回転行列と三角凾数の加法定理

角度$\alpha$と角度$\beta$の回転行列を掛ける

$$\begin{array}{rcl}\left(\begin{array}{cc}cos(\alpha )& -sin(\alpha )\\ sin(\alpha )& cos(\alpha )\end{array}\right)\left(\begin{array}{cc}cos(\beta )& -sin(\beta )\\ sin(\beta )& cos(\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)& =& \left(\begin{array}{cc}cos(\alpha )cos(\beta )-sin(\alpha )sin(\beta )& -cos(\alpha )sin(\beta )-sin(\alpha )cos(\beta )\\ sin(\alpha )cos(\beta )+cos(\alpha )sin(\beta )& -sin(\alpha )sin(\beta )+cos(\alpha )cos(\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =& \left(\begin{array}{cc}cos(\alpha )cos(\beta )-sin(\alpha )sin(\beta )& -(sin(\alpha )cos(\beta )+cos(\alpha )sin(\beta ))\\ sin(\alpha )cos(\beta )+cos(\alpha )sin(\beta )& -sin(\alpha )sin(\beta )+cos(\alpha )cos(\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =& \left(\begin{array}{cc}cos(\alpha +\beta )& -sin(\alpha +\beta )\\ sin(\alpha +\beta )& cos(\alpha +\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\dots \mathrm{こ}\mathrm{れ}\mathrm{が}\mathrm{角}\mathrm{度}(\alpha +\beta )\mathrm{の}\mathrm{回}\mathrm{転}\mathrm{行}\mathrm{列}\mathrm{と}\mathrm{等}\mathrm{し}\mathrm{い}．\end{array}$$ 対応する要素同士が加法定理となる． $$\begin{array}{rcl}cos(\alpha +\beta )& =& cos(\alpha )cos(\beta )-sin(\alpha )sin(\beta )\\ sin(\alpha +\beta )& =& sin(\alpha )cos(\beta )+cos(\alpha )sin(\beta )\end{array}$$

角度$\alpha$と角度$-\beta$の回転行列を掛ける

$$\begin{array}{rcl}\left(\begin{array}{cc}cos(\alpha )& -sin(\alpha )\\ sin(\alpha )& cos(\alpha )\end{array}\right)\left(\begin{array}{cc}cos(-\beta )& -sin(-\beta )\\ sin(-\beta )& cos(-\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)& =& \left(\begin{array}{cc}cos(\alpha )cos(-\beta )-sin(\alpha )sin(-\beta )& -cos(\alpha )sin(-\beta )-sin(\alpha )cos(-\beta )\\ sin(\alpha )cos(-\beta )+cos(\alpha )sin(-\beta )& -sin(\alpha )sin(-\beta )+cos(\alpha )cos(-\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =& \left(\begin{array}{cc}cos(\alpha )cos(\beta )+sin(\alpha )sin(\beta )& cos(\alpha )sin(\beta )-sin(\alpha )cos(\beta )\\ sin(\alpha )cos(\beta )-cos(\alpha )sin(\beta )& sin(\alpha )sin(\beta )+cos(\alpha )cos(\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\dots cos(-\beta )=cos(\beta ),sin(-\beta )=-sin(\beta )\mathrm{を}\mathrm{適}\mathrm{用}．\\ & =& \left(\begin{array}{cc}cos(\alpha )cos(\beta )+sin(\alpha )sin(\beta )& -(sin(\alpha )cos(\beta )-cos(\alpha )sin(\beta ))\\ sin(\alpha )cos(\beta )-cos(\alpha )sin(\beta )& cos(\alpha )cos(\beta )+sin(\alpha )sin(\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =& \left(\begin{array}{cc}cos(\alpha -\beta )& -sin(\alpha -\beta )\\ sin(\alpha -\beta )& cos(\alpha -\beta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\dots \mathrm{こ}\mathrm{れ}\mathrm{が}\mathrm{角}\mathrm{度}(\alpha -\beta )\mathrm{の}\mathrm{回}\mathrm{転}\mathrm{行}\mathrm{列}\mathrm{と}\mathrm{等}\mathrm{し}\mathrm{い}．\end{array}$$ $$\begin{array}{rcl}cos(\alpha -\beta )& =& cos(\alpha )cos(\beta )+sin(\alpha )sin(\beta )\\ sin(\alpha -\beta )& =& sin(\alpha )cos(\beta )-cos(\alpha )sin(\beta )\end{array}$$

角度$\alpha$と角度$\beta$が一致($\alpha =\beta =\theta$)

$$\begin{array}{rcl}\left(\begin{array}{cc}cos(\theta )& -sin(\theta )\\ sin(\theta )& cos(\theta )\end{array}\right)\left(\begin{array}{cc}cos(\theta )& -sin(\theta )\\ sin(\theta )& cos(\theta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)& =& \left(\begin{array}{cc}cos(\theta )cos(\theta )-sin(\theta )sin(\theta )& -cos(\theta )sin(\theta )-sin(\theta )cos(\theta )\\ sin(\theta )cos(\theta )+cos(\theta )sin(\theta )& -sin(\theta )sin(\theta )+cos(\theta )cos(\theta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =& \left(\begin{array}{cc}cos(\theta {)}^2-sin(\theta {)}^2& -2sin(\theta )cos(\theta )\\ 2sin(\theta )cos(\theta )& cos(\theta {)}^2-sin(\theta {)}^2\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\\ & =& \left(\begin{array}{cc}cos(2\theta )& -sin(2\theta )\\ sin(2\theta )& cos(2\theta )\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\dots \mathrm{こ}\mathrm{れ}\mathrm{が}\mathrm{角}\mathrm{度}(2\theta )\mathrm{の}\mathrm{回}\mathrm{転}\mathrm{行}\mathrm{列}\mathrm{と}\mathrm{等}\mathrm{し}\mathrm{い}．\end{array}$$ 対応する要素同士が倍角公式となる． $$\begin{array}{rcl}cos(2\theta )& =& cos(\theta {)}^2-sin(\theta {)}^2\\ sin(2\theta )& =& 2sin(\theta )cos(\theta )\end{array}$$