Half Angle Calculator
Compute half-angle values for sine, cosine, and tangent using the standard trig identities. Enter any angle and get sin(θ/2), cos(θ/2), and tan(θ/2) instantly.
Half-angle identities let you find the trig functions of half an angle when you know the trig functions of the full angle. They come directly from the double-angle formulas.
The Half-Angle Formulas:
- sin(θ/2) = ±sqrt((1 - cos θ) / 2)
- cos(θ/2) = ±sqrt((1 + cos θ) / 2)
- tan(θ/2) = sin θ / (1 + cos θ) = (1 - cos θ) / sin θ
The ± sign depends on the quadrant where θ/2 falls.
Example: θ = 60°
- cos(60°) = 0.5
- sin(30°) = sqrt((1 - 0.5) / 2) = sqrt(0.25) = 0.5 (correct!)
- cos(30°) = sqrt((1 + 0.5) / 2) = sqrt(0.75) = 0.8660...
- tan(30°) = sin(60°) / (1 + cos(60°)) = 0.8660 / 1.5 = 0.5774...
Where Half-Angle Formulas Are Used:
- Simplifying integrals in calculus (the Weierstrass substitution t = tan(θ/2) converts trig integrals into rational functions)
- Deriving exact values for angles like 15°, 22.5°, and 75°
- Signal processing and Fourier analysis
- Computer graphics for rotation calculations
Deriving From Double-Angle:
The double-angle formula says cos(2α) = 1 - 2sin²(α). Solving for sin(α) gives sin(α) = sqrt((1 - cos(2α)) / 2). Replace α with θ/2 and 2α with θ, and you have the half-angle formula.