Half Angle Calculator

The Half Angle Calculator evaluates the trigonometric functions of half a given angle using the half-angle identities. These formulas express $$\sin\!\left(\frac{\theta}{2}\right)$$, $$\cos\!\left(\frac{\theta}{2}\right)$$, and $$\tan\!\left(\frac{\theta}{2}\right)$$ in terms of $$\cos\theta$$, allowing you to find exact trig values of angles that are half of known angles.

The Half-Angle Formulas

The three primary half-angle identities are:

$$\sin\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$

$$\cos\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$

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

The $$\pm$$ sign depends on the quadrant in which $$\frac{\theta}{2}$$ lies. There are also two rationalized forms for the tangent half-angle that avoid the square root:

$$\tan\!\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$$

Derivation

The half-angle formulas are derived from the double-angle identity for cosine. Starting with $$\cos(2\alpha) = 1 - 2\sin^2\alpha$$, substitute $$\alpha = \theta/2$$:

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

Solving for $$\sin(\theta/2)$$ gives the half-angle sine formula. Similarly, using $$\cos(2\alpha) = 2\cos^2\alpha - 1$$ yields the half-angle cosine formula.

Finding Exact Values

Half-angle formulas let you compute exact trigonometric values for many angles. For example, since $$\cos(60°) = \frac{1}{2}$$, you can find exact values at 30°. More usefully, knowing $$\cos(45°) = \frac{\sqrt{2}}{2}$$ lets you find exact values for 22.5°, and $$\cos(30°)$$ gives exact values for 15°.

The Weierstrass Substitution

The tangent half-angle substitution $$t = \tan(\theta/2)$$ is one of the most powerful techniques in calculus. Under this substitution:

$$\sin\theta = \frac{2t}{1+t^2}, \quad \cos\theta = \frac{1-t^2}{1+t^2}, \quad d\theta = \frac{2\,dt}{1+t^2}$$

This converts any rational expression of trigonometric functions into a rational function of $$t$$, which can always be integrated using partial fractions.

How to Use This Calculator

Enter angle $$\theta$$ in degrees. The calculator computes $$\theta/2$$ and evaluates $$\sin(\theta/2)$$, $$\cos(\theta/2)$$, and $$\tan(\theta/2)$$ both directly and using the half-angle identity forms. The alternative tangent calculation via $$\sin\theta/(1+\cos\theta)$$ serves as a cross-check.