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Chapter 5: Problem 10

Evaluate \(\sin ^{-1}\left(\sin \frac{9 \pi}{4}\right)\).

Short Answer

Expert verified
The given angle \(\frac{9 \pi}{4}\) is equivalent to \(\frac{\pi}{4}\) within the interval \([0, 2 \pi)\) since the sine function is periodic with period \(2 \pi\). Thus, we have \(\sin^{-1}(\sin \frac{9 \pi}{4}) = \sin^{-1}(\sin \frac{\pi}{4}) = \frac{\pi}{4}.\)

Step by step solution

01

Find the angle between 0 and \(2\pi\).

To find the equivalent angle between 0 and \(2\pi\), we need to find the remainder of the division by \(2\pi\). Calculate: \[\frac{9\pi}{4} \mod 2\pi.\]

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