# Fourier Series

1. Apr 11, 2010

### twotaileddemon

1. The problem statement, all variables and given/known data

Show that the fourier series f(x) = $$\sum$$ansin(nx) + bncos(nx) can be written as $$\sum$$kn(cos(nx+$$\vartheta$$)) and define kn and $$\vartheta$$

where the summation is from 0 to $$\infty$$

2. Relevant equations
sin $$\vartheta$$ = cos (90 - $$\vartheta$$) ??

3. The attempt at a solution
Well what I originally did was replace the sin term by cos (90 - nx), put cosine in terms of complex exponentials, and then try to solve the equation, but I only got what I was given in the first place and not the solution (i.e. I went in a circle).

Is there some kind of property of sin or cos I could use?

2. Apr 11, 2010

### LCKurtz

$$a \cos x + b \sin x = \sqrt{a^2+b^2}\ \left(\frac a {\sqrt{a^2+b^2}}\cos x +\frac b {\sqrt{a^2+b^2}}\sin x\right )$$

and then think about what the expansion of

$$\cos{(x -\phi)}$$

looks like.

3. Apr 11, 2010

### vela

Staff Emeritus
Try expanding $k_n\cos(nx+\theta_n)$ using the angle addition trig identity.

4. Apr 19, 2010

### twotaileddemon

Thank you for the responses - I was able to derive the proof exactly.