Chapter 4.3, Problem 45E

(a)

To determine

To find: The value of given function $\tan \alpha$ using the trigonometric identities.

Answer to Problem 45E

The value of given function $\tan \alpha$ is $\frac{1}{5}$ .

Explanation of Solution

Given information:

The given function is $\cot \alpha =5$ .

Calculation:

Using the trigonometric identity.

  $\tan \alpha =\frac{1}{\cot \alpha }$

Calculate the value of $\tan \alpha$ .

  $\begin{array}{c}\tan \alpha =\frac{1}{\cot \alpha }\\ =\frac{1}{5}\end{array}$

Therefore, the value of given function $\tan \alpha$ is $\frac{1}{5}$ .

(b)

To determine

To find: The value of given function $\csc \alpha$ using the trigonometric identities.

Answer to Problem 45E

The value of given function $\csc \alpha$ is $\sqrt{26}$ .

Explanation of Solution

Given information:

The given function is $\cot \alpha =5$ .

Calculation:

Using the trigonometric identity.

  $1+{\cot }^2\alpha ={\csc }^2\alpha$

Calculate the value of $\tan \alpha$ .

  $\begin{array}{c}{\csc }^2\alpha =1+{\cot }^2\alpha \\ =1+5^2\\ =1+25\\ \csc \alpha =\sqrt{26}\end{array}$

Therefore, the value of given function $\csc \alpha$ is $\sqrt{26}$ .

(c)

To determine

To find: The value of given function $\cot \left(90°-\alpha \right)$ using the trigonometric identities.

Answer to Problem 45E

The value of given function $\cot \left(90°-\alpha \right)$ is $\frac{1}{5}$ .

Explanation of Solution

Given information:

The given function is $\cot \alpha =5$ .

Calculation:

Using the trigonometric identity.

  $\cot \left(90°-\alpha \right)=\tan \alpha$

Calculate the value of $\tan \alpha$ .

  $\begin{array}{c}\cot \left(90°-\alpha \right)=\tan \alpha \\ =\frac{1}{5}\end{array}$

Therefore, the value of given function $\cot \left(90°-\alpha \right)$ is $\frac{1}{5}$ .

(d)

To determine

To find: The value of given function $\cos \alpha$ using the trigonometric identities.

Answer to Problem 45E

The value of given function $\cos \alpha$ is $\frac{5\sqrt{26}}{26}$ .

Explanation of Solution

Given information:

The given function is $\cot \alpha =5$ .

Calculation:

Using the trigonometric identity.

  $\begin{array}{c}1+{\tan }^2\alpha ={\sec }^2\alpha \\ \cos \alpha =\frac{1}{\sec \alpha }\end{array}$

Calculate the value of $\sec \alpha$ .

  $\begin{array}{c}{\sec }^2\alpha =1+{\tan }^2\alpha \\ =1+{\left(\frac{1}{5}\right)}^2\\ =1+\frac{1}{25}\\ \sec \alpha =\sqrt{\frac{26}{25}}\end{array}$

Calculate the value of $\cos \alpha$ .

  $\begin{array}{c}\cos \alpha =\frac{1}{\sec \alpha }\\ =\frac{1}{\sqrt{\frac{26}{25}}}\\ =\frac{5\sqrt{26}}{26}\end{array}$

Therefore, the value of given function $\cos \alpha$ is $\frac{5\sqrt{26}}{26}$ .