Exercise · ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-004 · ★☆☆ Basic
Which is a period of \(\sin z\) on \(\mathbb{C}\)?
Choose one option.
Show answer
- C. \(2\pi\)
Explanation
Since \(e^{i(z+2\pi)}=e^{iz}\) and \(e^{-i(z+2\pi)}=e^{-iz}\), \(\sin(z+2\pi)=\sin z\). With \(2\pi i\) it does not repeat: \(e^{i(z+2\pi i)}=e^{-2\pi}e^{iz}\).
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-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)|.
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-009 What are all the zeros of sin z in ℂ?
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