Exercise · ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-009 · ★★★ Advanced
What are all the zeros of \(\sin z\) in \(\mathbb{C}\)?
Choose one option.
Show answer
- A. Only \(z=k\pi\) with \(k\in\mathbb{Z}\)
Explanation
\(\sin z=0\) is equivalent to \(e^{2iz}=1\), that is \(2iz=2\pi i k\) and \(z=k\pi\). No new zeros appear off the real axis.
Review the concept: Complex trigonometric functions
The sine \(\sin z\) and cosine \(\cos z\) of a complex number \(z\) are the functions that extend the real ones through the complex exponential \(e^{iz}\).
\[\sin z=\frac{e^{iz}-e^{-iz}}{2i},\qquad \cos z=\frac{e^{iz}+e^{-iz}}{2}\]
More practice problems: complex trigonometric functions
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-001 Compute cos (i).
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-002 For real y, what is sin (iy) equal to?
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-003 What is sin ^2 z+cos ^2 z for any z∈ℂ?
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-004 Which is a period of sin z on ℂ?
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-005 Simplify sin (z+π).
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-006 Compute sin \!((π)/(2)+iln 2).
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-007 For positive real y, which value of y satisfies cos (iy)=2?
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-008 Compute |sin (iln 2)|.
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