---
title: "Combining a Sine and a Cosine into a Single Wave"
description: "Write a sin x + b cos x as R sin(x + α), so two waves become one with a single amplitude and shift. Maximum and minimum values, and the x where they happen, then read off directly."
canonical: https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq
source: https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq.md
retrieved: 2026-09-12
---

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# Combining a Sine and a Cosine into a Single Wave

Write a sin x + b cos x as R sin(x + α), so two waves become one with a single amplitude and shift. Maximum and minimum values, and the x where they happen, then read off directly.

Subject: Mathematics · Area: Geometry · Ages 16 to 17
Page: https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq

## Ready when they can

- Write 3 sin x + 4 cos x in the form R sin(x + α)
- State the maximum value of that expression and the x where it occurs
- Solve an equation by rewriting the left side as a single wave

## Lesson: Two waves in, one wave out

A sum like a sin x plus b cos x is one wave in disguise. Write it as R sin of x plus alpha. Then R squared equals a squared plus b squared, and tan alpha equals b over a. Two waves become one amplitude with one shift.

**Example.** Take 3 sin x plus 4 cos x. Then R is the root of 9 plus 16, which is 5. The angle has tan alpha 4 over 3. So the sum is 5 sin of x plus alpha. One pair of numbers now does the work of two.

Maximum and minimum read off directly. The sine part peaks at 1, so the max is R and the min is minus R. For the example above the largest value is 5. It happens where x plus alpha equals 90 degrees.

**Tip.** Equations yield the same way. Divide 3 sin x plus 4 cos x equals 2.5 by 5 to get sin of x plus alpha equals 0.5. First check the right side sits within minus 1 and 1. Then solve the plain sine equation.

**Recap.** Find R and alpha, then read the max, the min, and the solutions off the single wave.

## Practice

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

## Needs first

- [Double Angle Formulae](https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam)
- [The Sine Rule](https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho)

## Opens up

- [Phasor Addition and the Mathematics of Interference](https://lightmysky.com/learn/science/phasor-addition-and-the-mathematics-of-interference-mt_dxh7HHbIUw)
- [Coupled Oscillators and Normal Modes](https://lightmysky.com/learn/science/coupled-oscillators-and-normal-modes-mt_vnTGOIqZzP)
- [Impedance and Phasors in AC Circuits](https://lightmysky.com/learn/science/impedance-and-phasors-in-ac-circuits-mt_vQVlVWv2ln)
- [Damped and Driven Oscillations](https://lightmysky.com/learn/mathematics/damped-and-driven-oscillations-mt_Y8K1njb1PH)