Chapter 4.3, Problem 70E

(a)

To determine

To find: The value of the angle $\csc \theta =\frac{2\sqrt{3}}{3}$ in degrees and in radians.

Answer to Problem 70E

The value of the angle $\csc \theta =\frac{2\sqrt{3}}{3}$ in degrees and in radians are $60°$ and $\frac{\pi }{3}\text{rad}$ .

Explanation of Solution

Given information:

The given angle is $\csc \theta =\frac{2\sqrt{3}}{3}$

Calculation:

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

  $\begin{array}{c}\csc \theta =\frac{2\sqrt{3}}{3}\\ \theta ={\csc }^{-1}\left(\frac{2\sqrt{3}}{3}\right)\therefore \left(\csc \theta =\frac{1}{\sin \theta }\right)\\ \theta =60°\end{array}$

Convert the value of $\theta$ into radians.

  $\begin{array}{c}\theta =60\mathrm{deg}\left(\frac{\pi \text{rad}}{180\mathrm{deg}}\right)\\ =\frac{\pi }{3}\text{rad}\end{array}$

Therefore, the value of the angle $\csc \theta =\frac{2\sqrt{3}}{3}$ in degrees and in radians are $60°$ and $\frac{\pi }{3}\text{rad}$ .

(b)

To determine

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

Answer to Problem 70E

The value of the angle $\sin \theta =\frac{1}{2}$ in degrees and in radians are $30°$ and $\frac{\pi }{6}\text{rad}$ .

Explanation of Solution

Given information:

The given angle is $\sin \theta =\frac{1}{2}$ .

Calculation:

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

  $\begin{array}{c}\sin \theta =\frac{1}{2}\\ \theta ={\sin }^{-1}\left(\frac{1}{2}\right)\\ \theta =30°\end{array}$

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 $\sin \theta =\frac{1}{2}$ in degrees and in radians are $30°$ and $\frac{\pi }{6}\text{rad}$ .