Chapter 5.1, Problem 18E

To determine

To evaluate: the remaining trigonometric function of the angle.

Answer to Problem 18E

  $\sin \theta =1,\cos \theta =0,\sec \theta =\text{undefined},\csc \theta =\text{1},\cot \theta =0$ .

Explanation of Solution

Given information:

Given trigonometric function is

  $\tan \theta =\text{undefined},\sin \theta >0$ .

Calculation:

Now, use quotient identity $\tan u=\frac{\sin u}{\cos u}$ :

As, $\tan \theta =\text{undefined},\sin \theta >0$

So, $\cos \theta =0$

Now, apply Pythagorean Identity ${\cos }^2\theta =1-{\sin }^2\theta$ to find $\sin \theta$ .

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

As $\sin \theta >0$

Hence, $\sin \theta =1$

Find the trigonometric function $\csc \theta$ by applying reciprocal identity;

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

Substitute $\sin \theta =1$ in the above function;

  $\begin{array}{c}\csc \theta =\frac{1}{\sin \theta }\\ \csc \theta =\frac{1}{1}\\ \csc \theta =1\end{array}$

Apply reciprocal identity $\sec u=\frac{1}{\cos u}$

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

Substitute $\cos \theta =0$ in the above expression;

  $\begin{array}{c}\sec \theta =\frac{1}{0}\\ =\text{undefined}\end{array}$

Hence, $\sec \theta =\text{undefined}$ .

Substitute $\tan \theta =\text{undefined}$ in the reciprocal identity $\cot \theta =\frac{1}{\tan \theta }$ ;

  $\begin{array}{c}\cot \theta =\frac{1}{\text{undefined}}\\ =0\end{array}$

Hence, $\cot \theta =0$ .

Therefore remaining trigonometric functions are $\sin \theta =1,\cos \theta =0,\sec \theta =\text{undefined},\csc \theta =\text{1},\cot \theta =0$ .