Chapter 4.3, Problem 71E

(a)

To determine

To find: The value of the angle $\tan \theta =\sqrt{3}$ in degrees and in radians.

Answer to Problem 71E

The value of the angle $\tan \theta =\sqrt{3}$ in degrees and in radians are $30°$ and $\frac{\pi }{6}\text{rad}$ .

Explanation of Solution

Given information:

The given angle is $\tan \theta =\sqrt{3}$

Calculation:

Use the trigonometric identities to calculate the value of $\theta$ .

  $\begin{array}{c}\tan \theta =\sqrt{3}\\ \theta ={\tan }^{-1}\left(\sqrt{3}\right)\\ \theta =30°\end{array}$

Convert the value of $\theta$ into radians.

  $\begin{array}{c}\theta =30\mathrm{deg}\left(\frac{\pi \text{rad}}{180\mathrm{deg}}\right)\\ =\frac{\pi }{6}\text{rad}\end{array}$

Therefore, the value of the angle $\tan \theta =\sqrt{3}$ in degrees and in radians are $30°$ and $\frac{\pi }{6}\text{rad}$ .

(b)

To determine

The value of the angle $\sin \theta =\frac{1}{2}$ in degrees and in radians.

Answer to Problem 71E

The value of the angle $\cos \theta =\frac{\sqrt{2}}{2}$ in degrees and in radians are $45°$ and $\frac{\pi }{4}\text{rad}$ .

Explanation of Solution

Given information:

The given angle is $\cos \theta =\frac{\sqrt{2}}{2}$ .

Calculation:

Use the trigonometric identities to calculate the value of $\theta$ .

  $\begin{array}{c}\cos \theta =\frac{\sqrt{2}}{2}\\ \theta ={\cos }^{-1}\left(\frac{\sqrt{2}}{2}\right)\\ \theta =45°\end{array}$

Convert the value of $\theta$ into radians.

  $\begin{array}{c}\theta =45\mathrm{deg}\left(\frac{\pi \text{rad}}{180\mathrm{deg}}\right)\\ =\frac{\pi }{4}\text{rad}\end{array}$

Therefore, the value of the angle $\cos \theta =\frac{\sqrt{2}}{2}$ in degrees and in radians are $45°$ and $\frac{\pi }{4}\text{rad}$ .