---
title: "Double Angle Formulae"
description: "Set B = A in the compound angle formulae to get sin 2A, cos 2A and tan 2A, and pick between the three forms of cos 2A to remove either a squared term or a doubled angle."
canonical: https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam
source: https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam.md
retrieved: 2026-09-12
---

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# Double Angle Formulae

Set B = A in the compound angle formulae to get sin 2A, cos 2A and tan 2A, and pick between the three forms of cos 2A to remove either a squared term or a doubled angle.

Subject: Mathematics · Area: Geometry · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam

## Ready when they can

- Derive sin 2A by writing it as sin(A + A)
- Choose the form of cos 2A that clears a sin²x from an equation
- Solve an equation containing both sin 2x and sin x

## Lesson: Doubling angles, halving work

Set B to A in the addition rules. Sin of A plus A is sin A cos A plus cos A sin A, which is 2 sin A cos A. One substitution gives the famous formula. Every double angle rule is born this way.

Cosine offers three versions: cos squared minus sin squared, 2 cos squared minus 1, and 1 minus 2 sin squared. Pick the version whose squares match your equation. An equation in sin squared calls for 1 minus 2 sin squared.

**Example.** Solve sin 2x equals sin x. Expand the left to 2 sin x cos x, then bring all across: sin x times 2 cos x minus 1 equals 0. Each factor gives angles: 0, 60, 180, and 300 degrees in a full turn.

**Tip.** Never divide by sin x to simplify, since that deletes the 0 and 180 degree solutions. Factorise and set each factor to zero. With sin A 0.6 and cos A 0.8, sin 2A is 0.96.

**Recap.** Derive from the sum rule, match the cosine form to the equation, and factorise instead of dividing.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Compound Angle Formulae](https://lightmysky.com/learn/mathematics/compound-angle-formulae-mt_ANDmX8exwg)
- [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh)

## Opens up

- [Combining a Sine and a Cosine into a Single Wave](https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq)