Double Angle Formulas
Reference for double angle formulas: sin(2θ)=2sinθcosθ, cos(2θ)=cos²θ-sin²θ, and tan(2θ).
Includes worked examples and derivations from the angle sum formulas.
The Formulas
sin(2θ) = 2 × sin θ × cos θ
cos(2θ) = cos²θ - sin²θ
cos(2θ) = 2cos²θ - 1
cos(2θ) = 1 - 2sin²θ
tan(2θ) = 2tan θ / (1 - tan²θ)
Double angle formulas express trig functions of 2θ in terms of trig functions of θ.
There are three equivalent forms for cos(2θ) — use whichever is most convenient.
Variables
| Symbol | Meaning |
|---|---|
| θ | The original angle |
| 2θ | Double the original angle |
| sin, cos, tan | The standard trigonometric functions |
Example 1
If sin θ = 3/5 and cos θ = 4/5, find sin(2θ)
sin(2θ) = 2 × sin θ × cos θ
sin(2θ) = 2 × (3/5) × (4/5)
sin(2θ) = 2 × 12/25
sin(2θ) = 24/25 = 0.96
Example 2
Find cos(60°) using the double angle formula with θ = 30°
cos(2θ) = 2cos²θ - 1
cos(60°) = 2cos²(30°) - 1
cos(60°) = 2 × (√3/2)² - 1 = 2 × (3/4) - 1
cos(60°) = 3/2 - 1
cos(60°) = 1/2 = 0.5 ✓
When to Use It
Use double angle formulas when:
- Simplifying expressions that contain sin(2θ), cos(2θ), or tan(2θ)
- Solving trig equations involving double angles
- Deriving other identities (half angle formulas are derived from these)
- Working with wave equations, oscillations, or signal processing
Key Notes
- The three forms of cos(2A): cos(2A) = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1. All three are equivalent. Choose the form that eliminates the variable you don't need in a given problem.
- sin(2A) = 2 sinA cosA: This form appears frequently in integration and simplification. Recognizing it in the reverse direction (2 sinA cosA → sin(2A)) is often the key to solving trigonometric integrals.
- tan(2A) = 2tanA / (1 − tan²A): Undefined when tan²A = 1, i.e., when A = 45° + n×90°. At these angles, the double angle is 90° + n×180°, where tan is undefined.
- Derived from angle addition: Setting B = A in sin(A+B) gives sin(2A) directly. Setting B = A in cos(A+B) gives cos(2A) = cos²A − sin²A.
- Power-reduction application: Rearranging the double-angle identity gives sin²A = (1 − cos2A)/2 and cos²A = (1 + cos2A)/2. These are essential for integrating sin² and cos² in calculus.