Exercise · ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-007 · ★★☆ Intermediate
For positive real \(y\), which value of \(y\) satisfies \(\cos(iy)=2\)?
Choose one option.
Show answer
- D. \(\ln(2+\sqrt3)\)
Explanation
\(\cos(iy)=\cosh y=2\) gives \(e^{y}=2+\sqrt3\) (the root of \(t^2-4t+1=0\) with \(t>1\)). Although \(|\cos x|\le1\) on the real line, in the plane the cosine does reach the value 2.
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-008 Compute |sin (iln 2)|.
- ZW-COMPLEX-TRIGONOMETRIC-FUNCTIONS-009 What are all the zeros of sin z in ℂ?
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