Trigonometric Functions | Sahithyan's Notes

For any complex number $z$ ,

\sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!}, \qquad \cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}

Exponential Form

$e^{iz} = \cos z + i\sin z$ for each $z \in \mathbb{C}$ , so

\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \qquad \cos z = \frac{e^{iz} + e^{-iz}}{2}

$\tan z$ , $\cot z$ , $\sec z$ , and $\csc z$ are defined from $\sin z$ and $\cos z$ as in the real case.

$\sin z$ and $\cos z$ are analytic on $\mathbb{C}$ .

\frac{\text{d}}{\text{d}z}\sin z = \cos z, \qquad \frac{\text{d}}{\text{d}z}\cos z = -\sin z

The other 4 functions are quotients, so each is analytic wherever its denominator is non-zero.

Function	Analytic on
$\tan z$ , $\sec z$	$\mathbb{C} \setminus \{\frac{\pi}{2} + n\pi\}$
$\cot z$ , $\csc z$	$\mathbb{C} \setminus \{n\pi\}$

Here $n \in \mathbb{Z}$ .

$\sin z = 0$ iff $z = n\pi$ , and $\cos z = 0$ iff $z = \frac{\pi}{2} + n\pi$ . All zeros are real.