Question

Angle x is a third quadrant angle such that cos x= −2/5 .

What is the exact value of cos(x/2) ?

Enter your answer, in simplest radical form, in the box.
cos(x/2) =

Answer 1

Answer:

-√(3/10)

Step-by-step explanation:

cos(x) = 2[cos(x/2)]² - 1

-2/5 = 2[cos(x/2)]² - 1

3/5 = 2[cos(x/2)]²

3/10 = [cos(x/2)]²

cos(x/2) = +/- sqrt(3/10)

Since x is in the 3rd quadrant, x/2 would be in the second quadrant.. so cos(x/2) is negative

Answer 2

$x$ is in quadrant III, so $\pi<x<\frac{3\pi}2$.

This makes $\frac\pi2<\frac x2<\frac{3\pi}4$, which means $\frac x2$ lies in quadrant II, for which we expect $\cos\frac x2<0$.

Recall the double-angle identity:

$\cos^2\dfrac x2=\dfrac{1+\cos x}2$

$\implies\cos\dfrac x2=-\sqrt{\dfrac{1-\frac25}2}=-\sqrt{\dfrac3{10}}$