Chapter 13.1, Problem 9WE

To determine

To find:The value of the six trigonometric functions.

Answer to Problem 9WE

The value of the six trigonometric functions

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\cos ec\theta$	$\sec \theta$	$\cot \theta$
${30}^{\circ }$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{2}$	$2$	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
${45}^{\circ }$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
${60}^{\circ }$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	$2$	$\frac{\sqrt{3}}{2}$

Explanation of Solution

Given information:

The given information is $-\frac{5\pi }{6}$ .

Calculation:

We know that $-\frac{5\pi }{6}$ is co terminal with $-\frac{\pi }{6}$ or $-{30}^{\circ }$ .Find the six trigonometric functions corresponding to $-{30}^{\circ }$ .

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\cos ec\theta$	$\sec \theta$	$\cot \theta$
${30}^{\circ }$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{2}$	$2$	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
${45}^{\circ }$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
${60}^{\circ }$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	$2$	$\frac{\sqrt{3}}{2}$

From the table, we get the values of six trigonometric functions as follows.

  $\begin{array}{l}\sin \left(-\frac{5\pi }{6}\right)=-\frac{1}{2}\cos \left(-\frac{5\pi }{6}\right)=-\frac{\sqrt{3}}{2}\tan \left(-\frac{5\pi }{6}\right)=\frac{\sqrt{3}}{3}\\ \cos ec\left(-\frac{5\pi }{6}\right)=-2\sec \left(-\frac{5\pi }{6}\right)=-\frac{2\sqrt{3}}{3}\cot \left(-\frac{5\pi }{6}\right)=\sqrt{3}\\ \end{array}$