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Half Angle Calculator

Select the parameter and enter its value to calculate the corresponding half angle using this half angle calculator.

Calculate With:

Angle (x)

deg ▾

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Half-Angle Calculator

Use this online half-angle calculator to calculate half-angle values for sine (sin(θ/2)), cosine (cos(θ/2)), and tangent (tan(θ/2)) for any given angle. The calculator applies half-angle formulas to simplify trigonometric computations and reduce errors in manual calculations.

What is a Half-Angle?

Half-angle formulas are derived from double-angle identities and are used to compute trigonometric values for half of a given angle. These formulas allow you to determine sin(θ/2), cos(θ/2), and tan(θ/2) using the known value of θ.

Half-Angle Identities

The standard half-angle formulas are:

sin(θ/2) = ±√((1 − cosθ)/2)
cos(θ/2) = ±√((1 + cosθ)/2)
tan(θ/2) = (1 − cosθ)/sinθ = sinθ/(1 + cosθ)

For related calculations, see the Double Angle Calculator.

How to Calculate Half-Angle Values

Example: Find the half-angle values for θ = 60°.

1. Sine Half-Angle

sin(60°/2) = √((1 − cos 60°)/2) = √((1 − 0.5)/2) = √0.25 = 0.5

2. Cosine Half-Angle

cos(60°/2) = √((1 + cos 60°)/2) = √((1 + 0.5)/2) = √0.75 ≈ 0.866

3. Tangent Half-Angle

tan(60°/2) = (1 − cos 60°)/sin 60° = (1 − 0.5)/0.866 ≈ 0.577
Alternatively: tan(θ/2) = sinθ / (1 + cosθ) ≈ 0.577

For right triangle applications, check the Pythagorean Theorem Calculator.

How the Half-Angle Calculator Works

Input:

Select the trigonometric function (sin, cos, or tan).
Enter the angle θ in degrees or radians.
Click Calculate.

Output:

Displays the half-angle value for the selected function.
Shows the formula applied and step-by-step calculations.
Optionally, provides a table of half-angle values for multiple angles.

FAQ

What are the real-life applications of trigonometry?

Trigonometry is widely used in aviation, navigation, cartography (map-making), satellite systems, and construction (e.g., roof slopes).

How were trigonometric functions discovered?

The Greek mathematician Hipparchus created the first table of chords around 140 BC, which evolved into modern trigonometric functions.

Conclusion

An online half-angle calculator simplifies the process of finding sin(θ/2), cos(θ/2), and tan(θ/2). Manual calculations can be tricky, but this free tool provides instant, step-by-step results.

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