Double-Angle and Half-Angle Formulas

By the end of this lesson, you will be able to derive and apply double-angle formulas for sine, cosine, and tangent, and half-angle formulas with the correct sign determination.

The Big Idea

Double-angle and half-angle formulas derive directly from the sum formulas — they are not memorized independently once you see the derivation. They enable computation of exact values at many more angles and are essential for integration in calculus (e.g., $\int \sin^2 x , dx$ uses the half-angle formula).

Double-Angle Formulas

Derived from $\sin(\alpha + \alpha)$, $\cos(\alpha + \alpha)$, $\tan(\alpha + \alpha)$:

Sine:

$$\sin(2\alpha) = 2\sin\alpha\cos\alpha$$

Cosine (three forms):

$$\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha = 2\cos^2\alpha - 1$$

Tangent:

$$\tan(2\alpha) = \frac{2\tan\alpha}{1-\tan^2\alpha}$$

The three cosine forms follow from the Pythagorean identity substitution into the first form.

Power-Reducing (Half-Angle Equivalents)

From the cosine double-angle formulas:

$$\sin^2\alpha = \frac{1-\cos(2\alpha)}{2}, \quad \cos^2\alpha = \frac{1+\cos(2\alpha)}{2}$$

These power-reducing formulas convert squares into first-power expressions — crucial for integration.

Half-Angle Formulas

Let $\alpha = \theta/2$ in the power-reducing formulas and take square roots:

$$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}$$

$$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}$$

$$\tan\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}} = \frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta}$$

Sign determination: depends on the quadrant of $\theta/2$, not of $\theta$.

Example 1 — Using Double-Angle

$\sin\theta = 4/5$ ($\theta$ in QI). Find $\sin(2\theta)$, $\cos(2\theta)$.

$\cos\theta = 3/5$

$\sin(2\theta) = 2(4/5)(3/5) = 24/25$

$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 9/25 - 16/25 = -7/25$

Example 2 — Exact Value: $\cos(22.5°)$

$22.5° = 45°/2$; $45°$ is in QI so $22.5°$ is in QI → cosine positive.

$$\cos(22.5°) = +\sqrt{\frac{1+\cos 45°}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2+\sqrt{2}}{4}} = \frac{\sqrt{2+\sqrt{2}}}{2}$$