角度 ( \( \theta \) ) と 複素角 ( \( i\theta \) ) の中を取り持つのが \(e\) つまり \(\exp\left(\ \right) \) です。
\[ \begin{flalign} \ \ \ \ \ \ \cos\left(\theta\right)=\frac{\ e^{i\theta}+e^{-i\theta}\ }{2}&& \end{flalign} \] \[ \begin{flalign} \ \ \ \ \ \ \sin\left(\theta\right)=\frac{\ e^{i\theta}-e^{-i\theta}\ }{2i}&& \end{flalign} \] \[ \begin{flalign} \ \ \ \ \ \ \cosh\left(\theta\right)=\frac{\ e^{\theta}+e^{-\theta}\ }{2}&& \end{flalign} \] \[ \begin{flalign} \ \ \ \ \ \ \sinh\left(\theta\right)=\frac{\ e^{\theta}-e^{-\theta}\ }{2}&& \end{flalign} \] \[ \begin{flalign} \ \ \ \ \ \ \exp\left(i\theta\right)=\cos\left(\theta\right)+i\sin\left(\theta\right)&& \end{flalign} \] \[ \begin{flalign} \ \ \ \ \ \ \exp\left(\theta\right)=\cos\left(i\theta\right)+\sin\left(i\theta\right)&& \end{flalign} \]
Desmosコピペ用テキストエリア（ Ctrl＋v で貼り付け ）：

\cos\left(\theta\right)=\frac{\ e^{i\theta}+e^{-i\theta}\ }{2} \sin\left(\theta\right)=\frac{\ e^{i\theta}-e^{-i\theta}\ }{2i} \cosh\left(\theta\right)=\frac{\ e^{\theta}+e^{-\theta}\ }{2} \sin\left(\theta\right)=\frac{\ e^{\theta}-e^{-\theta}\ }{2} \exp\left(i\theta\right)=\cos\left(\theta\right)+i\sin\left(\theta\right) \exp\left(\theta\right)=\cos\left(i\theta\right)+\sin\left(i\theta\right)