Double-Angle and Half-Angle Formulas
From addition formulas to powerful tools for simplification, exact values, and beyond
Prerequisites
- Addition (sum) formulas: $\sin(\alpha+\beta)$, $\cos(\alpha+\beta)$, $\tan(\alpha+\beta)$
- Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$
- Quadrant sign rules for sine, cosine, and tangent
1. Derivation of the Double-Angle Formulas
The double-angle formulas are a special case of the addition (sum) formulas, obtained by setting both arguments equal: $\alpha = \beta = \theta$.
1.1 Sine of a Double Angle
Start with the addition formula for sine:
$$ \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta $$Set $\alpha = \beta = \theta$:
$$ \sin(\theta + \theta) = \sin\theta\cos\theta + \cos\theta\sin\theta $$
$$ \boxed{\sin(2\theta) = 2\sin\theta\cos\theta} $$
1.2 Cosine of a Double Angle — Three Equivalent Forms
Start with the addition formula for cosine:
$$ \cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta $$Set $\alpha = \beta = \theta$:
$$ \cos(2\theta) = \cos^2\theta - \sin^2\theta \quad \text{(Form 1)} $$Form 2 (cosine only): Replace $\sin^2\theta = 1 - \cos^2\theta$ using the Pythagorean identity:
$$ \cos(2\theta) = \cos^2\theta - (1 - \cos^2\theta) = 2\cos^2\theta - 1 \quad \text{(Form 2)} $$Form 3 (sine only): Replace $\cos^2\theta = 1 - \sin^2\theta$:
$$ \cos(2\theta) = (1 - \sin^2\theta) - \sin^2\theta = 1 - 2\sin^2\theta \quad \text{(Form 3)} $$
$$ \boxed{\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta} $$
1.3 Tangent of a Double Angle
From the addition formula for tangent: