Integrating a Product of Cosines

Date: 05/04/2005 at 04:21:24
From: Peter
Subject: An integral involving cosine

Let n and m be natural numbers.  My book says that the integral from
-pi to pi of the function f(x) = cos(nx) * cos(mx) is 

  1) pi if n = m

  2) 0 else. 

I can't figure out how to prove this. 

Date: 05/04/2005 at 09:09:13
From: Doctor Jerry
Subject: Re: An integral involving cosine

Hello Peter,

What you need is the "product to sum formula" from trigonometry:

  cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)

  cos(a-b) = cos(a)*cos(b) + sin(a)*sin(b)

Adding,

  cos(a)*cos(b) = (1/2)(cos(a+b) + cos(a-b))
 
Use this to change the integrand to a sum.

Please write back if my comments are not clear or if you need more
help with this problem.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/04/2005 at 09:29:34
From: Peter
Subject: Thank you (An integral involving cosine)

Hi, Doctor Jerry. 

Thank you for the perfect answer.  Everything is now clear. 

Peter