Summing sinusoids of different frequency

[funnyguy]

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.
https://www.physicsforums.com/showthread.php?t=372263

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

 

[HallsofIvy]

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

 

[funnyguy]

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 there's 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]

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.

 

[funnyguy]

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]

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.

 

[LuizHP]

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

 

[NascentOxygen]

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]

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.

 

[FactChecker]

The short answer is no. Two different pure sine wave frequencies can not be summed to another pure sine wave frequency.