Double Angle Formula Calculator
Calculate sin(2θ), cos(2θ), and tan(2θ) using double angle formulas with step-by-step solutions
Calculate Double Angle Formulas
Enter the angle for which you want to calculate double angle formulas
Double Angle Results
Given Angle
θ = 0°
2θ = 0.00°
θ = 0.00° = 0.0000 rad
2θ = 0.00° = 0.0000 rad
Original Trigonometric Values
Alternative Cosine Double Angle Formulas
Step-by-step Solutions
sin(2θ) = 2 · sin(θ) · cos(θ)
sin(2θ) = 2 · 0.000000 · 1.000000
sin(2θ) = 2 · 0.000000
sin(2θ) = 0.000000
cos(2θ) = cos²(θ) - sin²(θ)
cos(2θ) = (1.000000)² - (0.000000)²
cos(2θ) = 1.000000 - 0.000000
cos(2θ) = 1.000000
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
tan(2θ) = (2 · 0.000000) / (1 - (0.000000)²)
tan(2θ) = 0.000000 / (1 - 0.000000)
tan(2θ) = 0.000000 / 1.000000
tan(2θ) = 0.000000
Example Calculation
Calculate double angle formulas for θ = 30°
Given: θ = 30°, so 2θ = 60°
Known values: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = √3/3
sin(2θ)
sin(60°) = 2 · sin(30°) · cos(30°)
= 2 · (1/2) · (√3/2)
= 2 · √3/4 = √3/2
cos(2θ)
cos(60°) = cos²(30°) - sin²(30°)
= (√3/2)² - (1/2)²
= 3/4 - 1/4 = 1/2
tan(2θ)
tan(60°) = 2tan(30°) / (1 - tan²(30°))
= 2(√3/3) / (1 - (√3/3)²)
= (2√3/3) / (2/3) = √3
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))
Common Double Angles
Quick Tips
Double angle formulas are derived from sum formulas
Cosine has three equivalent double angle forms
Useful for integration and solving trig equations
Check answers by direct calculation of sin(2θ)
Understanding Double Angle Formulas
What are Double Angle Formulas?
Double angle formulas express trigonometric functions of 2θ in terms of trigonometric functions of θ. They are derived from the sum formulas by setting both angles equal to θ in expressions like sin(θ + θ) and cos(θ + θ).
Derivation from Sum Formulas
Sine: sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ)
Cosine: cos(θ + θ) = cos(θ)cos(θ) - sin(θ)sin(θ) = cos²(θ) - sin²(θ)
Key Properties
- •Sin double angle formula is unique and straightforward
- •Cosine has three equivalent forms for different applications
- •Tangent formula may be undefined when tan²(θ) = 1
Applications
- •Simplifying trigonometric expressions
- •Solving trigonometric equations
- •Integration in calculus
- •Signal processing and wave analysis
Important Identities
sin²(θ) = (1 - cos(2θ))/2
cos²(θ) = (1 + cos(2θ))/2
tan(2θ) = sin(2θ)/cos(2θ)
1 + cos(2θ) = 2cos²(θ)
1 - cos(2θ) = 2sin²(θ)
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