Double Angle Calculator

Calculate sin(2θ), cos(2θ), and tan(2θ) using double angle formulas. Input any angle and get step-by-step solutions.

Enter Angle

Result

sin(2θ)
0.866025
for θ = 30 deg

Step-by-Step Solution

sin(2 x 30 deg) = 2 sin(30 deg) cos(30 deg) = 0.866025

Understanding Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of twice an angle (2θ) in terms of functions of the original angle (θ). They are derived from the angle addition formulas by setting both angles equal, and are among the most important identities in trigonometry.

The Double Angle Formulas

sin(2θ)

The sine double angle formula. There is only one standard form.

sin(2θ) = 2 sin(θ) cos(θ)

cos(2θ) - Form 1

Expressed in terms of both sine and cosine.

cos(2θ) = cos2(θ) - sin2(θ)

cos(2θ) - Form 2

Expressed using only cosine.

cos(2θ) = 2cos2(θ) - 1

cos(2θ) - Form 3

Expressed using only sine.

cos(2θ) = 1 - 2sin2(θ)

tan(2θ)

The tangent double angle formula.

tan(2θ) = 2tan(θ) / (1 - tan2(θ))

Derivation

All double angle formulas come from the angle sum identities with A = B = θ.

sin(A+B) with A = B = θ

Derivation from Angle Sum Formulas

The double angle formulas are special cases of the angle sum (addition) formulas. Starting from sin(A + B) = sin(A)cos(B) + cos(A)sin(B) and setting A = B = θ, we get sin(2θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ).

Similarly, from cos(A + B) = cos(A)cos(B) - sin(A)sin(B), setting A = B = θ gives cos(2θ) = cos2(θ) - sin2(θ). Using the Pythagorean identity sin2(θ) + cos2(θ) = 1, we can rewrite this as either 2cos2(θ) - 1 or 1 - 2sin2(θ).

Applications of Double Angle Formulas

  • Simplifying trigonometric expressions and solving equations.
  • Deriving half-angle formulas (by solving double angle formulas for the single angle).
  • Integration of trigonometric functions (e.g., integrating sin2(x) using cos(2x)).
  • Physics applications including wave interference, oscillations, and signal processing.
  • Engineering problems involving alternating current (AC) circuits.

Common Values

  • θ = 30 deg: sin(60 deg) = sqrt(3)/2, cos(60 deg) = 1/2, tan(60 deg) = sqrt(3)
  • θ = 45 deg: sin(90 deg) = 1, cos(90 deg) = 0, tan(90 deg) = undefined
  • θ = 60 deg: sin(120 deg) = sqrt(3)/2, cos(120 deg) = -1/2, tan(120 deg) = -sqrt(3)
  • θ = 90 deg: sin(180 deg) = 0, cos(180 deg) = -1, tan(180 deg) = 0

Related Identities

Double angle formulas are closely related to other trigonometric identities including half-angle formulas, power-reduction formulas, and product-to-sum formulas. They form part of a larger family of multiple angle formulas including triple angle formulas (sin(3θ), cos(3θ)) and beyond.