Half Angle Calculator

How to Calculate Half Angles

Half angle formulas are important in trigonometry for simplifying expressions and solving complex problems. These formulas express trigonometric functions of half an angle in terms of functions of the original angle.

Formulas

The half angle formulas are:

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

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 half angle formulas:
   • For sin(θ/2), take the square root of (1 - cos(θ)) / 2
   • For cos(θ/2), take the square root of (1 + cos(θ)) / 2
   • For tan(θ/2), divide sin(θ) by 1 + cos(θ)

Example

Let's calculate the half angle values for θ = 60°:

1. Convert 60° to radians: 60° × (π / 180°) ≈ 1.0472 rad
2. Calculate sin(60°) = √3/2 ≈ 0.866 and cos(60°) = 1/2 = 0.5
3. Apply the formulas:
   • sin(30°) = √((1 - 0.5) / 2) = 0.5
   • cos(30°) = √((1 + 0.5) / 2) ≈ 0.866
   • tan(30°) = 0.866 / (1 + 0.5) ≈ 0.577

Visual Representation

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