2️⃣ Double & Half Angle Calculator

Calculate double angle and half angle formulas for trigonometric functions. Verify formulas and explore angle relationships with step-by-step solutions.

📊 12,000+ formulas calculated 🔄 Updated: February 2026 ⭐ 4.8/5 Rating

Understanding Double and Half Angle Formulas

Double and half angle formulas are powerful tools in trigonometry that express trigonometric functions of multiple angles in terms of functions of the original angle. These formulas are essential for solving complex trigonometric equations, simplifying expressions, and integrating trigonometric functions in calculus.

Double angle formulas allow you to find sin(2θ), cos(2θ), and tan(2θ) when you know the values of sin(θ), cos(θ), and tan(θ). Half angle formulas work in reverse, helping you find trigonometric values of θ/2 when you know the values for θ. These relationships are fundamental in physics for analyzing wave interference, in engineering for signal processing, and in mathematics for solving advanced problems.

Our calculator not only computes these formulas but also shows multiple equivalent forms and verifies results by comparing direct calculation with formula-based calculation, helping you understand the mathematical relationships and build confidence in using these important identities.

Calculate Double or Half Angle Formulas

Formula Reference

Double Angle Formulas

sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)
cos(2θ) = 2cos²(θ) - 1
cos(2θ) = 1 - 2sin²(θ)
tan(2θ) = 2tan(θ) / (1 - tan²(θ))

Half Angle Formulas

sin(θ/2) = ±√[(1 - cos(θ))/2]
cos(θ/2) = ±√[(1 + cos(θ))/2]
tan(θ/2) = ±√[(1 - cos(θ))/(1 + cos(θ))]
tan(θ/2) = sin(θ) / (1 + cos(θ))
tan(θ/2) = (1 - cos(θ)) / sin(θ)

Applications and Uses

📐 Trigonometric Equations

Double and half angle formulas are essential for solving complex trigonometric equations that can't be solved with basic identities.

∫ Calculus Integration

These formulas are crucial for integrating powers of trigonometric functions and solving differential equations in calculus.

🌊 Wave Analysis

In physics and engineering, these formulas help analyze wave interference, frequency doubling, and harmonic analysis.

📡 Signal Processing

Digital signal processing uses these formulas for frequency analysis, filter design, and signal modulation techniques.

Frequently Asked Questions

How do I determine the correct sign for half angle formulas?

The sign depends on which quadrant the half angle falls in. For sin(θ/2), use positive if θ/2 is in quadrants I or II, negative if in III or IV. For cos(θ/2), use positive if θ/2 is in quadrants I or IV, negative if in II or III. The tan(θ/2) formulas without ± are always correct and avoid sign ambiguity.

Why are there multiple formulas for cos(2θ)?

All three formulas for cos(2θ) are equivalent and derived from the basic cos(2θ) = cos²θ - sin²θ using the Pythagorean identity sin²θ + cos²θ = 1. Different forms are useful in different situations: use 2cos²θ - 1 when you know cos(θ), use 1 - 2sin²θ when you know sin(θ).

When should I use these formulas instead of a calculator?

Use these formulas when you need exact values (like √2/2 instead of 0.707), when solving equations algebraically, when working with variables instead of specific numbers, or when you need to simplify expressions before substituting values. They're also essential for proving other identities and solving calculus problems.