Orthogonality of sine and cosine question

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Discussion Overview

The discussion revolves around the properties of sine and cosine functions in the context of Fourier Series, specifically focusing on their even and odd characteristics and implications for integrals involving these functions.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant questions the implications of sine being odd and cosine being even when integrating or summing these functions.
  • Another participant provides specific integral results for sine and cosine functions over a defined interval, indicating that the Fourier series typically involves integrals of a function multiplied by sine or cosine.
  • A participant clarifies their initial question by focusing on the specific values of cos(nπ) and sin(nπ), noting that these values are relevant in a particular context of the Fourier Series.
  • A later reply acknowledges the clarification and expresses appreciation for the shared insights.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants explore different aspects of the properties of sine and cosine without resolving the initial question about integrals.

Contextual Notes

Participants reference specific integral results and properties, but the discussion does not fully address the implications of these results in a broader context of Fourier Series.

Tikkelsen
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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?
 
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$$\int_a^a \sin kx \; dx = 0\\
\int_a^a \cos kx \; dx = 2\int_0^a \cos kx\; dx$$
... with the Fourier series you are more interested in the integral of f(x) multiplied by a sine or a cosine though.
 
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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.
 
No worries, and well done.
Thanks for sharing too.
:)
 

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