Developping sin(x) on [0, pi] as a serie of cosines

[quasar987]

Am I crazy or this is impossible to do if we require that the period of the extended even function be 2pi ?

Yet, this question comes in my textbook before the section concerning Fourier extension of functions of period other than 2pi. So there must be a way?

 

[LeonhardEuler]

I'm pretty sure it's possible. Extend sin(x) before x=0 as -sin(x) so the function is even. Now do a cosine series on the interval [-pi,pi] and it will be valid on [0,pi] and have period 2pi.

 

[quasar987]

This is what I did, but the period of this function (|sin(x)|) is pi.

 

[quasar987]

Arr, you're right.