Fourier Series: Rewriting with k_n and θ

[twotaileddemon]

Homework Statement

Show that the Fourier series f(x) = [tex]\sum[/tex]a_nsin(nx) + b_ncos(nx) can be written as [tex]\sum[/tex]k_n(cos(nx+[tex]\vartheta[/tex])) and define k_n and [tex]\vartheta[/tex]

where the summation is from 0 to [tex]\infty[/tex]

Homework Equations

sin [tex]\vartheta[/tex] = cos (90 - [tex]\vartheta[/tex]) ??

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?

 

[LCKurtz]

Think about something like this:

[tex] 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 )[/tex]

and then think about what the expansion of

[tex]\cos{(x -\phi)}[/tex]

looks like.

 

[vela]

Try expanding [itex]k_n\cos(nx+\theta_n)[/itex] using the angle addition trig identity.

 

[twotaileddemon]

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