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) =

-√(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

is in quadrant III, so .

This makes , which means lies in quadrant II, for which we expect .

Recall the double-angle identity:

∠acb = 8°

step-by-step explanation:

∠xba lies on the circle and is half the measure of the intercepted arc ax

hence

measure of arc ax = 2 × 42° = 84°

∠acb is outside the circle and is

∠acb = (arcab - arcax ), hence

∠acb = 0.5 (100 - 84) = 0.5 × 16° = 8°