Half Angle Formulas
Reference for half-angle formulas for sin, cos, and tan.
Derived from double-angle identities for integration, exact values, and simplifying trig expressions.
The Formulas
sin(θ/2) = ±√((1 - cos θ) / 2)
cos(θ/2) = ±√((1 + cos θ) / 2)
tan(θ/2) = sin θ / (1 + cos θ) = (1 - cos θ) / sin θ
Half angle formulas express trig functions of θ/2 in terms of trig functions of θ.
The ± sign depends on which quadrant θ/2 is in.
Variables
| Symbol | Meaning |
|---|---|
| θ | The original angle |
| θ/2 | Half the original angle |
| ± | Choose + or - based on the quadrant of θ/2 |
Choosing the Sign
- If θ/2 is in Quadrant I (0° to 90°): sin and cos are both positive
- If θ/2 is in Quadrant II (90° to 180°): sin is positive, cos is negative
- If θ/2 is in Quadrant III (180° to 270°): sin and cos are both negative
- If θ/2 is in Quadrant IV (270° to 360°): sin is negative, cos is positive
Example 1
Find sin(15°) using the half angle formula with θ = 30°
sin(15°) = sin(30°/2) = √((1 - cos 30°) / 2)
cos 30° = √3/2 ≈ 0.8660
sin(15°) = √((1 - 0.8660) / 2) = √(0.1340 / 2) = √0.0670
sin(15°) ≈ 0.2588
Example 2
Find cos(22.5°) using the half angle formula with θ = 45°
cos(22.5°) = cos(45°/2) = √((1 + cos 45°) / 2)
cos 45° = √2/2 ≈ 0.7071
cos(22.5°) = √((1 + 0.7071) / 2) = √(1.7071 / 2) = √0.8536
cos(22.5°) ≈ 0.9239
When to Use It
Use half angle formulas when:
- You need to find the trig value of an angle that is half of a known angle
- Finding exact values for angles like 15°, 22.5°, or 75°
- Simplifying integrals in calculus that involve sin² or cos²
- Solving trig equations that involve half angles
Key Notes
- Formulas: sin(θ/2) = ±√((1−cosθ)/2); cos(θ/2) = ±√((1+cosθ)/2); tan(θ/2) = sinθ/(1+cosθ) = (1−cosθ)/sinθ. The ± for sine and cosine depends on which quadrant θ/2 falls in; the tangent forms avoid the ± ambiguity and are preferred.
- Derived from double-angle formulas: cos(2A) = 1 − 2sin²A → sin²A = (1−cos2A)/2 → sin(A) = ±√((1−cos2A)/2). Substituting A = θ/2 gives the half-angle identity. The derivation path through cos 2A is the standard approach.
- Exact values for non-standard angles: Half-angle formulas allow computing exact trig values for angles like 22.5° = 45°/2 or 15° = 30°/2 that are not in the standard table. For example, cos(22.5°) = √((1+cos45°)/2) = √((1+√2/2)/2).
- Weierstrass substitution t = tan(θ/2): Setting t = tan(θ/2) converts sinθ = 2t/(1+t²), cosθ = (1−t²)/(1+t²), dθ = 2dt/(1+t²). This substitution transforms trigonometric integrals into rational functions — solvable by partial fractions.
- Applications: Half-angle formulas are used in evaluating integrals of trig functions (via Weierstrass substitution), proving identities, computing exact values for non-standard angles, and in digital signal processing where half-band filters are designed using related properties.