Discussion
Which of the following sets is mapped onto the right half plane under the map \(f(z) = \sin z\) ?
Answer: A
The set is \(\{z : 0 \lt \mathrm{Re}(z) \le \pi/2\}\).
With \(z = x + iy\),
\[ \sin z = \sin x \cosh y + i \cos x \sinh y. \]
For \(0 \lt x \lt \pi/2\) the real part \(\sin x \cosh y\) is positive, and the half-strip maps one-to-one onto the right half plane except the ray \([1, \infty)\). That ray is the image of the edge \(x = \pi/2\), where \(\sin z = \cosh y \ge 1\). Including the edge \(\mathrm{Re}(z) = \pi/2\) therefore fills the right half plane exactly.
The full strip \(-\pi/2 \le \mathrm{Re}(z) \le \pi/2\) also covers the left half plane, and the horizontal strips in Im(z) contain points with \(\sin x \lt 0\).
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