How do you find cos(x/2) given sin(x)=1/4?

Oct 17, 2017

$\cos \left(\frac{x}{2}\right) = 0.99$
$\cos \left(\frac{x}{2}\right) = 0.127$

Explanation:

$\sin x = \frac{1}{4}$ . Find cos x
${\cos}^{2} x = 1 - {\sin}^{2} x = 1 - \frac{1}{16} = \frac{15}{16}$
$\cos x = \pm \frac{\sqrt{15}}{4}$
There are 2 values of cos x because if $\sin x = \frac{1}{4}$, x could either be in Quadrant 1 or in Quadrant 2
Use trig identity:
$2 {\cos}^{2} \left(\frac{x}{2}\right) = 1 - \cos 2 a$
In this case:
${\cos}^{2} \left(\frac{x}{2}\right) = \frac{1}{2} \pm \frac{\sqrt{15}}{8} = \frac{1}{2} \pm 0.484$
a. ${\cos}^{2} \left(\frac{x}{2}\right) = 0.984$
b. ${\cos}^{2} \left(\frac{x}{2}\right) = 0.016$
a. $\cos \left(\frac{x}{2}\right) = \sqrt{0.984} = 0.99$
b. $\cos \left(\frac{x}{2}\right) = \sqrt{0.016} = 0.127$
Check with calculator.
a. $\cos \left(\frac{x}{2}\right) = 0.99$ --> $\frac{x}{2} = {7}^{\circ} 27$ --> $x = {14}^{\circ} 53$
$\sin \left({14}^{\circ} 52\right) = 0.25$. Proved
b. $\cos \left(\frac{x}{2}\right) = 0.127$ --> $\frac{x}{2} = {82}^{\circ} 70$ --> $x = {165}^{\circ} 41$
$\sin \left({165}^{\circ} 41\right) = 0.25$. Proved