# What is the Half-Angle Identities?

Dec 18, 2014

The half-angle identities are defined as follows:

$\setminus m a t h b f \left(\sin \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{2}}\right)$

$\left(+\right)$ for quadrants I and II
$\left(-\right)$ for quadrants III and IV

$\setminus m a t h b f \left(\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}}\right)$

$\left(+\right)$ for quadrants I and IV
$\left(-\right)$ for quadrants II and III

$\setminus m a t h b f \left(\tan \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\right)$

$\left(+\right)$ for quadrants I and III
$\left(-\right)$ for quadrants II and IV

We can derive them from the following identities:

${\sin}^{2} x = \frac{1 - \cos \left(2 x\right)}{2}$

${\sin}^{2} \left(\frac{x}{2}\right) = \frac{1 - \cos \left(x\right)}{2}$

$\textcolor{b l u e}{\sin \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos \left(x\right)}{2}}}$

Knowing how $\sin x$ is positive for $0 - {180}^{\circ}$ and negative for $180 - {360}^{\circ}$, we know that it is positive for quadrants I and II and negative for III and IV.

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

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

$\textcolor{b l u e}{\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos \left(x\right)}{2}}}$

Knowing how $\cos x$ is positive for $0 - {90}^{\circ}$ and $270 - {360}^{\circ}$, and negative for $90 - {270}^{\circ}$, we know that it is positive for quadrants I and IV and negative for II and III.

$\tan \left(\frac{x}{2}\right) = \sin \frac{\frac{x}{2}}{\cos \left(\frac{x}{2}\right)} = \frac{\pm \sqrt{\frac{1 - \cos \left(x\right)}{2}}}{\pm \sqrt{\frac{1 + \cos \left(x\right)}{2}}}$

$\textcolor{b l u e}{\tan \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos \left(x\right)}{1 + \cos \left(x\right)}}}$

We can see that if we take the conditions for positive and negative values from $\sin x$ and $\cos x$ and divide them, we get that this is positive for quadrants I and III and negative for II and IV.