Question

Find x and y

cos x + i sin x = cosh(y -1) + ixy

Hint: Sketch the cosine and cosh functions

Answer 1

We start with solving

cos x = cosh(y-1)                [eq.1]

and

sin x = xy                             [eq.2]

plot of cosh(y-1)

plot of cos(x)

This shows that maximum value of cos x is equal to 1 at x=2nπ while the minimum value of cosh(y-1) is equal to 1 at y=1. Therefore we get the solution x=2nπ and y=1 from eq.1

While Solving eq.2, we use y=1 and hence the eq.2 becomes

sin x = x and the plot of it is given below:

This gives us the solution x=0 which is a subset of the solution x=2nπ from solving eq.1.

Hence the final solution is x=0 and y=1.