# How do you find the value of tan of θ/2 given that cos a = -4/5 and a is in Quadrant II?

May 30, 2016

tan (x/2) = 3

#### Explanation:

To find tan (x/2), first, find $\sin \left(\frac{x}{2}\right)$ and $\cos \left(\frac{x}{2}\right)$
Use the trig identities:
$\cos 2 a = 2 {\cos}^{2} a - 1$
$\cos 2 a = 1 - 2 {\sin}^{2} a$
a. $\cos x = - \frac{4}{5} = 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)$
$2 {\sin}^{2} \left(\frac{x}{2}\right) = 1 + \frac{4}{5} = \frac{9}{5}$
${\sin}^{2} \left(\frac{x}{2}\right) = \frac{9}{10}$
$\sin \frac{x}{2} = \pm \frac{3}{\sqrt{10.}}$
b. $\cos x = - \frac{4}{5} = 2 {\cos}^{2} \left(\frac{x}{2}\right) - 1$
$2 {\cos}^{2} \left(\frac{x}{2}\right) = 1 - \frac{4}{5} = \frac{1}{5}$
${\cos}^{2} \left(\frac{x}{2}\right) = \frac{1}{10}$
$\cos \left(\frac{x}{2}\right) = \pm \frac{1}{\sqrt{10.}}$
x is in Quadrant II, then $\frac{x}{2}$ is in Quadrant I, and both $\sin \left(\frac{x}{2}\right)$ and $\cos \left(\frac{x}{2}\right)$ are positive.
Therefor,
$\tan \left(\frac{x}{2}\right) = \sin \frac{\frac{x}{2}}{\cos \left(\frac{x}{2}\right)} = \left(\frac{3}{\sqrt{10}}\right) \left(\frac{\sqrt{10}}{1}\right) = 3$