# How do you find the exact value of sin (x/2) and cos (x/2) if cos x= - 1/4 and pi< x<2pi?

May 9, 2015

Use the trig identity: $\cos x = 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)$

$- \frac{1}{4} = 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right) \to {\sin}^{2} \left(\frac{x}{2}\right) = \frac{5}{8} = 0.625$

sin x/2 = -0.79

Next, use trig identity: ${\cos}^{2} \left(\frac{x}{2}\right) - {\sin}^{2} \left(\frac{x}{2}\right) = \cos x$

${\cos}^{2} \left(\frac{x}{2}\right) = - \frac{1}{4} + \frac{5}{8} = \frac{3}{8} \to \ast \cos \frac{x}{2} = - 0.61 \ast$

Check:
${\sin}^{2} \left(\frac{x}{2}\right) + {\cos}^{2} \left(\frac{x}{2}\right) = \frac{5}{8} + \frac{3}{8} = \frac{8}{8} = 1$ Correct.