Double Angle Formula Calculator

Calculate sin(2θ), cos(2θ), and tan(2θ) using double angle formulas. Verify trig identities and simplify expressions. Also covers half-angle formulas.

📐 Trigonometry📐 sin(2θ) = 2sinθcosθ🔢 Math
Angle θ (degrees)
Or: sin(θ) value directly
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Double Angle Formula Calculator—sin(2θ) = 2sinθcosθdimensionless

Step-by-Step Examples

Example 1
θ=30°

Find sin(60°) using double angle.

  • sin(2×30°)=2×sin30°×cos30°=2×½×√3/2=√3/2≈0.866
  • Direct: sin(60°)=0.866 ✓
✓ sin(60°) = √3/2
Example 2
θ=45°

Find cos(90°) and tan(90°).

  • cos(2×45°)=cos²45°−sin²45°=½−½=0
  • cos(90°)=0 ✓
  • tan(90°) is undefined
✓ cos(90°)=0, tan(90°)=undefined
Example 3
From sin value

sin(θ)=0.6. Find sin(2θ).

  • cos(θ)=√(1−0.36)=0.8
  • sin(2θ)=2×0.6×0.8=0.96
✓ sin(2θ) = 0.96

Real-World Applications

📐
Trigonometry
Double angle formulas are used in every trig identity proof and simplification.
🧪
Physics
Rotational kinematics, optics (Brewster's angle), and wave interference.
💻
Signal Processing
Double frequency terms appear in products of sinusoidal signals (beat frequency).
📚
Calculus
Power-reducing formulas (from double angle) needed for integrating sin²x and cos²x.

Common Mistakes to Avoid

⚠️
Three forms of cos(2θ)

All three are equivalent: cos²θ−sin²θ, 1−2sin²θ, 2cos²θ−1. Choose whichever is most useful.

⚠️
sin(2θ) ≠ 2sin(θ)

sin(2×30°)=sin(60°)≈0.866 ≠ 2×sin(30°)=1. This is the most common mistake.

⚠️
Half angle: be careful with signs

sin(θ/2) = ±√((1−cosθ)/2). The ± depends on which quadrant θ/2 is in.

Frequently Asked Questions

What are double angle formulas? ▾
sin(2θ)=2sinθcosθ. cos(2θ)=cos²θ−sin²θ=1−2sin²θ=2cos²θ−1. tan(2θ)=2tanθ/(1−tan²θ).
Why do three forms of cos(2θ) exist? ▾
All use sin²θ+cos²θ=1. Substituting sin²θ=1−cos²θ gives cos(2θ)=2cos²θ−1. Substituting cos²θ=1−sin²θ gives 1−2sin²θ. Each form is useful in different simplification contexts.
What are half-angle formulas? ▾
sin(θ/2)=±√((1−cosθ)/2), cos(θ/2)=±√((1+cosθ)/2). Derived by substituting θ/2 for θ in double-angle identities.
What are power-reducing formulas? ▾
From cos(2θ)=1−2sin²θ: sin²θ=(1−cos2θ)/2. From cos(2θ)=2cos²θ−1: cos²θ=(1+cos2θ)/2. Used for integrating powers of trig functions.
When is tan(2θ) undefined? ▾
When cos(2θ)=0, i.e., 2θ=90°,270°,..., i.e., θ=45°,135°.... Also when tan(θ) is undefined.

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Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

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