Double Angle Formula Calculator
Calculate trigonometric functions of double angles
How to Use This Calculator
Enter the Angle
Input the angle (θ) you want to calculate the double angle functions for.
Select Unit
Choose whether your angle is in degrees or radians.
Calculate
Click "Calculate Double Angle" to see sin(2θ), cos(2θ), and tan(2θ) calculated both directly and using the formulas.
Formula
Double Angle Formulas:
sin(2θ) = 2sin(θ)cos(θ)
Sine double angle formula
cos(2θ) = cos²(θ) - sin²(θ)
Alternative forms: cos(2θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Tangent double angle formula (undefined when tan²(θ) = 1)
Derivation:
These formulas come from the angle addition formulas. For example, sin(2θ) = sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ).
About Double Angle Formula Calculator
The Double Angle Formula Calculator helps you find trigonometric values for double angles (2θ) using double angle identities. These formulas are essential in trigonometry, calculus, and various engineering applications. Instead of calculating 2θ directly, you can use these formulas to compute sin(2θ), cos(2θ), and tan(2θ) from the values of sin(θ), cos(θ), and tan(θ).
When to Use This Calculator
- Trigonometric Simplification: Simplify expressions involving double angles
- Integration: Solve integrals involving trigonometric functions
- Problem Solving: Calculate double angle values without computing 2θ directly
- Verification: Verify your manual calculations using double angle formulas
Why Use Our Calculator?
- ✅ Formula Verification: Shows both direct calculation and formula-based results
- ✅ Complete Results: Calculates all three main double angle functions
- ✅ Educational: Helps understand double angle identities
- ✅ 100% Free: No registration required
Frequently Asked Questions
What are double angle formulas used for?
Double angle formulas are used to simplify trigonometric expressions, solve equations, compute integrals in calculus, and in various physics and engineering problems involving periodic functions.
Why are there multiple forms of cos(2θ)?
cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ). These forms are equivalent using the Pythagorean identity sin²(θ) + cos²(θ) = 1. Different forms are useful in different contexts.
When is tan(2θ) undefined?
tan(2θ) is undefined when 1 - tan²(θ) = 0, which means tan²(θ) = 1, so tan(θ) = ±1. This occurs when θ = 45° + n×90° (or π/4 + n×π/2 in radians).