Trigonometric functions.

Definition 4.2.1       

1. $\cos z = \frac 12 (e^{iz}+e^{-iz})$ .
2. $\sin z = \frac {1}{2i} (e^{iz}-e^{-iz})$ .
3. $\tan z = \frac {\sin z}{\cos z}$ .

Example 4.2.2  

$$\begin{cases}\cos \; i = \frac 12 (e^{i^2}+e^{-i^2}) = \frac ... ...i^2}) = \frac {1}{2i} ( e^{-1} - e) \approx 1.1752i \end{cases}$$

Proposition 4.2.3       

1. $\forall z \in \mathbb{C}, \cos^2 z + \sin^2 z =1$ .
2. $\forall z_1, z_2 \in \mathbb{C}, \sin (z_1 \pm z_2) = \sin z_1 \cos z_2 \pm \cos z_1 \sin z_2$
3. $\forall z_1, z_2 \in \mathbb{C}, \cos (z_1 \pm z_2) = \cos z_1 \cos z_2 \mp \sin z_1 \sin z_2$
4. $$\begin{cases}\frac{d}{dz} (\sin z) = \cos z \ \frac{d}{dz} (\cos z) = - \sin z \end{cases}$$
5. $\forall z \in \mathbb{C}, \begin{cases}\cos (z + 2 \pi) = \cos z \ \sin (z + 2 \pi) = \sin z \end{cases}.$

Example 4.2.4  

Proposition 4.2.5       

1. $\forall z \in \mathbb{C}-\{ \pi / 2 + k \pi \}, \; \tan (z+ \pi )= \tan z$ .
2. $\forall z_1,z_2 \in \mathbb{C}-\{ \pi / 2 + k \pi \}, \; \tan (z_1 \pm z_2) = \frac {\tan z_1 \pm \tan z_2}{1 \mp \tan z_1 \tan z_2}$ .
3. $\forall z \in \mathbb{C}-\{ \pi / 2 + k \pi \}, \; \tan (2z)= \frac {2 \tan z}{1- \tan ^2 z}$ .

Example 4.2.6