Half Angle Formula Calculator Degrees

This half angle formula calculator helps you compute trigonometric values for half angles in degrees. Whether you're solving geometry problems, physics equations, or engineering calculations, understanding half angle formulas is essential for accurate results.

What is the Half Angle Formula?

The half angle formulas allow you to find trigonometric values (sine, cosine, tangent) for half of a given angle. These formulas are derived from the double angle formulas and are particularly useful when you need to find intermediate values between known angles.

Half angle formulas are commonly used in:

Solving right triangles
Physics problems involving waves and oscillations
Engineering calculations for mechanical systems
Computer graphics for smooth interpolations

Note: All angles in this calculator are in degrees. For radians, you would use different formulas.

How to Use the Calculator

Enter the angle in degrees for which you want to find the half angle
Select the trigonometric function you need (sine, cosine, or tangent)
Click "Calculate" to see the result
Review the explanation and chart visualization

The calculator will display the result in both decimal and fraction forms, along with a visual representation of the angle relationship.

Half Angle Formulas

The half angle formulas for sine, cosine, and tangent are:

sin(θ/2) = ±√[(1 - cosθ)/2]

cos(θ/2) = ±√[(1 + cosθ)/2]

tan(θ/2) = (1 - cosθ)/sinθ = sinθ/(1 + cosθ)

The sign (positive or negative) depends on the quadrant in which the half angle lies.

Worked Examples

Example 1: Finding sin(30°)

Using the half angle formula for sine:

sin(30°) = sin(60°/2) = ±√[(1 - cos60°)/2] = ±√[(1 - 0.5)/2] = ±√[0.25] = ±0.5

Since 30° is in the first quadrant, we take the positive value: sin(30°) = 0.5

Example 2: Finding cos(15°)

Using the half angle formula for cosine:

cos(15°) = cos(30°/2) = ±√[(1 + cos30°)/2] = ±√[(1 + √3/2)/2] ≈ ±√[0.933] ≈ ±0.968

Since 15° is in the first quadrant, we take the positive value: cos(15°) ≈ 0.968

Frequently Asked Questions

What is the difference between half angle and double angle formulas?: The half angle formulas allow you to find trigonometric values for half of an angle, while double angle formulas find values for double an angle. They are mathematically related but solve different problems.
When should I use half angle formulas?: Use half angle formulas when you need to find trigonometric values for angles that are half of known angles, such as in problems involving bisected angles or intermediate values.
How do I know when to use the positive or negative sign in half angle formulas?: The sign depends on the quadrant in which the half angle lies. You can determine this by examining the original angle and applying standard trigonometric sign rules.
Can I use half angle formulas for angles in radians?: No, the formulas provided are specifically for degrees. For radians, you would use different formulas that account for the radian measure.
What if my angle is negative?: Negative angles are handled by the formulas, but you should ensure the resulting half angle is in the correct quadrant for the sign to be properly determined.