Chapter 4.3, Problem 56E

(a)

To determine

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

Answer to Problem 56E

The value of $\cos \theta$ is $\frac{1}{5}$ .

Explanation of Solution

Given Information:

The value of $\sec \theta =5$ and $\tan \theta =2\sqrt{6}$ .

Formula Used:

Trigonometric identity $\cos \theta =\frac{1}{\sec \theta }$ .

Calculation:

Substitute $5$ for $\sec \theta$ in above identity.

  $\begin{array}{c}\cos \theta =\frac{1}{\sec \theta }\\ =\frac{1}{5}\end{array}$

Therefore the value of $\cos \theta$ is $\frac{1}{5}$ .

(b)

To determine

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

Answer to Problem 56E

The value of $\cot \theta$ is $\frac{\sqrt{6}}{12}$ .

Explanation of Solution

Given Information:

The value of $\sec \theta =5$ and $\tan \theta =2\sqrt{6}$ .

Formula Used:

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

Calculation:

Substitute $2\sqrt{6}$ for $\tan \theta$ in above identity.

  $\begin{array}{c}\cot \theta =\frac{1}{\tan \theta }\\ =\frac{1}{2\sqrt{6}}\\ =\frac{\sqrt{6}}{12}\end{array}$

Therefore the value of $\csc \theta$ is $\frac{1}{3}$ .

(c)

To determine

The value of $\cot \left(90°-\theta \right)$ with the help of the trigonometric identities.

Answer to Problem 56E

The value of $\cot \left(90°-\theta \right)$ is $2\sqrt{6}$ .

Explanation of Solution

Given Information:

The value of $\sec \theta =5$ and $\tan \theta =2\sqrt{6}$ .

Formula Used:

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

Calculation:

Substitute $2\sqrt{6}$ for $\tan \theta$ in above identity.

  $\begin{array}{c}\cot \left(90°-\theta \right)=\tan \theta \\ =2\sqrt{6}\end{array}$

Therefore the value of $\cot \left(90°-\theta \right)$ is $2\sqrt{6}$ .

(d)

To determine

The value of $\sin \theta$ with the help of the trigonometric identities.

Answer to Problem 56E

The value of $\sin \theta$ is $\frac{2\sqrt{6}}{5}$ .

Explanation of Solution

Given Information:

The value of $\sec \theta =5$ and $\tan \theta =2\sqrt{6}$ .

Formula Used:

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

Calculation:

Substitute $2\sqrt{6}$ for $\tan \theta$ and $5$ for $\sec \theta$ in above identity.

  $\begin{array}{l}\sin \theta =\frac{\tan \theta }{\sec \theta }\\ =\frac{2\sqrt{6}}{5}\end{array}$

Therefore the value of $\sin \theta$ is $\frac{2\sqrt{6}}{5}$ .