problem 5 consider fz sin z 1 show that maps is one to one and onto fr

Question

Problem 5: Consider f(z) = sin z. (1) Show that ƒ maps is one-to-one and onto from the rectangular region -T/2 < x < T/2,0 < y < b to thesemi-elliptical region shown in the following picture. (2) In the picture, A' = f(A),B' = f(B) and so on. Find C' (i.e. find f(C)).
(3) If L is the line segment y = c2,-T/2 < x <T/2, find its image L' (i.e. find f(L)). (Hint: From the picture,we see that L' is the top half of an ellipse.)