Half-Angle Formulae in Trigonometry: Derivation & Examples

The half-angle formulae give sin(θ/2), cos(θ/2) and tan(θ/2) in terms of θ: sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2), and tan(θ/2) = sin θ/(1 + cos θ) = (1 − cos θ)/sin θ. They come from the power-reduction forms of cos 2θ with θ replaced by θ/2.

They matter because they produce exact values for angles like 15° and 22.5°, which appear in geometry and design but sit outside the standard table. They also run the double-angle process in reverse, which calculus exploits when integrating.

Key Concepts

• Derivation: start from cos 2A = 1 − 2 sin²A, set A = θ/2, and solve for sin(θ/2); similarly for cosine.
• The ± sign: the square root needs a sign, chosen by the quadrant containing θ/2, not the quadrant containing θ.
• Tangent needs no ±: the forms sin θ/(1 + cos θ) and (1 − cos θ)/sin θ carry the correct sign automatically.
• Quadrant mapping: if θ is between 0° and 360°, then θ/2 lies between 0° and 180°, covering quadrants I and II.

Formulas

• Sine: sin(θ/2) = ±√((1 − cos θ)/2); take + if θ/2 is in Q1 or Q2, − if in Q3 or Q4.
• Cosine: cos(θ/2) = ±√((1 + cos θ)/2); take + if θ/2 is in Q1 or Q4, − if in Q2 or Q3.
• Tangent: tan(θ/2) = sin θ/(1 + cos θ) = (1 − cos θ)/sin θ, valid where the denominators are non-zero.
• Assumptions: θ is any real angle; the sign rule depends on the quadrant of the half angle.

Worked Example

Use the half-angle formulae to find sin 30° and cos 30° from θ = 60°.

1. Here θ = 60°, so θ/2 = 30°, which lies in the first quadrant where both sine and cosine are positive.
2. sin 30° = +√((1 − cos 60°)/2) = √((1 − 1/2)/2) = √(1/4) = 1/2.
3. cos 30° = +√((1 + cos 60°)/2) = √((1 + 1/2)/2) = √(3/4) = √3/2.
4. Both match the standard table: sin 30° = 1/2 and cos 30° = √3/2.
5. Check consistency: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1, confirming the pair.

Common Misconceptions

• The sign follows θ/2, not θ. If θ is in Q3, θ/2 is in Q2 (for θ under 360°), where sine is positive — using θ's quadrant gives the wrong sign.
• Never drop the ±. Writing sin(θ/2) = √((1 − cos θ)/2) without choosing a sign is incomplete.
• Half-angle is not half the value. sin(θ/2) ≠ (sin θ)/2; the square-root formula is the correct relationship.

Real-World Applications

• Exact values: sin 15° = (√6 − √2)/4 and cos 22.5° = √(2 + √2)/2 come straight from these formulae.
• Calculus: reversing double-angle steps to integrate products of trigonometric functions.
• Optics: half-angle relations appear in formulas for reflection and refraction at interfaces.
• Engineering design: computing dimensions for 15° and 22.5° joints exactly, without rounding.

The half-angle formulae — sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2), and the sign-free tangent forms — are the double-angle identities run backwards. Always choose the ± sign from the quadrant of θ/2, and verify against known values like sin 30° = 1/2.