# Sinusoid sum of cosine and sine

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.

• That is correct. – Ron Gordon Oct 7 '13 at 9:47
• Thank you. So I can also use this a proof yes? – Marc Ourens Oct 7 '13 at 9:49
• Ummm...less comfortable with that; sorry could not be more help. – 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.
• 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. – Marc Ourens Oct 7 '13 at 10:15
• 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. – Marc Ourens Oct 7 '13 at 10:37
• @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 $2\pi$. – hmakholm left over Monica Oct 7 '13 at 11:33