Double Angle Calculator

Compute sin(2θ), cos(2θ), and tan(2θ) for any angle. Visualize on the unit circle and see the exact identities.

0° 30° (π/6) 45° (π/4) 60° (π/3) 90° (π/2) 120° (2π/3) 180° (π)
Double Angle Values
sin(2θ)0.866025
cos(2θ)0.500000
tan(2θ)1.732051
cot(2θ)0.577350
sec(2θ)2.000000
csc(2θ)1.154701
θ (degrees)30.00°
θ (radians)0.523599
2θ (degrees)60.00°
2θ (radians)1.047198

All double-angle identities:

  • sin(2θ) = 2 sin θ cos θ
  • cos(2θ) = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ
  • tan(2θ) = 2 tan θ / (1 – tan²θ)
  • cot(2θ) = (cot²θ – 1) / (2 cot θ)
  • sec(2θ) = 1 / cos(2θ)
  • csc(2θ) = 1 / sin(2θ)

Understanding Double-Angle Formulas

Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. They are derived from the sum formulas and are fundamental in simplifying expressions, solving equations, and analyzing periodic phenomena.

Derivation from Sum Formulas

Starting with the sine and cosine addition formulas:

  • sin(α + β) = sin α cos β + cos α sin β
  • cos(α + β) = cos α cos β – sin α sin β
  • tan(α + β) = (tan α + tan β) / (1 – tan α tan β)

Setting α = β = θ gives the double-angle identities for sin, cos, and tan. The remaining three functions follow from reciprocals:

  • cot(2θ) = 1 / tan(2θ) = (cot²θ – 1) / (2 cot θ)
  • sec(2θ) = 1 / cos(2θ)
  • csc(2θ) = 1 / sin(2θ)

Geometric Interpretation

On the unit circle, the angle θ corresponds to a point (cos θ, sin θ). The double angle 2θ rotates this point further. The coordinates of the point at 2θ are (cos 2θ, sin 2θ). The double-angle formulas show how these coordinates relate to the original ones algebraically.

Common Applications

  • Calculus: Integration of powers of sine and cosine (e.g., ∫ sin²x dx uses cos(2x)).
  • Physics: Wave interference, harmonic oscillators, and projectile motion.
  • Engineering: Signal modulation (e.g., frequency doubling in electronics).
  • Geometry: Calculating chord lengths, area of circular sectors.
  • Trigonometric equations: Solving equations like sin 2θ = cos θ.
  • Numerical analysis: Reduction formulas for efficient computation.

Related Identities

Double-angle formulas are part of a larger family of trigonometric identities:

  • Half-angle formulas: sin(θ/2), cos(θ/2), tan(θ/2) – used in integral calculus and geometry.
  • Power-reduction formulas: Express sin²θ, cos²θ in terms of cos(2θ).
  • Triple-angle formulas: sin(3θ), cos(3θ) – extensions to multiples.

Tip: The unit circle visualization above shows how the angle doubles. Notice that the red radius (2θ) is exactly twice the angle of the blue radius (θ). This geometric relationship is mirrored in the algebraic identities.