# Summing sinusoids of different frequency

I'm curious if there is a way to combine sinusoids of multiple frequency into a single sinusoid?

For example, I'm looking for a way to combine:

Acos(2παt + φ) + Bcos(2πβt + χ) where A, α, φ, B, β, χ are scalar constants and α & β represent the frequency.

I'd even be interested in a less complex solution to:
cos(2παt) + cos(2πβt)
if this is much simpler than the general form above.

I found an incomplete explanation on here that does a transform and then recombines using complex numbers ending up with a cosine with an arctangent inside.

However, explanations of the process are probably better than just a discrete example alone.

HallsofIvy
Homework Helper
cos(a+ b)= cos(a)cos(b)- sin(a)sin(b)

$cos(2\pi\alpha t+\phi)= cos(2\pi\alpha t)cos(\phi)- sin(2\pi\alpha t)sin(\phi)$ and
$cos(2\pi\beta t+\chi)= cos(2\pi\beta t)cos(\chi)- sin(2\pi\beta t)sin(\chi)$

cos(a+ b)= cos(a)cos(b)- sin(a)sin(b)

$cos(2\pi\alpha t+\phi)= cos(2\pi\alpha t)cos(\phi)- sin(2\pi\alpha t)sin(\phi)$ and
$cos(2\pi\beta t+\chi)= cos(2\pi\beta t)cos(\chi)- sin(2\pi\beta t)sin(\chi)$

I don't think trig identities are going to be much help.

Acos(2παt + φ) + Bcos(2πβt + χ) = A(cos(2παt)cos(φ) - sin(2παt)sin(φ)) + B(cos(2πβt)cos(χ) - sin(2πβt)sin(χ))

Which has turned a sum of 2 sinusoids into 4 products and 3 summing operations on 8 sinusoids. This is kind of the opposite direction of where I was looking to go. I'm trying to figure out if theres a transformation I can use to get this all inside of a single sinusoid function, even if there is a trig function inside the sinusoid.

Integral
Staff Emeritus
Gold Member
Summing sines of different frequencies can yield very NON sinusoidal shapes. Why do you expect to use a single sine function to describe non sine shaped functions?

i.e. The sum of all odd harmonics yields a square wave.

Summing sines of different frequencies can yield very NON sinusoidal shapes. Why do you expect to use a single sine function to describe non sine shaped functions?

i.e. The sum of all odd harmonics yields a square wave.

I don't expect to, I asked if it was possible.

I know it is possible to write any repeating function as a sum of sinusoids (Fourier?) but I was just curious if there is a process to combine a sum of 2 into 1. I'm mainly interested for representing purposes, I just wanted to see if there was a way to represent the sum mathematically without both sinusoids needing to be shown. Maybe it could be represented as a series?

arildno
Homework Helper
Gold Member
Dearly Missed
Functions that are called linearly independent cannot be summed into a single one.

Sines of different frequencies are no different in this regard than trying to sum together, say, the functions Bz and Bz^2 into a SINGLE power in z.

• FactChecker
What about the period of the sum of sinusoids of different periods? What would that be?

NascentOxygen
Staff Emeritus
What about the period of the sum of sinusoids of different periods? What would that be?
It would be a repeating waveform, of period equal to the LCM of their periods. Try some sketches and discover for yourself.

Samy_A
Homework Helper
It would be a repeating waveform, of period equal to the LCM of their periods. Try some sketches and discover for yourself.
To give an example, if the ratio of the periods is not a rational number, the sum will not be periodic.
##f(x)=\sin(x) + sin(\pi x)## is not periodic.

• NascentOxygen
FactChecker