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Half Angle Calculator

Calculate sin(θ/2), cos(θ/2), and tan(θ/2) using the half-angle formulas.

Reviewed by Chase FloiedUpdated

This free online half angle calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

sin(θ/2)

0.5

cos(θ/2)

0.866025

tan(θ/2)

0.57735

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Half Angle Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Half Angle Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Half Angle Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Half Angle Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Half Angle Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate sin(θ/2), cos(θ/2), and tan(θ/2) using the half-angle formulas. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Half Angle Calculator

The half-angle formulas allow you to express the sine, cosine, and tangent of half an angle in terms of the cosine of the full angle. These formulas are: sin(θ/2) = ±√((1 − cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2), and tan(θ/2) = sin θ/(1 + cos θ). The sign depends on the quadrant in which θ/2 lies. Half-angle formulas are derived from the double-angle cosine identity and are widely used in calculus (especially for integrating trigonometric expressions), computer graphics, and signal processing. They also provide exact values for angles that are half of standard angles, such as 15° = 30°/2 and 22.5° = 45°/2. This calculator assumes the positive root (first quadrant) and computes all three half-angle values.

The Math Behind It

The half-angle formulas are derived from the double-angle identity for cosine. Starting with cos(2α) = 1 − 2sin²(α), we set θ = 2α (so α = θ/2) to get cos θ = 1 − 2sin²(θ/2), which rearranges to sin²(θ/2) = (1 − cos θ)/2. Taking the square root gives sin(θ/2) = ±√((1 − cos θ)/2). Similarly, from cos(2α) = 2cos²(α) − 1, we get cos(θ/2) = ±√((1 + cos θ)/2). The tangent half-angle has three useful forms: tan(θ/2) = sin θ/(1 + cos θ) = (1 − cos θ)/sin θ = ±√((1 − cos θ)/(1 + cos θ)). The Weierstrass substitution t = tan(θ/2) transforms trigonometric integrals into rational function integrals: sin θ = 2t/(1+t²), cos θ = (1−t²)/(1+t²), dθ = 2dt/(1+t²). This technique can integrate any rational function of sin θ and cos θ. In geometry, the tangent half-angle formula relates the incircle radius to the triangle: r = (s − a)tan(A/2).

Formula Reference

Half Angle – Sine

sin(θ/2) = ±√((1 − cos θ)/2)

Variables: θ = original angle

Half Angle – Cosine

cos(θ/2) = ±√((1 + cos θ)/2)

Variables: θ = original angle

Half Angle – Tangent

tan(θ/2) = sin θ / (1 + cos θ)

Variables: θ = original angle

Worked Examples

Example 1: Finding sin(15°)

Use the half-angle formula with θ = 30°.

Step 1:sin(15°) = sin(30°/2) = √((1 − cos 30°)/2)
Step 2:cos 30° = √3/2 ≈ 0.8660
Step 3:sin(15°) = √((1 − 0.8660)/2) = √(0.1340/2) = √0.0670

sin(15°) ≈ 0.2588

Common Mistakes & Tips

  • !Forgetting to choose the correct sign (±) based on the quadrant of θ/2.
  • !Using cos(θ/2) when you need cos(θ) — the formulas use cos of the full angle.
  • !Confusing the half-angle formulas with the double-angle formulas.

Related Concepts

Double Angle Calculator

Compute sin(2θ), cos(2θ), and tan(2θ).

Unit Circle

Understand angle positions and quadrant signs.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Double Angle Formula Calculator

Frequently Asked Questions

How do I know which sign (+ or −) to use?

The sign depends on the quadrant where θ/2 falls. If θ/2 is in Q1 or Q2, sine is positive. If Q1 or Q4, cosine is positive.

What is the Weierstrass substitution?

Setting t = tan(θ/2) converts trig integrals into rational function integrals, making any rational trig expression integrable.