Chapter 13.2, Problem 11WE

To determine

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

Answer to Problem 11WE

The value of six trigonometric function are $\sin \left(\frac{3\pi }{2}\right)=-1$ , $\cos \left(\frac{3\pi }{2}\right)=0$ , $\tan \left(\frac{3\pi }{2}\right)=\infty$ , $\text{cot}\left(\frac{3\pi }{2}\right)=0$ , $\sec \left(\frac{3\pi }{2}\right)=\infty$ , $\text{cosec}\left(\frac{3\pi }{2}\right)=-1$ .

Explanation of Solution

Given information:

The given value of angle is $\frac{3\pi }{2}$ .

Calculation:

The value of angle is $\frac{3\pi }{2}=270°$ . The $270°$ lies in the third quadrant

The value of all trigonometric functions except tangent and cotangent is negative in this quadrant.

Calculate the values of trigonometric functions.

  $\begin{array}{l}\sin \left(\frac{3\pi }{2}\right)=-1\\ \cos \left(\frac{3\pi }{2}\right)=0\\ \tan \left(\frac{3\pi }{2}\right)=\infty \\ \cot \left(\frac{3\pi }{2}\right)=0\end{array}$

  $\begin{array}{l}\sec \left(\frac{3\pi }{2}\right)=\infty \\ \text{cosec}\left(\frac{3\pi }{2}\right)=-1\\ \cot \left(\frac{3\pi }{2}\right)=0\end{array}$

Therefore, the value of six trigonometric function are $\sin \left(\frac{3\pi }{2}\right)=-1$ , $\cos \left(\frac{3\pi }{2}\right)=0$ , $\tan \left(\frac{3\pi }{2}\right)=\infty$ , $\text{cot}\left(\frac{3\pi }{2}\right)=0$ , $\sec \left(\frac{3\pi }{2}\right)=\infty$ , $\text{cosec}\left(\frac{3\pi }{2}\right)=-1$ .