[Math] Sum of Sinusoids with Same Frequency = Sinusoid (proof)

fourier analysistrigonometry

I am studying Fourier analysis on my own, I realised that probably the first thing you want to proof in Fourier transform is that the sum of 2 sinuoids (namely a sine and cosine) with the same frequency gives another sinusoid. So I am trying to find a proof of this. In this document, I found this identity:

$$ A\cos(\omega t + \alpha) + B\sin(\omega t + \beta)
= \color{red}{\sqrt{(A\cos\alpha + \beta\sin\beta)^2 + (A \sin\alpha – B\cos\beta)^2}} \cdot \cos\left(\omega t + \color{green}{\arctan \frac{A\sin\alpha – B\cos\beta}{A\cos\alpha+B\sin\beta}}\right)
$$

EDIT: sorry I made mistake in equation.

Assuming I know how to go from the equation on the left to the equation on the right, would it be good enough as a proof since I can say that the terms that I highlighted with color are constants thus that the sum of the cosine and sine is equal to a constant multiplied by a cosine of the same frequency with some constant phase shift.

It would be great to have the confirmation from an expert.

Thank you.

Best Answer

To avoid any confusion, let us state that

  • a (pure) sine has the form $A\sin(\omega t)$,
  • a (pure) cosine has the form $A\cos(\omega t)$,
  • a sinusoid has an arbitrary phase and one of the equivalent forms $A\sin(\omega t+\phi)$ or $A\cos(\omega t+\psi)$ - where $\phi$ and $\psi$ differ by a quarter turn.

So the sine and cosine are special cases of the sinusoid.

By the well-known addition formula, $$A\sin(\omega t+\phi)=A\sin(\omega t)\cos(\phi)+A\cos(\omega t)\sin(\phi)=A'\sin(\omega t)+A''\cos(\omega t).$$

This means that

  1. a sinusoid can be expressed as a linear combination of a sine and a cosine,
  2. conversely, a linear combination of sine and cosine can be represented as single sinusoid$^*$,
  3. a linear combination of two or more sinusoids can be expressed as a linear combination of a sine and a cosine, hence can be expressed as a single sinusoid.

These properties no more hold if you mix sinusoids of different periods.

$^*$This is done by solving the system $$A\cos(\phi)=A'\\A\sin(\phi)=A'',$$ i.e. $$A=\sqrt{A'^2+A''^2}\\\tan(\phi)=\frac{A''}{A'}.$$

You will soon discover that complex numbers are intensively used in harmonic analysis, based on Euler's formula $e^{ix}=\cos(x)+i\sin(x)$.

Sinusoids can be represented as the imaginary part of $Ae^{i(\omega t+\phi)}=Ae^{i\phi}e^{i\omega t}=Ze^{i\omega t}$, where $Z$ is a complex number, that carries both the amplitude and the phase ($Z=A$ is real for a sine, $Z=iA$ is imaginary for a cosine).

Using this notation, adding sinusoids becomes a trivial matter:

$$Z_0e^{i\omega t}+Z_1e^{i\omega t}+Z_2e^{i\omega t}=(Z_0+Z_1+Z_2)e^{i\omega t}.$$

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