Summing sines/cosines (Harmonic Addition Theorem)

[omission9]

I have a mathematical model which, in part, does a calculation based on the location of certain points on a unit circle.
I am just working in 2d so for some arbitrary values of a,b,c,d,e,f
In the case of the unit circle equally divided so that I have three points on the unit circle (120° apart) this would look like this:
x=a cos (θ) + b cos (θ) + c cos (θ)
y=d sin (θ) + e sin (θ) + f sin (θ)
Now, I want to examine what happens if any two of these points are "free". That is, only one of the points is fixed and the others may individually take on any value from 0° to 360°.
Here is my question:
In this case I believe the model must use the Harmonic Addition Theorem, yes?
I believe this is the case since the two free positions on the circle have the same period but are out of phase with each other since they are moving independent of each other and can take on any arbitrary value. Is that right?
In this case my model would then look like this (where A and B represent the differences in phase)
x=a cos (θ) + (b cos (θ-A) + c cos (θ-B) )
y=d sin (θ) + (e sin (θ-A) + f sin (θ-B))
Is this correct?

 

[theorem4.5.9]

In the case of the unit circle equally divided so that I have three points on the unit circle (120° apart) this would look like this:
x=a cos (θ) + b cos (θ) + c cos (θ)
y=d sin (θ) + e sin (θ) + f sin (θ)

If you are using different points on the circle, you need different arguments i.e. $\theta_1, \theta_2, \theta_3$.

In any case I believe the formulas you are asking for are the sum-to-product formulas. Look at the bottom of this link: trig formulas, or if you really want the harmonic addition theorem, it's here harmonic addition

 

[omission9]

If you are using different points on the circle, you need different arguments i.e. $\theta_1, \theta_2, \theta_3$.

In any case I believe the formulas you are asking for are the sum-to-product formulas.

I am not asking for the formulas. I already know those! What am I asking is if this is an appropriate application of the Harmonic Addition Theorem. Is it?

 

[theorem4.5.9]

In that case I'm not sure what you're asking. I'm not sure what your model is suppose to be or what symmetry you're trying to take advantage of. The harmonic addition theorem is just a formula, so there's not really a wrong application of it.

 

[CellCoree]

Yes, you are correct in using the Harmonic Addition Theorem for this scenario. The Harmonic Addition Theorem states that the sum of two or more sines or cosines with the same period but different phase shifts can be written as a single sine or cosine with a combined phase shift.

In this case, your model would be correct in using the Harmonic Addition Theorem to combine the two free points on the unit circle. The phase shifts A and B represent the differences in phase between the two points and are necessary for accurately representing the location of the points on the circle. This approach allows for the independent movement of the two points while still maintaining the overall pattern of the unit circle.

As a scientist, it is important to carefully consider and utilize mathematical models to accurately represent and understand the phenomena we are studying. It is clear that you have a strong understanding of the Harmonic Addition Theorem and its application in this scenario. Well done!