anonymous
  • anonymous
Show that for all integers m and n, with m ≠ +/-n, the integral from -π to π of cos(mθ)cos(nθ) dθ = 0
Mathematics
chestercat
  • chestercat
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myininaya
  • myininaya
okay and we also know that \[\cos(m \theta - n \theta)=\cos(m \theta) \cos(n \theta)+\sin(m \theta) \sin(n \theta)\] so this means we have \[\sin(m \theta) \sin(n \theta)=\cos(m \theta-n \theta)-\cos(m \theta)\cos(n \theta)\] and remember that \[\cos(m \theta) \cos(n \theta)=\cos(m \theta+n \theta)+\sin(m \theta)\sin(n \theta) \] \[\cos(m \theta) \cos(n \theta)=\cos(m \theta+n \theta)+\cos(m \theta-n \theta)-\cos(m \theta)\cos(n \theta)\] \[2 \cos(m \theta) \cos(n \theta)= \cos(m \theta+n \theta)+\cos(m \theta-n \theta)\]
myininaya
  • myininaya
we are almost there now
myininaya
  • myininaya
\[\cos(m \theta) \cos(n \theta)= \frac{1}{2} (\cos(m \theta+n \theta)+\cos(m \theta-n \theta))\]

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myininaya
  • myininaya
so this means we have \[\frac{1}{2}\int\limits_{-\pi}^{\pi}(\cos([m+n] \theta)+\cos([m-n] \theta) d \theta\]
myininaya
  • myininaya
\[[\frac{1}{2}\frac{1}{m+n} \sin([m+n] \theta)+\frac{1}{2} \frac{1}{m-n}\sin([m-n] \theta)]_{-\pi}^{\pi}\]
myininaya
  • myininaya
i will assume m and n are integers
myininaya
  • myininaya
this means m+n is an integer and m-n is an integer sin(integer * pi) is 0 sin(integer * (-pi)) is 0 m+n can't be zero m-n can't be zero so the answer is 0 as long as m does not equal -n or m equals n
myininaya
  • myininaya
:)
anonymous
  • anonymous
in first part didnt u forget one sin(mt)sin(nt) ?
myininaya
  • myininaya
? go to the your post and read what i posted there
myininaya
  • myininaya
i put some more steps in the one i posted for you

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