Chapter 5.1, Problem 17E

To determine

To calculate: The values of all six trigonometric functions.

Answer to Problem 17E

The required values of $\sin \theta$ is 0, $\cos \theta$ is $-1$ , $\tan \theta$ is 0, $\sec \theta$ is $-1$ and $\cot \theta$ is undefined.

Explanation of Solution

Given information:

The function: $\csc \theta$ is undefined, $\cos \theta <0$

Formula used:

Trigonometric identities:

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

  $\sec \theta =\frac{1}{\cos \theta }$

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

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

  ${\sin }^2\theta +{\cos }^2\theta =1$

  ${\sec }^2\theta =1+{\tan }^2\theta$

  $\cot \theta =\frac{\cos \theta }{\sin \theta }$

Calculation:

The given function is $\csc \theta$ is undefined.

Since, $\csc \theta$ is undefined.

So, $\sin \theta =0$

Use the ${\sin }^2\theta +{\cos }^2\theta =1$ to find the value of $\cos \theta$ .

  $\begin{array}{c}{\cos }^2\theta =1-{\sin }^2\theta \\ \cos \theta =\sqrt{1-{\left(0\right)}^2}\\ \cos \theta =1\end{array}$

Since, $\cos \theta <0$ ;

So, $\cos \theta =-1$

Use the formula $\cot \theta =\frac{\cos \theta }{\sin \theta }$ to find the value of $\cot \theta$ .

  $\cot \theta =\frac{-1}{0}$

Thus, $\cot \theta$ is undefined.

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

Now, $\cot \theta$ is undefined; so, $\tan \theta =0$ .

Hence, the required values of $\sin \theta$ is 0, $\cos \theta$ is $-1$ , $\tan \theta$ is 0, $\sec \theta$ is $-1$ and $\cot \theta$ is undefined.