Half Angle Formulas/Tangent

Theorem
where $\tan$ denotes tangent and $\cos$ denotes cosine.

When $\theta = \left({2 k + 1}\right) \pi$, $\tan \dfrac \theta 2$ is undefined.

Proof
Since $\cos \theta \ge -1$, it follows that $\cos \theta + 1 \ge 0$.

When $\cos \theta = -1$ it follows that $\cos \theta + 1 \ge 0$ and then $\tan \dfrac \theta 2$ is undefined.

This happens when $\theta = \left({2 k + 1}\right) \pi$.

We have that:
 * $\tan \dfrac \theta 2 = \dfrac {\sin \dfrac \theta 2}{\cos \dfrac \theta 2}$

Quadrant I
In quadrant I, we have:
 * Sine in first quadrant: $\sin \dfrac \theta 2 > 0$
 * Cosine in first quadrant: $\cos \dfrac \theta 2 > 0$

and so in quadrant I:

Quadrant II
In quadrant II, we have:
 * Sine in second quadrant: $\sin \dfrac \theta 2 > 0$
 * Cosine in second quadrant: $\cos \dfrac \theta 2 < 0$

and so in quadrant II:

Quadrant III
In quadrant III, we have:
 * Sine in third quadrant: $\sin \dfrac \theta 2 < 0$
 * Cosine in third quadrant: $\cos \dfrac \theta 2 < 0$

and so in quadrant III:

Quadrant IV
In quadrant IV, we have:
 * Sine in fourth quadrant: $\sin \dfrac \theta 2 < 0$
 * Cosine in fourth quadrant: $\cos \dfrac \theta 2 > 0$

and so in quadrant IV:

Also see

 * Half Angle Formula for Sine
 * Half Angle Formula for Cosine