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{displaymath}\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}\end{displaymath}    

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{displaymath}\begin{cases}\frac{d}{dz} (\sin z) = \cos z \ \frac{d}{dz}
(\cos z) = - \sin z \end{cases}\end{displaymath}
  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  



Noah Dana-Picard 2007-12-24