Complex Analysis - Trigonometric Functions

Question

I do not understand the first question because, I do not know how to find the $ \sin (L) $ because $L$ is a line, that is $L = \{ - : x + iy_0 ; x \in \mathbb{R} \}$. Regarding the second question, I do not know which is the maximum. Any help ?, I'm grabbing two books quite applied, and I can not find any way to solve those two exercises.

1. If $L$ is a line parallel to the real axis (respectively imaginary), then $\sin(L)=?$.
2. Calculate $\max \{|\cos(z)|\}=?$ when $0 \leq x \leq 2 \pi$ and $0 \leq y \leq 2 \pi$.

Answer 1

Hints: 1. $\sin(L) $ is a set ! $\sin(L)=\{\sin(z): z \in L\}$.

2. Let $R=\{x+iy: 0 \le x \le 2 \pi, 0 \le y \le 2 \pi \}$. Since $\cos(z)$ is holomorphic, the Maximum Principle gives

$\max \{|\cos(z)| :z \in R\}= \max \{|\cos(z)| :z \in \partial R\}$.