I'm not asking for a full solution or answer. Please give me a direction in which to get the right answer.
I found in a university book that if a complex-valued function $f$ is holomorphic, then (as $r$ increases) the value of $M_f(r)$ increases monotonically. I will not prove this statement, but just use it for our complex-valued functions $e^{\sin z},$ $e^{\cos z}$, $\sin z,$ $\cos z,$ which are obviously holomorphic.
This is easy to find the maximum modulus of the functions $\cos z$ and $e^{\cos z}$ as a function of $r$ (modulus of a complex-valued number $z$). But solving a similar example for $e^{\sin z}$ becomes much more difficult. Here is my attempt to solve this problem (it did not lead to the final answer), you can see it in the two attached photos: 1st image 2nd image
I will also give backup links to these 2 images on google drive:
1st - drive.google.com/file/d/1kChJ1cX0TOcR9d7U_ufVm6_HyyImxtYm/view
2nd - drive.google.com/file/d/1T6wey4qsyRhbA1wX2HjGMV6gcEtcHN2v/view