Chapter 3.6, Problem 1E

  $\left(a\right)$

To determine

To calculate: The value of trigonometric expression ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$ .

Answer to Problem 1E

The value of trigonometric expression ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is $\frac{\pi }{3}$ .

Explanation of Solution

Given information:

The trigonometric expression ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$ .

Formula used:

Inverse sine trigonometric function is expressed as ${\sin }^{-1}\left(x\right)=y$ which is equivalent to $x=\sin y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Calculation:

Consider trigonometric expression ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$ .

Recall that the inverse sine trigonometric function is expressed as ${\sin }^{-1}\left(x\right)=y$ which is equivalent to $x=\sin y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Recall that value of sine function at an angle $\frac{\pi }{3}$ is $\frac{\sqrt{3}}{2}$ .

That is, $\sin \left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}$ .

Take sine function to the right hand side,

  ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi }{3}$ .

Thus, the value of trigonometric expression ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is $\frac{\pi }{3}$ .

  $\left(b\right)$

To determine

To calculate: The value of trigonometric expression ${\cos }^{-1}\left(-1\right)$ .

Answer to Problem 1E

The value of trigonometric expression ${\cos }^{-1}\left(-1\right)$ is $\pi$ .

Explanation of Solution

Given information:

The trigonometric expression ${\cos }^{-1}\left(-1\right)$ .

Formula used:

Inverse cosine trigonometric function is expressed as ${\cos }^{-1}\left(x\right)=y$ which is equivalent to $x=\cos y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Calculation:

Consider trigonometric expression ${\cos }^{-1}\left(-1\right)$ .

Recall that the inverse cosine trigonometric function is expressed as ${\cos }^{-1}\left(x\right)=y$ which is equivalent to $x=\cos y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Recall that value of cosine function at an angle $\pi$ is $-1$ .

That is, $\cos \left(\pi \right)=-1$ .

Take cosine function to the right hand side,

  ${\cos }^{-1}\left(-1\right)=\pi$ .

Thus, the value of trigonometric expression ${\cos }^{-1}\left(-1\right)$ is $\pi$ .