Double Angle Identities Calculator
Calculate sin(2θ), cos(2θ), and tan(2θ) using double angle trigonometric identities
Calculate Double Angle Identities
Current: 30° = 0.5236 rad
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Single Angle Values
Double Angle Results (2θ = 60°)
Standard Formulas
sin(2θ) = 2sin(θ)cos(θ) = 2 × 0.5000 × 0.8660 = 0.866025
cos(2θ) = cos²(θ) - sin²(θ) = 0.7500 - 0.2500 = 0.500000
tan(2θ) = 2tan(θ)/(1 - tan²(θ)) = 2×0.5774/(1 - 0.3333) = 1.732051
Verification (Direct Calculation)
sin(60°) = 0.866025
cos(60°) = 0.500000
tan(60°) = 1.732051
Example Calculation
Finding cos(120°) using double angle identity
Given: We want to find cos(120°)
Note: 120° = 2 × 60°, so we can use the double angle formula
Known: sin(60°) = √3/2 ≈ 0.866, cos(60°) = 1/2 = 0.5
Solution
cos(120°) = cos(2 × 60°)
Using: cos(2θ) = cos²(θ) - sin²(θ)
cos(120°) = cos²(60°) - sin²(60°)
cos(120°) = (1/2)² - (√3/2)²
cos(120°) = 1/4 - 3/4 = -2/4 = -0.5
Double Angle Formulas
Sine
sin(2θ) = 2sin(θ)cos(θ)
Cosine
cos(2θ) = cos²(θ) - sin²(θ)
= 2cos²(θ) - 1
= 1 - 2sin²(θ)
Tangent
tan(2θ) = 2tan(θ)/(1-tan²(θ))
Common Double Angles
Tips & Notes
Double angle identities are derived from compound angle formulas
Multiple forms exist for cos(2θ) - choose the most convenient
tan(2θ) is undefined when tan²(θ) = 1
These identities are useful for integration and equation solving
Understanding Double Angle Identities
What are Double Angle Identities?
Double angle identities are trigonometric equations that express the trigonometric functions of twice an angle (2θ) in terms of the trigonometric functions of the original angle (θ). They are special cases of the compound angle formulas.
Applications
- •Simplifying trigonometric expressions
- •Solving trigonometric equations
- •Integration and differentiation in calculus
- •Physics problems involving oscillations
Derivation from Compound Angles
Starting with sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
Let α = β = θ:
sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ)
sin(2θ) = 2sin(θ)cos(θ)
Key Properties
- Periodicity: These identities maintain the periodicity of trig functions
- Symmetry: sin(2θ) is odd, cos(2θ) is even
- Range: Results follow standard trig function ranges