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Chapter 5: Problem 1

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

Short Answer

Expert verified
The angle within the principal range \([0, \pi]\) whose cosine is \(\frac{1}{2}\) is \(\frac{\pi}{3}\) radians. Therefore, \(\cos^{-1} \frac{1}{2} = \frac{\pi}{3}\).

Step by step solution

01

Determine the principal range

The principal range for the inverse cosine function (arccos) is \([0, \pi]\). Therefore, we should find an angle within this interval that has a cosine of \(\frac{1}{2}\).
02

Identify the angle

Recall the unit circle or the 30-60-90 right triangle to identify at which angle the cosine is equal to \(\frac{1}{2}\). In this case, the angle is \(60掳\) or \(\frac{\pi}{3}\) radians.
03

Write the result

The result is the angle in radians which is: \(\cos^{-1} \frac{1}{2} = \frac{\pi}{3}\)

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