Half-Angle Calculator

Calculate sin(θ/2), cos(θ/2), and tan(θ/2) using half-angle identities with sign determination.

Enter Angle

Result

Half-Angle (θ/2)
60
degrees
sin(θ/2)
cos(θ/2)
tan(θ/2)
Sign (±)

Step-by-Step Solution

sin(θ/2) = ±√((1 - cosθ) / 2)

Understanding Half-Angle Formulas

The half-angle formulas are trigonometric identities that express sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the full angle. These formulas are derived from the double-angle identities and are essential tools in trigonometry, calculus, and engineering.

The Half-Angle Formulas

Sine Half-Angle

Express sin of half the angle using cosine of the full angle.

sin(θ/2) = ±√((1 - cosθ) / 2)

Cosine Half-Angle

Express cos of half the angle using cosine of the full angle.

cos(θ/2) = ±√((1 + cosθ) / 2)

Tangent Half-Angle (Form 1)

Using sine and cosine of the full angle.

tan(θ/2) = sinθ / (1 + cosθ)

Tangent Half-Angle (Form 2)

Alternative expression using sine and cosine.

tan(θ/2) = (1 - cosθ) / sinθ

Determining the Sign (±)

The sign of the half-angle formulas for sine and cosine depends on which quadrant θ/2 falls in:

  • Quadrant I (0° to 90°): sin and cos are both positive.
  • Quadrant II (90° to 180°): sin is positive, cos is negative.
  • Quadrant III (180° to 270°): sin and cos are both negative.
  • Quadrant IV (270° to 360°): sin is negative, cos is positive.

Applications of Half-Angle Formulas

  • Evaluating exact trig values: Find exact values of trig functions for angles like 15°, 22.5°, 75°, etc.
  • Integration: Simplify integrals involving even powers of sine or cosine.
  • Signal processing: Used in Fourier analysis and frequency domain computations.
  • Physics: Optical and wave interference calculations often use half-angle identities.
  • Computer graphics: Rotation interpolation and quaternion computations leverage these formulas.

Derivation from Double-Angle Identity

The half-angle formulas are derived from the double-angle identity for cosine: cos(2α) = 1 - 2sin2(α) = 2cos2(α) - 1. By substituting α = θ/2 and solving for sin(θ/2) and cos(θ/2), we obtain the half-angle formulas.

Tips for Using Half-Angle Formulas

  • Always determine the quadrant of θ/2 before choosing the sign.
  • Convert between degrees and radians as needed for your problem context.
  • For tangent, the alternative forms avoid the ambiguity of the ± sign entirely.
  • When θ is known, first compute cos(θ), then apply the half-angle formula.