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Chapter 5: Problem 9

Evaluate \(\cos ^{-1}(\cos 3 \pi)\).

Short Answer

Expert verified
The short answer is: \(\cos^{-1}(\cos(3\pi)) = \cos^{-1}(-1) = \boxed{\pi}\).

Step by step solution

01

Evaluate \(\cos(3\pi)\)

First, we need to find the value of \(\cos(3\pi)\). Since \(\cos(x)\) has a period of \(2\pi\), we know that \(\cos(x + 2n\pi) = \cos(x)\) for any integer \(n\). In this case, we can consider \(3\pi = \pi + 2\pi\), so we get \(\cos(3\pi) = \cos(\pi)\). Recall that \(\cos(\pi) = -1\). Therefore, \(\cos(3\pi) = -1\). Now we have to find the angle \(\theta\) such that \(\cos^{-1}(-1)=\theta\).
02

Find the angle \(\theta\) using inverse cosine function

To find the angle \(\theta\) that satisfies \(\cos(\theta) = -1\), we must recall the range of the inverse cosine function, which is \(0 \le \theta \le \pi\). Considering this range, there is only one angle whose cosine is equal to -1, and that is \(\theta = \pi\). So, \(\cos^{-1}(\cos(3\pi)) = \cos^{-1}(-1) = \pi\). Therefore, the final answer is \(\boxed{\pi}\).

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