Double Angle Identities Calculator

Explore and verify all double angle identities with interactive numerical computation.

Select Identity & Enter Angle

Identity Verification

All identities verified for θ
45°
all forms produce identical results
sin(2θ) = 2sinθcosθ 1
cos(2θ) = cos²θ - sin²θ 0
cos(2θ) = 2cos²θ - 1 0
cos(2θ) = 1 - 2sin²θ 0
tan(2θ) = 2tanθ/(1-tan²θ) Undefined

Step-by-Step Verification

All 3 forms of cos(2θ) produce the same value: 0

Complete Reference: Double Angle Identities

Double angle identities are trigonometric equations that relate the trigonometric functions of a doubled angle (2θ) to those of the original angle (θ). They are among the most frequently used identities in trigonometry and are derived from the angle addition formulas by setting both angles equal.

All Double Angle Identity Forms

sin(2θ) Identity

Only one standard form exists for the sine double angle identity.

sin(2θ) = 2 sin(θ) cos(θ)

cos(2θ) - Form 1

Uses both sine and cosine. Derived directly from cos(A+B).

cos(2θ) = cos²(θ) - sin²(θ)

cos(2θ) - Form 2

Uses only cosine. Replace sin² with 1 - cos² in Form 1.

cos(2θ) = 2cos²(θ) - 1

cos(2θ) - Form 3

Uses only sine. Replace cos² with 1 - sin² in Form 1.

cos(2θ) = 1 - 2sin²(θ)

tan(2θ) Identity

Undefined when tan²(θ) = 1, i.e., θ = 45 + n(180).

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

Derived: Half-Angle

Solving cos(2θ) forms for sin(θ) and cos(θ) yields half-angle identities.

sin²(θ) = (1 - cos(2θ))/2

Derivation from Addition Formulas

Each double angle identity is derived by setting A = B = θ in the corresponding sum formula:

  1. sin(A+B) = sinA cosB + cosA sinB becomes sin(2θ) = sinθ cosθ + cosθ sinθ = 2sinθ cosθ
  2. cos(A+B) = cosA cosB - sinA sinB becomes cos(2θ) = cos²θ - sin²θ
  3. tan(A+B) = (tanA + tanB)/(1 - tanA tanB) becomes tan(2θ) = 2tanθ/(1 - tan²θ)

Why Three Forms for cos(2θ)?

The three forms exist because of the Pythagorean identity sin²θ + cos²θ = 1. By substituting sin²θ = 1 - cos²θ or cos²θ = 1 - sin²θ into the first form, we obtain the other two. Each form is useful in different contexts:

  • Form 1: Best when you know both sinθ and cosθ.
  • Form 2: Best when you only know cosθ, or when simplifying expressions with cos².
  • Form 3: Best when you only know sinθ, or when simplifying expressions with sin².

Applications of Double Angle Identities

  • Integration: Rewriting sin²x = (1 - cos2x)/2 simplifies integral computations.
  • Physics: The range of a projectile is R = (v² sin2θ)/g, which maximizes at θ = 45.
  • Power Reduction: Converting products like sinθcosθ into sin(2θ)/2.
  • Equation Solving: Transforming equations into solvable forms.
  • Fourier Analysis: Decomposing periodic signals into frequency components.

Common Pitfalls

  • Do not confuse sin(2θ) with 2sin(θ); they are not the same.
  • tan(2θ) is undefined when θ = 45 + n(180) because the denominator becomes zero.
  • Always verify your angle unit (degrees vs radians) before computing.
  • All three forms of cos(2θ) give the same numerical result for any θ.