Orthogonality of Sine and Cosine functions

[Hendrick]

Hi,

would anyone be able to explain how to evaluate a function using orthogonality (i.e. using orthogonality to solve a definite integration problem with sines/cosines)?

Thank you

 

[ExactlySolved]

It means to use the six formulas that are appear after the first paragraph of text on this page:

http://mathworld.wolfram.com/FourierSeries.html

 

[daviddoria]

Orthogonality in a vector space means that the inner product of two vectors is 0. In this case, your space is a function space so the inner product is defined as the integral of the product of the functions.

http://en.wikipedia.org/wiki/Orthogonal#Orthogonal_functions

 

[Bob S]

I would look in (already referenced in post #2 By ExactlySolved)
http://mathworld.wolfram.com/FourierSeries.html
And in particular look at Eqns (8) and (9) for solving for the coefficients in a Fourier series. Also look at (18) and (19). The sine-like and cosine-like terms are orthogonal, as proved by integrating their product over the interval 0 to 2 pi.