Chapter 12, Problem 4CT

To determine

Find the value of $\tan \theta$ such that $\cos \theta =-\frac{3}{4}\text{ and }\sin \theta <0$

Answer to Problem 4CT

The value of $\tan \theta$ is $\frac{\sqrt{7}}{4}$ .

Explanation of Solution

Given information: The given trigonometric function is,

  $\cos \theta =-\frac{3}{4}\text{ and }\sin \theta <0\mathrm{....}\left(1\right)$

Calculation: From (1) the cosine and sine functions are negative which means the angle $\left(\theta \right)$ lie in 3^rd quadrant. Therefore the tangent function evaluated in 3^rd quadrant will be positive. Therefore the value of $\tan \theta$ using (1) is evaluated as,

  $\begin{array}{c}\cos \theta =-\frac{3}{4}\text{ and }\sin \theta <0\mathrm{....}\left(\text{From }\left(1\right)\right)\\ \sin \theta =\pm \sqrt{1-{\cos }^2\theta }\\ =\pm \sqrt{1-{\left(-\frac{3}{4}\right)}^2}\\ =-\frac{\sqrt{7}}{4}\mathrm{....}\left(\text{Take negative value}\right)\end{array}$

Now evaluate for $\tan \theta$ ,

  $\begin{array}{c}\tan \theta =\frac{\sin \theta }{\cos \theta }\\ =\frac{\left(-\sqrt{7}/4\right)}{\left(-3/4\right)}\\ =\frac{\sqrt{7}}{4}\end{array}$

Hence the value of $\tan \theta$ is $\frac{\sqrt{7}}{4}$ .