Half-Angle Identity Calculator

Calculate half-angle identities for sine, cosine, and tangent. Supports both degrees and radians. Simplify trigonometric expressions with step-by-step solutions.

Half-Angle Formulas:

sin(θ/2) = ±√[(1 - cosθ)/2]
cos(θ/2) = ±√[(1 + cosθ)/2]
tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)] = sinθ/(1 + cosθ) = (1 - cosθ)/sinθ

The ± sign depends on the quadrant of θ/2.

Degrees
Radians
Select which half-angle identity to calculate
Enter the original angle. Negative angles and values greater than 360° (or 2π rad) are supported.
30°
45°
60°
90°
120°
180°
270°
360°
-30°
450°
π/6
π/4
π/3
π/2
2π/3
π
3π/2
2π
-π/6
5π/2
I
II
III
IV
Auto
Auto-selected based on angle
Select the quadrant where the half-angle θ/2 lies. Choose "Auto" to automatically determine based on the angle value.
Calculating...

Understanding Half-Angle Identities

Half-angle identities are trigonometric identities that express the sine, cosine, or tangent of half of an angle in terms of trigonometric functions of the original angle. These identities are derived from the double-angle formulas and are useful for simplifying expressions and solving trigonometric equations.

Half-Angle Formulas:

sin(θ/2) = ±√[(1 - cosθ)/2]
cos(θ/2) = ±√[(1 + cosθ)/2]
tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)] = sinθ/(1 + cosθ) = (1 - cosθ)/sinθ

The sign (±) depends on the quadrant in which θ/2 lies.

Sign Determination by Quadrant

Quadrant of θ/2 sin(θ/2) cos(θ/2) tan(θ/2)
I (0° to 90°) + + +
II (90° to 180°) + - -
III (180° to 270°) - - +
IV (270° to 360°) - + -

Derivation of Half-Angle Formulas

1

From Double-Angle Formulas: Starting with cos(2α) = 1 - 2sin²(α), let α = θ/2. Then cos(θ) = 1 - 2sin²(θ/2). Solving for sin(θ/2) gives sin(θ/2) = ±√[(1 - cosθ)/2].

2

Similarly for cosine: From cos(2α) = 2cos²(α) - 1, we get cos(θ) = 2cos²(θ/2) - 1. Solving for cos(θ/2) gives cos(θ/2) = ±√[(1 + cosθ)/2].

3

For tangent: Using the identity tan(θ/2) = sin(θ/2)/cos(θ/2) and substituting the half-angle formulas gives tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)].

Applications of Half-Angle Identities

  • Integration: Simplifying integrals containing trigonometric functions
  • Physics: Solving wave equations and oscillation problems
  • Engineering: Signal processing and electrical circuit analysis
  • Geometry: Calculating areas and lengths in triangles
  • Computer Graphics: Rotation transformations and 3D rendering

Calculator Features:

  • Calculates half-angle identities for sine, cosine, and tangent
  • Supports both degrees and radians
  • Handles negative angles and large angle values
  • Provides exact values and decimal approximations
  • Includes quadrant selection for correct sign determination
  • Visualizes angles on the unit circle
  • Shows step-by-step derivation of results

Frequently Asked Questions

The ± sign appears because when we take the square root in the derivation, we get both positive and negative solutions. The correct sign depends on which quadrant the half-angle (θ/2) lies in. For example, if θ/2 is in the first quadrant, all trigonometric functions are positive, so we use the positive sign.

Yes, this calculator supports both degrees and radians. Use the unit selector above the angle input to switch between units. The calculator will automatically adjust all calculations and visualizations.

Negative angles are converted to their positive equivalents by adding 360° (or 2π radians) until they fall in the range of 0° to 360° (or 0 to 2π radians). This is standard practice in trigonometry as trigonometric functions are periodic.

When tan(θ/2) is undefined (for example, when θ/2 = 90° or π/2), the calculator will display "undefined" instead of a numerical value. This occurs when cos(θ) = -1 in the formula tan(θ/2) = ±√[(1 - cosθ)/(1 + cosθ)], which leads to division by zero.

Yes, half-angle formulas work for any angle. For angles greater than 180°, you may need to adjust the quadrant selection accordingly. Remember that trigonometric functions are periodic, so you can also reduce the angle modulo 360° first. The calculator handles this automatically.