# Half Angle Calculator

Calculate sin(θ/2), cos(θ/2), and tan(θ/2) using half-angle identities. Supports degrees and radians. Free online half angle calculator.

## What this calculates

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.

## Inputs

- **Angle (θ)** — The full angle. Half-angle values will be computed for θ/2.
- **Angle Unit** — options: Degrees, Radians — Choose whether the angle is in degrees or radians.

## Outputs

- **Half Angle (θ/2)** — formatted as text — The half angle in both degrees and radians.
- **sin(θ/2)** — The sine of the half angle.
- **cos(θ/2)** — The cosine of the half angle.
- **tan(θ/2)** — formatted as text — The tangent of the half angle.
- **Formulas Applied** — formatted as text — The half-angle identities used in the calculation.

## Details

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.

## Frequently Asked Questions

**Q: How do I know whether to use + or - in the half-angle formula?**

A: The sign depends on the quadrant where the half angle (θ/2) lands. If θ/2 is in Quadrant I or II, sine is positive. If θ/2 is in Quadrant I or IV, cosine is positive. For example, if θ = 300°, then θ/2 = 150°, which is in Quadrant II, so sin(150°) is positive but cos(150°) is negative.

**Q: What is the Weierstrass substitution?**

A: The Weierstrass substitution uses t = tan(θ/2) to convert trigonometric integrals into rational functions. Under this substitution, sin θ = 2t/(1+t²), cos θ = (1-t²)/(1+t²), and dθ = 2dt/(1+t²). It can handle any integral involving trig functions, though the algebra can get messy.

**Q: Can I find exact values for 15 degrees using half-angle formulas?**

A: Yes. Since 15° = 30°/2, you can use the half-angle formula with θ = 30°. cos(30°) = sqrt(3)/2, so sin(15°) = sqrt((1 - sqrt(3)/2) / 2) = sqrt((2 - sqrt(3)) / 4). This gives the exact value without a calculator.

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Source: https://vastcalc.com/calculators/math/half-angle
Category: Math
Last updated: 2026-04-08