Question

Angle x is a third quadrant angle such that cos x = -1/5.

Answer 1

Starting from the formula

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

We can divide all the angles by 2 and we get

$\cos^2\left(\dfrac{x}{2}\right)=\dfrac{1+\cos(x)}{2}$

And finally

$\cos\left(\dfrac{x}{2}\right)=\pm\sqrt{\dfrac{1+\cos(x)}{2}}$

Angle x is a third quadrant angle, meaning that it is somewhere between 180 and 270. This implies that x/2 lies between 90 and 135. So, x/2 is a second quadrant angle, and its cosine is negative. This means that we can update the previous equation to

$\cos\left(\dfrac{x}{2}\right)=-\sqrt{\dfrac{1+\cos(x)}{2}}$

You know that

$\cos(x)=-\dfrac{1}{5}$

so, plug that value into the expression and you'll get cos(x/2).