I am studying Fourier series right now. I asked a question before of math.statckexchange regarding Fourier series. This question is related and hopefully quite simple:
Generally Fourier series works because a sinusoid can be recomposed from a linear combination of a cosine and sine. In a book (Fourier Analysis Stein & Shakarchi) that was kindly pointed to me by someone here on Stack there is this definition:
If you can't access the image, I will reproduce the formula:
$a \cos(ct) + b \sin(ct) = A\cos(ct - \phi)$
What I like is that it says this can be easily verified. $\phi$ is the phase, A the amplitude. However considering I am trying to learn maths myself, I was wondering if someone could confirm my findings (at least I tried to "verify" it myself).
I used trigonometry identity:
$A\cos(ct - \phi) = A\cos(ct) * \cos(\phi) + A\sin(ct) * \sin(\phi)$
Because $\phi$ is a constant then we can write $a = A * cos(\phi)$ and $b = B * sin(\phi)$.
I would like to know if this is correct? Also can this be used as a proof (?) that any sinusoid can be recomposed from a combination of cosine and sine functions?
Thanks a lot.