Chapter 13.2, Problem 12WE

To determine

To find: The exact values of six trigonometric functions and declare undefined if so.

Answer to Problem 12WE

The value of six trigonometric function are $\sin \frac{3\pi }{4}=\frac{1}{\sqrt{2}}$ , $\cos \frac{3\pi }{4}=-\frac{1}{\sqrt{2}}$ , $\tan \left(\frac{3\pi }{4}\right)=-1$ , $\cot \left(\frac{3\pi }{4}\right)=-1$ , $\sec \left(\frac{3\pi }{4}\right)=-\sqrt{2}$ , $\csc \left(\frac{3\pi }{4}\right)=\sqrt{2}$ .

Explanation of Solution

Given information:

The given value of angle is $\frac{3\pi }{4}$ .

Calculation:

The value of $\sin \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$

Calculate $\sin \left(\frac{3\pi }{4}\right)$ .

  $\begin{array}{l}\sin \frac{3\pi }{4}=\sin \left(\pi -\frac{3\pi }{4}\right)\\ \sin \frac{3\pi }{4}=\sin \frac{\pi }{4}\\ \sin \frac{3\pi }{4}=\frac{1}{\sqrt{2}}\end{array}$

Calculate $\cos \left(\frac{3\pi }{4}\right)$ .

  $\begin{array}{l}\cos \frac{3\pi }{4}=\cos \left(\pi -\frac{3\pi }{4}\right)\\ \cos \frac{3\pi }{4}=-\cos \frac{\pi }{4}\\ \cos \frac{3\pi }{4}=-\frac{1}{\sqrt{2}}\end{array}$

Calculate $\tan \left(\frac{3\pi }{4}\right)$ .

  $\begin{array}{l}\tan \left(\frac{3\pi }{4}\right)=\tan \left(\pi -\frac{3\pi }{4}\right)\\ \tan \left(\frac{3\pi }{4}\right)=-\tan \frac{\pi }{4}\\ \tan \left(\frac{3\pi }{4}\right)=-1\end{array}$

Calculate $\cot \left(\frac{3\pi }{4}\right)$ .

  $\begin{array}{l}\cot \left(\frac{3\pi }{4}\right)=\frac{1}{\tan \left(\frac{3\pi }{4}\right)}\\ \cot \left(\frac{3\pi }{4}\right)=\frac{1}{-1}\\ \cot \left(\frac{3\pi }{4}\right)=-1\end{array}$

Calculate $\csc \left(\frac{3\pi }{4}\right)$ .

  $\begin{array}{l}\csc \left(\frac{3\pi }{4}\right)=\frac{1}{\sin \frac{3\pi }{4}}\\ \csc \left(\frac{3\pi }{4}\right)=\frac{1}{\frac{1}{\sqrt{2}}}\\ \csc \left(\frac{3\pi }{4}\right)=\sqrt{2}\end{array}$

Calculate $\sec \left(\frac{3\pi }{4}\right)$ .

  $\begin{array}{l}\sec \left(\frac{3\pi }{4}\right)=\frac{1}{\cos \frac{3\pi }{4}}\\ \sec \left(\frac{3\pi }{4}\right)=\frac{1}{-\frac{1}{\sqrt{2}}}\\ \sec \left(\frac{3\pi }{4}\right)=-\sqrt{2}\end{array}$

Therefore, the value of six trigonometric function are $\sin \frac{3\pi }{4}=\frac{1}{\sqrt{2}}$ , $\cos \frac{3\pi }{4}=-\frac{1}{\sqrt{2}}$ , $\tan \left(\frac{3\pi }{4}\right)=-1$ , $\cot \left(\frac{3\pi }{4}\right)=-1$ , $\sec \left(\frac{3\pi }{4}\right)=-\sqrt{2}$ , $\csc \left(\frac{3\pi }{4}\right)=\sqrt{2}$ .