和積の公式

最終更新日： 2023年3月3日

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