Sine, Cosine, and Tangent of a Half Angle
🎯 Big Idea
Half-angle formulas allow us to find the exact values of trigonometric functions for angles that are half of a known angle. For example, if we know the trig values for $60°$, we can find the exact values for $30°$. More powerfully, we can find exact values for angles like $22.5°$ (half of $45°$) or $15°$ (half of $30°$) that aren't standard angles on the unit circle.
📐 Deriving the Half-Angle Formulas
We start from the double-angle formula for cosine:
Now, let $\theta = 2\alpha$, which means $\alpha = \frac{\theta}{2}$. Substituting:
Starting with: $\cos\theta = 1 - 2\sin^2\!\left(\frac{\theta}{2}\right)$
$2\sin^2\!\left(\frac{\theta}{2}\right) = 1 - \cos\theta$
$\sin^2\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}$
$$\sin\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$
Starting with: $\cos\theta = 2\cos^2\!\left(\frac{\theta}{2}\right) - 1$
$2\cos^2\!\left(\frac{\theta}{2}\right) = 1 + \cos\theta$
$\cos^2\!\left(\frac{\theta}{2}\right) = \frac{1 + \cos\theta}{2}$
$$\cos\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$
Since $\tan\!\left(\frac{\theta}{2}\right) = \frac{\sin(\theta/2)}{\cos(\theta/2)}$, we get:
$$\tan\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$$
There are also two very useful sign-free alternative forms:
These forms are derived by multiplying numerator and denominator by appropriate conjugates and using the identity $\sin^2\theta + \cos^2\theta = 1$.