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Chapter 4: Problem 15

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

Short Answer

Expert verified
The value of \(\arccos \left(-\frac{1}{2}\right)\) is \(\frac{2\pi}{3}\).

Step by step solution

01

Understand the property of arccosine function

The arccosine function, denoted as \(\arccos(x)\), is the inverse operation of the cosine function. It returns the arc (or angle) whose cosine is \(x\). The range of the \(\arccos\) function is \([0, \pi]\), which means the output of this function is always between \(0\) and \(\pi\).
02

Recall the special angles in a unit circle

In a unit circle, there are a few special angles at which we know the cosine values without using a calculator, like \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and \(\frac{2\pi}{3}\). Precisely, the cosine of \(\frac{\pi}{3}\) is \(\frac{1}{2}\) and the cosine of \(\frac{2\pi}{3}\) is \(-\frac{1}{2}\).
03

Apply the arccosine function

Given \(\arccos\left(-\frac{1}{2}\right)\), since the cosine of \(\frac{2\pi}{3}\) equals to \(-\frac{1}{2}\), it means the \(\arccos\) of \(-\frac{1}{2}\) equals to \(\frac{2\pi}{3}\).

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Most popular questions from this chapter

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