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I'm having some troubles determining the amplitude/magnitude of the following equation.

$$ A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3) $$ Since each part is at a different frequency, i cannot sum the magnitudes of each part.

I have also thought about using variations of the double/triple angle formulae and some basic trigonometric identities, so that I can write the equation under a single frequency, but by doing so, I am introducing some higher order terms, which seem to negate the ability to sum the amplitudes together.

For example, the term with $5\omega$ would look like $$16\cos^5(\omega t)-20\cos^3(\omega t)+5\cos(\omega t)$$

I suppose another method would be to plot the first equation and then record the amplitude, but i would like a more generalised approach to solve this problem. All input is welcome and appreciated.

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If you want the absolute maximum amplitude and the $\beta$s are such that the peaks can line up, you can just sum the individual waves. They don't line up perfectly very often and less often as more sine waves are included. Another approach is to do a root sum square of the individual amplitudes. This gives you a better feeling for the real amplitudes you will see, but will not be the maximum. If you go to Alpha you can plot it or again. The first has all the peaks lining up at $t=0$, the other has some delays added.

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You could always differentiate it.

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