Half Angle Formulas
Half angle formulas for sin(θ/2), cos(θ/2), and tan(θ/2).
Calculate trig functions of half angles with worked examples.
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