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.
:)

 

[blue_raver22]

The orthogonality of sine and cosine functions is a fundamental concept in mathematics and physics. It means that these two functions are perpendicular to each other, or at right angles, when plotted on a graph. This has important implications in many areas of science, including signal processing and wave analysis.

When taking the integral or sum of cos(nx) or sin(nx), they do indeed retain their form. This is because these functions are orthogonal, meaning they are independent of each other and do not affect each other's values when combined. In other words, the integral or sum of these functions will not change their shape or form, but rather add or subtract from their overall value.

This orthogonality property is what allows us to use Fourier Series to represent any periodic function as a sum of sine and cosine functions with different frequencies. By choosing the appropriate coefficients, we can accurately reconstruct the original function. So while cos(nx) and sin(nx) may seem simple on their own, their orthogonality allows for complex and powerful applications in mathematics and science. I hope this helps answer your question.