Discussion
Can \( \sin z \) exceed one in modulus?
Answer: A
On the imaginary axis, with \( z = iy \),
\[ \sin(iy) \;=\; i\sinh y, \]
whose modulus grows without bound as \( y \) does. The familiar bound of one holds only on the real axis; in the complex plane the trigonometric and hyperbolic functions are the same family seen in two directions
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