Mapping of a complex function from z-plane to w-plane

I have been tasked to find the image of a set $$U=\{z \in \mathbb{C} \mid \frac{-\pi}{2} \lt \Re(z) \lt \frac{\pi}{2} \}$$ under the function $$f(z)=\sin(z)$$ which I have been asked to do so answering a series of sub-questions that go as follows:

(a): What is the image of the line segment $$L_1=(\frac{-\pi}{2}, \frac{\pi}{2})$$ (the real axis) under $$f$$?

(b): What is the image of the Imaginary Axis $$L_2=\{iy \mid y \in \mathbb{R} \}$$ under $$f$$?

(c): What is the image of the vertical line $$L_3=\{\frac{-\pi}{2}+ iy \mid y \in \mathbb{R}\}$$ under $$f$$?

(d): What is the image of the vertical line $$L_3=\{\frac{\pi}{2}+iy \mid y \in \mathbb{R} \}$$ under $$f$$?

(e): Given your observations in the previous steps what do you guess the image of the set $$U$$ is under $$f$$?

I am done figuring out the image of the invidual parts from (a)-(d) but ultimately cannot combine the results to judge the image of the set $$U$$ under $$f$$.

For parts (a)-(d), I got the following results:

(a): $$(-1, 1)$$

(b): $$\{\iota \sinh(y) \mid y \in \mathbb{R}\}$$

(c): $$\{-\cosh(y) \mid y \in \mathbb{R} \}$$

(d): $$\{\cosh(y) \mid y \in \mathbb{R}\}$$

How do I combine all the information obtained above into finding the image of the set $$U$$ under $$f$$. I also replaced $$z$$ with $$x+\iota y$$ and substituted it in the definition formula of the complex sine function and the addition formula and obtained the equivalent of $$\sin(x+\iota y)$$ as $$\sin(x)\cosh(y)+\iota \cos(x)\sinh(y)$$. How can I restrict the $$x$$ i.e. the $$\Re(z)$$ to being in the set $$U$$ so that I can plug in that information in the obtained equivalent of the complex sine function to obtain something meaningful?

Thanks

• In particular, what's happening here is that the image $f(U)$ is the entire complex plane, except for "most of" the real axis; e.g. $\Bbb C-\{z\in\Bbb C : |\Re(z)|\geq 1\}$. [The part of the real axis that is included is the part calculated in (a), and parts (c) and (d) calculate the parts excluded.] – aleph_two Dec 19 '18 at 4:27