Chapter 12, Problem 11CR

To determine

To find: The value of ${\cos }^{-1}\left(-0.5\right)$ .

Answer to Problem 11CR

The value of ${\cos }^{-1}\left(-0.5\right)={\cos }^{-1}\left(-0.5\right)=\frac{2\pi }{3}$ .

Explanation of Solution

Given information:

The given equation is ${\cos }^{-1}\left(-0.5\right)$ .

Calculation:

Calculate the equation,

  $\theta ={\cos }^{-1}\left(-0.5\right)$ .we have to look for the angle $\theta ,\theta \le \theta \le \pi$ ,whose equals $-0.5$ .

Determine the angle $\theta$ whose cosine equals $-0.5$ for $0\le \theta \le \pi$ .

Table is given below of some angles and their cosine values.

  

From the table, we see that the angle $\theta$ for which the cosine equals $-0.5$ with in the interval $\left[0,\pi \right]$ is $\frac{2\pi }{3}$ .

Since $\cos \frac{2\pi }{3}=-0.5$ , and $\frac{2\pi }{3}$ is in the interval $0\le \theta \le \pi$ , we get ${\cos }^{-1}\left(-0.5\right)=\frac{2\pi }{3}$ .

Therefore, the value of ${\cos }^{-1}\left(-0.5\right)={\cos }^{-1}\left(-0.5\right)=\frac{2\pi }{3}$ .