Orthogonality of sine and cosine question

[Tikkelsen]

Hello,

I'm trying to solve Fourier Series, but I have a question.
I know that cos(nx) is even and sin(nx) is odd. But what does this mean when I take the integral or sum of cos(nx) or sin(nx)? Do they have a value or do they just keep their form?

 

[Simon Bridge]

$$\int_a^a \sin kx \; dx = 0\\
\int_a^a \cos kx \; dx = 2\int_0^a \cos kx\; dx$$
... with the Fourier series you are more interested in the integral of f(x) multiplied by a sine or a cosine though.

 

[Tikkelsen]

I found the answer to my own question.
I wasn't concerned about cos(nx), but by cos(npi) which is equal to (-1)^n and for sin(npi) it's equal to 0. I now understand that this is only for a particular result of the Fourier Series where the integral includes pi. Thank you for your answer though Simon.

 

[Simon Bridge]

No worries, and well done.
Thanks for sharing too.
:)