Chapter 13.2, Problem 13WE

To determine

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

Answer to Problem 13WE

The value of six trigonometric function are $\sin \left(\frac{11\pi }{6}\right)=\frac{-1}{2}$ , $\cos \left(\frac{11\pi }{6}\right)=\frac{\sqrt{3}}{2}$ , $\tan \left(\frac{11\pi }{6}\right)=-\frac{\sqrt{3}}{3}$ , $\cot \left(\frac{11\pi }{6}\right)=-\sqrt{3}$ , $\sec \left(\frac{11\pi }{6}\right)=\frac{2\sqrt{3}}{3}$ , $\text{cosec}\left(\frac{11\pi }{6}\right)=-2$ .

Explanation of Solution

Given information:

The given value of angle is $\frac{11\pi }{6}$ .

Calculation:

The value of angle is $\frac{11\pi }{6}=330°$ . The $330°$ is co terminal with $30°$ that lies in the fourth quadrant.

The value of all trigonometric functions except cosine and sec is negative in this quadrant.

Calculate the values of trigonometric functions.

  $\begin{array}{l}\sin \left(\frac{11\pi }{6}\right)=\frac{-1}{2}\\ \cos \left(\frac{11\pi }{6}\right)=\frac{\sqrt{3}}{2}\\ \tan \left(\frac{11\pi }{6}\right)=-\frac{\sqrt{3}}{3}\\ \cot \left(\frac{11\pi }{6}\right)=-\sqrt{3}\end{array}$

  $\begin{array}{l}\sec \left(\frac{11\pi }{6}\right)=\frac{2\sqrt{3}}{3}\\ \text{cosec}\left(\frac{11\pi }{6}\right)=-2\end{array}$

Therefore, the value of six trigonometric function are $\sin \left(\frac{11\pi }{6}\right)=\frac{-1}{2}$ , $\cos \left(\frac{11\pi }{6}\right)=\frac{\sqrt{3}}{2}$ , $\tan \left(\frac{11\pi }{6}\right)=-\frac{\sqrt{3}}{3}$ , $\cot \left(\frac{11\pi }{6}\right)=-\sqrt{3}$ , $\sec \left(\frac{11\pi }{6}\right)=\frac{2\sqrt{3}}{3}$ , $\text{cosec}\left(\frac{11\pi }{6}\right)=-2$ .