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Half Angle Calculator

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

By Christopher FloiedPublished Updated

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. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown 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 computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — 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.

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.

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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.

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