Double Angle Calculator
Calculate sin(2θ), cos(2θ), and tan(2θ) using double angle formulas
Calculate Double Angle Values
Enter the angle for which you want to calculate double angle values
Double Angle Results
Original Angle: θ = 0.000°
Double Angle: 2θ = 0.000°
Formula: sin(2θ) = 2sin(θ)cos(θ)
Verification: Direct calculation = 0.000000
Formula: cos(2θ) = cos²(θ) - sin²(θ)
Verification: Direct calculation = 1.000000
Formula: tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Verification: Direct calculation = 0.000000
Example Calculations
Example 1: θ = 30°
Given: θ = 30°, sin(30°) = 0.5, cos(30°) = 0.866
sin(60°): sin(2×30°) = 2×sin(30°)×cos(30°) = 2×0.5×0.866 = 0.866
cos(60°): cos(2×30°) = cos²(30°) - sin²(30°) = 0.866² - 0.5² = 0.5
tan(60°): tan(2×30°) = 2×tan(30°)/(1-tan²(30°)) = 2×0.577/(1-0.577²) = 1.732
Example 2: θ = 45°
Given: θ = 45°, sin(45°) = cos(45°) = 0.707, tan(45°) = 1
sin(90°): sin(2×45°) = 2×sin(45°)×cos(45°) = 2×0.707×0.707 = 1
cos(90°): cos(2×45°) = cos²(45°) - sin²(45°) = 0.707² - 0.707² = 0
tan(90°): tan(2×45°) = 2×1/(1-1²) = 2/0 = ∞ (undefined)
Double Angle Formulas
Sine
sin(2θ) = 2sin(θ)cos(θ)
Cosine
cos(2θ) = cos²(θ) - sin²(θ)
cos(2θ) = 2cos²(θ) - 1
cos(2θ) = 1 - 2sin²(θ)
Tangent
tan(2θ) = 2tan(θ)/(1-tan²(θ))
*Undefined when tan(θ) = ±1
Special Cases
Important Notes
Double angle formulas are used to simplify trigonometric expressions
tan(2θ) is undefined when tan(θ) = ±1 (θ = ±45° + n×180°)
Multiple forms of cos(2θ) formula provide flexibility in calculations
2sin(x) ≠ sin(2x) - these are different functions
Understanding Double Angle Formulas
What are Double Angle Formulas?
Double angle formulas are trigonometric identities that express trigonometric functions of double angles (2θ) in terms of trigonometric functions of the original angle (θ).
Derivation
These formulas are derived using the angle addition formulas. For example:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
Setting A = B = θ:
sin(2θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ)
Applications
- •Simplifying trigonometric expressions
- •Solving trigonometric equations
- •Calculus integration and differentiation
- •Physics: wave mechanics and oscillations
- •Engineering: signal processing
Remember: These formulas are fundamental identities that every student of trigonometry should memorize and understand.