I'm new into using the complex maximum principle so I'm stuck calculating the maximum of the function $f(z)$= $|\sin z|$ over the set $[0,1] \times [-1,1]$.
In which points or point the maximum holds. My line of thought goes as the following:
We can write
$$\sin(z) = \sin(x+iy) = \sin x \cos iy + \sin iy \cos x = \sin x \cosh y + i \sinh y \cos x.$$
Then we calculate the square norm so we get:
$$|\sin(z)|^2 = \sin^2 x \cosh^2 y + i \sinh^2 y \cos^2 x$$
And I know that because the complex maximum principle the supremum of $|\sin(z)|$ is attained on the boundary of the rectangle $[0,1] \times [-1,1]$.
Can someone please help me filling up the gaps in the calculations and proof so I can understand how to properly use the maximum principle over this function?? Thanks