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:enter image description here

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.

  • $\begingroup$ That is correct. $\endgroup$ – Ron Gordon Oct 7 '13 at 9:47
  • $\begingroup$ Thank you. So I can also use this a proof yes? $\endgroup$ – Marc Ourens Oct 7 '13 at 9:49
  • $\begingroup$ Ummm...less comfortable with that; sorry could not be more help. $\endgroup$ – Ron Gordon Oct 7 '13 at 9:55

What you do is correct so far as it goes -- if you know $A$ and $\varphi$, you can find corresponding $a$ and $b$ in the way you describe.

Note, however, that this is not what the part you quote from the book does. (And what you quote is not a definition, by the way). The book says you can go in the other direction: If you already know $a$ and $b$, it is always possible to find matching $A$ and $\varphi$. That is not much more difficult to prove, though it is slightly trickier because $\varphi$ will not be uniquely determined.

  • $\begingroup$ I see cool. Thank you. Okay for definition I won't change it because I can't cross text over. I will be more careful next time. Then from what you are suggesting is this why we say that in Fourier Series the coefficients $a$ and $b$ can be used to compute the phase (using complex numbers)? And how do you get to know A and $\phi$. Is this by using the Euler's Formula/complex numbers? I can try to find myself just want to be on the right track. Thanks. $\endgroup$ – Marc Ourens Oct 7 '13 at 10:15
  • $\begingroup$ So trying to see if I can come up with something but that seems maybe too easy. If $a = A\cos(\phi)$ and $b = A\sin(\phi)$ then if we consider this to be a complex number $z(A, \phi) = A cos( \phi ) + A i sin(\phi)$ then we can compute A as $A = \sqrt{a^2 + b^2}$ and $\phi$ as $\phi = atan^{-1}({b \over a})$. Is this correct? What did you say $\phi$ was not uniquely determined? thank you. $\endgroup$ – Marc Ourens Oct 7 '13 at 10:37
  • $\begingroup$ @MarcOurens: Yes, except that if $a$ is negative, the $\operatorname{atan}$ will give you a phase that differs by $\pi$ from the correct one. And it is not uniquely determined because you can always add any multple of $2\pi$. $\endgroup$ – hmakholm left over Monica Oct 7 '13 at 11:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.