Chapter 7, Problem 24RE

To determine

To calculate: The exact value of the expression ${\cos }^{-1}\left[\cos \left(\frac{7\pi }{6}\right)\right]$ .

Answer to Problem 24RE

The exact value of the expression ${\cos }^{-1}\left[\cos \left(\frac{7\pi }{6}\right)\right]$ is $\frac{5\pi }{6}$ .

Explanation of Solution

Given information:

The expression is ${\cos }^{-1}\left[\cos \left(\frac{7\pi }{6}\right)\right]$ .

Formula used:

For the cosine function and its inverse, the following holds:

  $\begin{array}{c}{f}^{-1}\left(f\left(x\right)\right)={\cos }^{-1}\left(\cos \left(x\right)\right)\\ =x\end{array}$

Where, $0\le x\le \pi$

Addition or subtraction of integer multiple of $2\pi$ to $\theta$ , the value of the cosine function remain

unchanged.

That is, for all $\theta$

  $\cos \left(2k\pi \pm \theta \right)=\cos \theta$

Where $k$ is any integer.

Calculation:

Consider the expression ${\cos }^{-1}\left[\cos \left(\frac{7\pi }{6}\right)\right]$ .

Recall that addition or subtraction of integer multiple of $2\pi$ to $\theta$ , the value of the cosine

function remain unchanged. That is, for all $\theta$

  $\cos \left(2k\pi \pm \theta \right)=\cos \theta$

Where $k$ is any integer.

Applied it,

  $\begin{array}{c}{\cos }^{-1}\left[\cos \left(\frac{7\pi }{6}\right)\right]={\cos }^{-1}\left[\cos \left(2\pi -\frac{5\pi }{6}\right)\right]\\ ={\cos }^{-1}\left[\cos \left(\frac{5\pi }{6}\right)\right]\end{array}$

Observe that the angle $\frac{5\pi }{6}$ is lies between $0$ and $\pi$ .

Recall that for the cosine function and its inverse, the following holds:

  $\begin{array}{c}{f}^{-1}\left(f\left(x\right)\right)={\cos }^{-1}\left(\cos \left(x\right)\right)\\ =x\end{array}$

Where, $0\le x\le \pi$

Simplify further,

  ${\cos }^{-1}\left[\cos \left(\frac{5\pi }{6}\right)\right]=\frac{5\pi }{6}$

Thus, the exact value of the expression ${\cos }^{-1}\left[\cos \left(\frac{7\pi }{6}\right)\right]$ is $\frac{5\pi }{6}$ .