Double Angle Calculator

How to Calculate Double Angles

Double angle formulas are essential in trigonometry for simplifying expressions and solving complex problems. These formulas express trigonometric functions of twice an angle in terms of functions of the original angle.

Formulas

The double angle formulas are:

\[ \sin(2\theta) = 2\sin\theta\cos\theta \] \[ \cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \] \[ \tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta} \]

Where θ is the original angle.

Calculation Steps

1. Ensure the angle is in radians. If it's in degrees, convert it using the formula: radians = degrees × (π / 180°).
2. Calculate sin(θ) and cos(θ) for the given angle.
3. Apply the double angle formulas:
   • For sin(2θ), multiply 2 × sin(θ) × cos(θ)
   • For cos(2θ), calculate cos²(θ) - sin²(θ)
   • For tan(2θ), divide 2tan(θ) by 1 - tan²(θ)

Example

Let's calculate the double angle values for θ = 30°:

1. Convert 30° to radians: 30° × (π / 180°) ≈ 0.5236 rad
2. Calculate sin(30°) = 0.5 and cos(30°) = √3/2 ≈ 0.866
3. Apply the formulas:
   • sin(60°) = 2 × 0.5 × 0.866 ≈ 0.866
   • cos(60°) = 0.866² - 0.5² = 0.5
   • tan(60°) = (2 × 0.577) / (1 - 0.577²) ≈ 1.732

Visual Representation

This diagram illustrates the double angle relationship for 30°. The blue arc represents the original angle (30°), while the red arc shows the double angle (60°).