Chapter 4.3, Problem 39E

(a)

To determine

Value of $\sin \theta$ .

Answer to Problem 39E

The value of $\sin \theta$ is $\frac{2\sqrt{2}}{3}$

Explanation of Solution

Given info:

  $\cos \theta =\frac{1}{3}$

Formula used:

We have the results

  $\begin{array}{l}\sin \theta =\cos (90-\theta )\\ \cos \theta =\sin (90-\theta )\\ \cos ec\theta =\sec (90-\theta )\\ \sec \theta =\cos ec(90-\theta )\\ \tan \theta =\cot (90-\theta )\\ \cot \theta =\tan (90-\theta )\\ tan\theta =\frac{\sin \theta }{\cos \theta }\\ \cot \theta =\frac{\cos \theta }{\sin \theta }\\ cosec\theta =\frac{1}{\sin \theta }\\ \sec \theta =\frac{1}{\cos \theta }\\ si{n}^2\theta +{\cos }^2\theta =1\\ {\sec }^2\theta -{\tan }^2\theta =1\\ cose{c}^2\theta -{\cot }^2\theta =1\end{array}$

Calculation:

Here we need to find $\sin \theta$ using $\cos \theta =\frac{1}{3}$

We have

  $\begin{array}{l}si{n}^2\theta +{\cos }^2\theta =1\\ si{n}^2\theta +\frac{1}{9}=1\\ {\sin }^2\theta =\frac{8}{9}\\ \sin \theta =\frac{2\sqrt{2}}{3}\end{array}$

Value of $\sin \theta$ is $\frac{2\sqrt{2}}{3}$

Conclusion:

Thus,the value of $\sin \theta$ is $\frac{2\sqrt{2}}{3}$

(b)

To determine

Value of $\tan \theta$ .

Answer to Problem 39E

The value of $\tan \theta$ is $2\sqrt{2}$

Explanation of Solution

Given info:

  $\cos \theta =\frac{1}{3}$

Formula used:

We have the results

  $\begin{array}{l}\sin \theta =\cos (90-\theta )\\ \cos \theta =\sin (90-\theta )\\ \cos ec\theta =\sec (90-\theta )\\ \sec \theta =\cos ec(90-\theta )\\ \tan \theta =\cot (90-\theta )\\ \cot \theta =\tan (90-\theta )\\ tan\theta =\frac{\sin \theta }{\cos \theta }\\ \cot \theta =\frac{\cos \theta }{\sin \theta }\\ cosec\theta =\frac{1}{\sin \theta }\\ \sec \theta =\frac{1}{\cos \theta }\\ si{n}^2\theta +{\cos }^2\theta =1\\ {\sec }^2\theta -{\tan }^2\theta =1\\ cose{c}^2\theta -{\cot }^2\theta =1\end{array}$

Calculation:

Here we need to find $\tan \theta$ using $\cos \theta =\frac{1}{3}$

From part (a) we know $\sin \theta =\frac{2\sqrt{2}}{3}$

We have

  $\begin{array}{l}tan\theta =\frac{\sin \theta }{\cos \theta }\\ tan\theta =\frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}}=2\sqrt{2}\end{array}$

Value of $\tan \theta$ is $2\sqrt{2}$

Conclusion:

Thus,the value of $\tan \theta$ is $2\sqrt{2}$

(c)

To determine

Value of $\sec \theta$ .

Answer to Problem 39E

The value of $\sec \theta$ is $3$

Explanation of Solution

Given info:

  $\cos \theta =\frac{1}{3}$

Formula used:

We have the results

  $\begin{array}{l}\sin \theta =\cos (90-\theta )\\ \cos \theta =\sin (90-\theta )\\ \cos ec\theta =\sec (90-\theta )\\ \sec \theta =\cos ec(90-\theta )\\ \tan \theta =\cot (90-\theta )\\ \cot \theta =\tan (90-\theta )\\ tan\theta =\frac{\sin \theta }{\cos \theta }\\ \cot \theta =\frac{\cos \theta }{\sin \theta }\\ cosec\theta =\frac{1}{\sin \theta }\\ \sec \theta =\frac{1}{\cos \theta }\\ si{n}^2\theta +{\cos }^2\theta =1\\ {\sec }^2\theta -{\tan }^2\theta =1\\ cose{c}^2\theta -{\cot }^2\theta =1\end{array}$

Calculation:

Here we need to find $\sec \theta$ using $\cos \theta =\frac{1}{3}$

We have

  $\begin{array}{l}\sec \theta =\frac{1}{\cos \theta }\\ \sec \theta =\frac{1}{\frac{1}{3}}=3\end{array}$

Value of $\sec \theta$ is $3$

Conclusion:

Thus,the value of $\sec \theta$ is $3$

(d)

To determine

Value of $\csc (90-\theta )$ .

Answer to Problem 39E

The value of $\csc (90-\theta )$ is $3$

Explanation of Solution

Given info:

  $\cos \theta =\frac{1}{3}$

Formula used:

We have the results

  $\begin{array}{l}\sin \theta =\cos (90-\theta )\\ \cos \theta =\sin (90-\theta )\\ \cos ec\theta =\sec (90-\theta )\\ \sec \theta =\cos ec(90-\theta )\\ \tan \theta =\cot (90-\theta )\\ \cot \theta =\tan (90-\theta )\\ tan\theta =\frac{\sin \theta }{\cos \theta }\\ \cot \theta =\frac{\cos \theta }{\sin \theta }\\ cosec\theta =\frac{1}{\sin \theta }\\ \sec \theta =\frac{1}{\cos \theta }\\ si{n}^2\theta +{\cos }^2\theta =1\\ {\sec }^2\theta -{\tan }^2\theta =1\\ cose{c}^2\theta -{\cot }^2\theta =1\end{array}$

Calculation:

Here we need to find $\csc (90-\theta )$ using $\cos \theta =\frac{1}{3}$

We have

  $\begin{array}{l}\csc (90-\theta )=\sec \theta =\frac{1}{\cos \theta }\\ \csc (90-\theta )=\frac{1}{\frac{1}{3}}=3\end{array}$

Value of $\csc (90-\theta )$ is $3$

Conclusion:

Thus,the value of $\csc (90-\theta )$ is $3$