---
title: "Compound Angle Formulae"
description: "Expand sin(A ± B), cos(A ± B) and tan(A ± B), and use them for exact values of angles built from known ones and for equations mixing two angles."
canonical: https://lightmysky.com/learn/mathematics/compound-angle-formulae-mt_ANDmX8exwg
source: https://lightmysky.com/learn/mathematics/compound-angle-formulae-mt_ANDmX8exwg.md
retrieved: 2026-09-12
---

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# Compound Angle Formulae

Expand sin(A ± B), cos(A ± B) and tan(A ± B), and use them for exact values of angles built from known ones and for equations mixing two angles.

Subject: Mathematics · Area: Geometry · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/compound-angle-formulae-mt_ANDmX8exwg

## Ready when they can

- Find an exact value for cos 15° by writing it as cos(45° - 30°)
- Expand sin(x + π/4) and simplify the constants
- Solve an equation after expanding its compound angle

## Lesson: Splitting angles to conquer them

Four expansions cover sums and differences. Sine keeps the middle sign, cosine flips it. Sin of A plus B is sin A cos B plus cos A sin B. Cos of A minus B is cos A cos B plus sin A sin B. Write the formula before substituting.

**Example.** Cos 15 degrees splits as cos of 45 minus 30. Expand: cos 45 cos 30 plus sin 45 sin 30. Special values give root 6 plus root 2, all over 4. Splitting turned an unknown angle into known ones.

Expanding helps with sums inside expressions. Sin of x plus 45 degrees is sin x cos 45 plus cos x sin 45. The constants simplify to root 2 over 2 each. Collapsing runs backwards: a sin-cos plus cos-sin pattern names one sum.

**Tip.** After collapsing, equations fall quickly. Sin of x plus 30 degrees equals 1 gives x plus 30 equals 90, so x is 60. Track the shifted range: x between 0 and 360 with a 30 shift solves across 30 to 390.

**Recap.** Split into known angles, expand with careful signs, and track the range through every shift.

## Practice

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

## Needs first

- [Solving Trigonometric Equations in a Given Interval](https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA)

## Opens up

- [Double Angle Formulae](https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam)