Chapter 4.3, Problem 52E

(a)

To determine

The value of $\csc 30°$ with the help of the trigonometric identities.

Answer to Problem 52E

The value of $\csc 30°$ is $2$ .

Explanation of Solution

Given Information:

The value of $\sin 30°=\frac{1}{2}$ and $\tan 30°=\frac{\sqrt{3}}{3}$ .

Formula Used:

Trigonometric identity $\csc \theta =\frac{1}{\sin \theta }$

Calculation:

Substitute $30°$ for $\theta$ in the identity $\csc \theta =\frac{1}{\sin \theta }$ .

  $\csc 30°=\frac{1}{\sin 30°}$

Substitute $\frac{1}{2}$ for $\sin 30°$ in above identity.

  $\begin{array}{c}\csc 30°=\frac{1}{\sin 30°}\\ =\frac{1}{\frac{1}{2}}\\ =2\end{array}$

Therefore, the value of $\csc 30°$ is $2$ .

(b)

To determine

The value of $\cot 60°$ with the help of the trigonometric identities.

Answer to Problem 52E

The value of $\cot 60°$ is $\frac{\sqrt{3}}{3}$ .

Explanation of Solution

Given Information:

The value of $\sin 30°=\frac{1}{2}$ and $\tan 30°=\frac{\sqrt{3}}{3}$ .

Formula Used:

Trigonometric identity $\tan \left(90°-\theta \right)=\cot \theta$

Calculation:

Substitute $60°$ for $\theta$ in the identity $\tan \left(90°-\theta \right)=\cot \theta$ .

  $\tan \left(90°-60°\right)=\cot 60°$ , therefore $\cot 60°=\tan 30°$

Substitute $\frac{\sqrt{3}}{3}$ for $\tan 30°$ in above identity.

  $\cot 60°=\frac{\sqrt{3}}{3}$

Therefore the value of $\cot 60°$ is $\frac{\sqrt{3}}{3}$ .

(c)

To determine

The value of $\cos 30°$ with the help of the trigonometric identities.

Answer to Problem 52E

The value of $\cos 30°$ is $\frac{\sqrt{3}}{2}$ .

Explanation of Solution

Given Information:

The value of $\sin 30°=\frac{1}{2}$ and $\tan 30°=\frac{\sqrt{3}}{3}$ .

Formula Used:

Trigonometric identity $\cos \theta =\frac{\sin \theta }{\tan \theta }$

Calculation:

Substitute $30°$ for $\theta$ in the identity $\cos \theta =\frac{\sin \theta }{\tan \theta }$ .

  $\cos 30°=\frac{\sin 30°}{\tan 30°}$

Substitute $\frac{1}{2}$ for $\sin 30°$ and $\frac{\sqrt{3}}{3}$ for $\tan 30°$ in above identity.

  $\begin{array}{c}\cos 30°=\frac{\sin 30°}{\tan 30°}\\ =\frac{\frac{1}{2}}{\frac{\sqrt{3}}{3}}\\ =\frac{\sqrt{3}}{2}\end{array}$

Therefore value of $\cos 30°$ is $\frac{\sqrt{3}}{2}$ .

(d)

To determine

The value of $\cot 30°$ with the help of the trigonometric identities.

Answer to Problem 52E

The value of $\cot 30°$ is $\sqrt{3}$ .

Explanation of Solution

Given Information:

The value of $\sin 30°=\frac{1}{2}$ and $\tan 30°=\frac{\sqrt{3}}{3}$ .

Formula Used:

Trigonometric identity $\cot \theta =\frac{1}{\tan \theta }$

Calculation:

Substitute $30°$ for $\theta$ in the identity $\cot \theta =\frac{1}{\tan \theta }$ .

  $\cot 30°=\frac{1}{\tan 30°}$

Substitute $\frac{\sqrt{3}}{3}$ for $\tan 30°$ in above identity.

  $\begin{array}{c}\cot 30°=\frac{1}{\tan 30°}\\ =\frac{1}{\frac{\sqrt{3}}{3}}\\ =\sqrt{3}\end{array}$

Therefore value of $\cot 30°$ is $\sqrt{3}$ .