Linear combination of sine and cosine as phase shift

The Endeavour 2024-11-12

Here’s a simple calculation that I’ve done often enough that I’d like to save the result for my future reference and for the benefit of anyone searching on this.

A linear combination of sines and cosines

a sin(x) + b cos(x)

can be written as a sine with a phase shift

A sin(x + φ).

Going between {a, b} and {A, φ} is the calculation I’d like to save. For completeness I also include the case

A cos(x + ψ).

Derivation

Define

f(x) = a sin(x) + b cos(x)

and

g(x) = A sin(x + φ).

Both functions satisfy the differential equation

y″ + y = 0

and so f = g if and only if f(0) = g(0) and f′(0) = g′(0).

Setting the values at 0 equal implies

b = A sin(φ)

and setting the derivatives at 0 equal implies

a = A cos(φ).

Taking the ratio of these two equations shows

b/a = tan(φ)

and adding the squares of both equations shows

a² + b² = A².

Equations

First we consider the case

a sin(x) + b cos(x) = A sin(x + φ).

Sine with phase shift

If a and b are given,

A = √(a² + b²)

and

φ = tan⁻¹(b / a).

If A and φ are given,

a = A cos(φ)

and

b = A sin(φ)

from the previous section.

Cosine with phase shift

Now suppose we want

a sin(x) + b cos(x) = A cos(x + ψ)

If a and b are given, then

A = √(a² + b²)

as before and

ψ = − tan⁻¹(a / b).

If A and ψ are given then

a = − A sin(ψ)

and

b = A cos(ψ).

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