Complex exponentials (simplify the expression)

  • Thread starter Luongo
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  • #1
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1. Simplify ei6x(1+e-i10x)/(1+ei2x)



2. i have no idea how to simplify this its supposed to be in terms of cosines



3. i dont how i can simplify this such that i can use the 1/2(e^x +e^-x) = cosx formula
 

Answers and Replies

  • #2
fzero
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You need the Euler formula [tex]e^{ix} = \cos x + i \sin x[/tex].
 
  • #3
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You need the Euler formula [tex]e^{ix} = \cos x + i \sin x[/tex].

i tried that, i got garbage it MUST be in terms of cosines
 
  • #4
fzero
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You might want to show some work to see where your problem is. (Also you might find [tex]e^{6ix} = e^{5ix} e^{ix}[/tex] useful.)
 
  • #5
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You might want to show some work to see where your problem is. (Also you might find [tex]e^{6ix} = e^{5ix} e^{ix}[/tex] useful.)


well, ive multiplied it out i got e^6x+e^-4x and then i did e^2x * e^4x + e^-4x
in the numerator to try to get it in cosine form but i cant get the e^2 out of there so i really have no idea how, can someone show me the steps of simplifying this
 
  • #6
fzero
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You also want to simplify the denominator, so you don't want to multiply the numerator through by the whole [tex]e^{6ix}[/tex] factor. (Also you're leaving out the factors of i in your exponentials, which is a bit confusing, but you shouldn't do it in anything you turn in to be graded.)
 
  • #7
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You might want to show some work to see where your problem is. (Also you might find [tex]e^{6ix} = e^{5ix} e^{ix}[/tex] useful.)


i got it! you're a genius, how on earth did you see that?!?!
 
  • #8
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Exponent rules :D...
 
  • #9
fzero
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i got it! you're a genius, how on earth did you see that?!?!

You know that

[tex]\frac{ e^{ia} + e^{-ia}}{2} = \cos a[/tex]

so if you see

[tex] 1 + e^{ib}[/tex]

you want to rewrite that as

[tex] e^{ib/2} ( e^{-ib/2} + e^{ib/2} ).[/tex]
 
  • #10
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You know that

[tex]\frac{ e^{ia} + e^{-ia}}{2} = \cos a[/tex]

so if you see

[tex] 1 + e^{ib}[/tex]

you want to rewrite that as

[tex] e^{ib/2} ( e^{-ib/2} + e^{ib/2} ).[/tex]


thanks alot i appreciate it i was hung up on this question for a while :redface:
 
  • #11
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thanks alot i appreciate it i was hung up on this question for a while :redface:


for simplifying in terms of sines... can i use the same formula except the negative sign is between the 2 expos?
 
  • #12
fzero
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for simplifying in terms of sines... can i use the same formula except the negative sign is between the 2 expos?

Yes.
 

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