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Chapter 5: Problem 8

Evaluate \(\cos ^{-1}\left(\cos \frac{1}{2}\right)\).

Short Answer

Expert verified
Evaluating the given expression, we find that \(\cos\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2}\). Since \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\), the inverse cosine function evaluates to \(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}\). Thus, \(\cos^{-1}\left(\cos \frac{1}{2}\right) = \frac{\pi}{6}\).

Step by step solution

01

Understand the inverse cosine function

First, we need to recall that the inverse cosine function, denoted as \(\cos^{-1}(x)\), represents the angle whose cosine is equal to \(x\). It's important to note that the range of the inverse cosine function is \([0, \pi]\). In other words, it only outputs angles in this interval.
02

Evaluate the cosine function within the expression

In our expression, we have \(\cos\left(\frac{1}{2}\right)\). Calculating the cosine of \(\frac{1}{2}\): \[\cos\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2}\]
03

Evaluate the inverse cosine function

Now we have to replace the cosine function within our expression with its result: \[\cos^{-1}\left(\cos\frac{1}{2}\right)=\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\] Since we know that the cosine of an angle \(\theta\) in the range \([0, \pi]\) is equal to \(\frac{\sqrt{3}}{2}\) when \(\theta = \frac{\pi}{6}\), we can evaluate the inverse cosine function: \[\cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}\]
04

Write the final solution

Combining all the steps, we find the solution for the given expression: \[\cos^{-1}\left(\cos \frac{1}{2}\right) = \frac{\pi}{6}\]

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