Q:

Finding $\sin(x/2)$ given a $\tan$ using half-angle identities

How can I find $\sin(x/2)$ given $\tan(x) = -5.099$ and $x$ is in Quadrant IV, assuming that $0 < x < 2\pi$?
I know I have to use half-angle identities in some way, but cannot figure it out.

A:

Your first goal is to find $\sin x$ and $\cos x$. You know the following things:

$\sin^2 x + \cos^2 x = 1$.
$\frac{\sin x}{\cos x} = -5.099$.
Because $x$ is in Quadrant IV, $\cos x > 0$ and $\sin x < 0$.

We will only need $\cos x$ for now, but finding $\sin x$ and finding $\cos x$ goes hand in hand.
Once you know that, you want to solve for $$\sin \frac x2 = \pm \sqrt{\frac{1 - \cos x}{2}}.$$
The sign of $\sin \frac x2$, once again, cannot be determined from the value of $\cos x$. You have to ask yourself: if $x$ is in Quadrant IV, what is the range of possible values of $x$ (as an angle)? What, then, is the range of possible values of $\frac x2$? Is $\sin x$ positive or negative for those values?

