Is the Set {cos x, cos 2x, cos 3x, ...} Orthogonal Using Integral Products?

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SUMMARY

The set {cos x, cos 2x, cos 3x, ...} is proven to be orthogonal using the integral product over the interval from -π to π. The integral of the product cos(mx)cos(nx)dx equals zero for any integers m and n, confirming orthogonality. This conclusion is reached by defining the integral product and the concept of an orthogonal set, then applying these definitions. Integration by parts can be utilized to set up the integral if needed.

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  • Understanding of integral calculus
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  • Knowledge of trigonometric functions, specifically cosine
  • Experience with integration techniques, including integration by parts
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How would you prove, using the integral product, that the set of {cos x, cos 2x, cos 3x, cos 4x, ...} is an orthogonal set?
 
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welcome to pf!

hi roto25! welcome to pf! :wink:

i] define the integral product

ii] define orthogonal set

iii] apply i and ii …

what do you get? :smile:​
 
over the interval -pi to pi, the integral of cos(mx)cos(nx)dx is zero, as long as m and n are integers. Therefore, if you select ANY pair of elements from the set, the 'integral of their product' will be zero, thereby satisfying the condition of orthogonality.
 
Bavid said:
over the interval -pi to pi, the integral of cos(mx)cos(nx)dx is zero, as long as m and n are integers. Therefore, if you select ANY pair of elements from the set, the 'integral of their product' will be zero, thereby satisfying the condition of orthogonality.

On top of what Bavid said if you don't know where to start set up the integral and use integration by parts.
 

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