$ \mathbb{C} $
Referring [@taketo1024](http://www.twitter.com/taketo1024)'s blog
# Trigonometric function
## Definition
Euler's Formula
$$
\begin{eqnarray}
e^{-ix} &=& \cos{(-x)} + i \sin{(-x)} \\
&=& \cos{x} - i \sin{x}
\end{eqnarray}
$$
leads to
$$
\begin{cases}
e^{ix} + e^{-ix} = 2\cos{x} \\\
e^{ix} - e^{-ix} = 2i \sin{x}
\end{cases}
\Leftrightarrow
\begin{cases}
\cos{x} = \frac{1}{2}(e^{ix} + e^{-ix}) \\\
\sin{x} = \frac{1}{2i}(e^{ix} - e^{-ix})
\end{cases}
\tag{3}
$$
then just extend x $ \mapsto $ z .
This is the definition of $ \sin(z) $ , $ \cos(z) $
$$
\begin{eqnarray}
\begin{cases}
\cos{z} : &=& \frac{1}{2}(e^{iz} + e^{-iz}) \\\
\sin{z} : &=& \frac{1}{2i}(e^{iz} - e^{-iz})
\end{cases}
\end{eqnarray}
$$
## visualization
$ w = \cos{z} $
![](/image/complex-trigonometric/complex-trigonometric-01.png width="600")
$ w = \sin{z} $
![](/image/complex-trigonometric/complex-trigonometric-02.png width="600")
## Euler's Formula in Complex Number
$$ \cos{z} + i \sin{z} = e^{iz} $$
$$ e^{iz} = e^{-y}(\cos x + i \sin x) $$
![](/image/complex-trigonometric/complex-trigonometric-03.gif width="300")
when $z = t + ci$ where $c$ is constant,
e.g. $\cos z$ is red, $\sin z$ is blue and $e^{iz}$ is green.
if $c = 0$, then like this
![](/image/complex-trigonometric/complex-trigonometric-04.gif width="300")
## 加法定理
the same with Real Number