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Chapter 5: Problem 5

Evaluate the expression without using a calculator. \(\arccos \frac{1}{2}\)

Short Answer

Expert verified
\(\arccos \frac{1}{2} = \frac{\pi}{3}\)

Step by step solution

01

Recall the Definition of \(\arccos\)

The \(\arccos\) function, or inverse cosine function, gives the arc (i.e., the angle) whose cosine is \(\frac{1}{2}\). This implies that we are looking for an angle \(\theta\) where \(cos(\theta) = \frac{1}{2}\).
02

Recognize the Necessary Values from the Unit Circle

From knowledge of the unit circle, we recognize that cosine equals \(\frac{1}{2}\) at two standard angles: \(\frac{\pi}{3}\) and \(-\frac{\pi}{3}\).
03

Apply the Range of \(\arccos\)

However, the range of the \(\arccos\) function is \([0, \pi]\). Therefore, the angle cannot be negative. Thus, the only acceptable answer is \(\theta = \frac{\pi}{3}\)

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Most popular questions from this chapter

In Exercises 106鈥�108, verify the differentiation formula. $$ \frac{d}{d x}\left[\sinh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}+1}} $$

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