Euler's Theorem Converting from Trignometric to Exponential Form

  • Thread starter jaylwood
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  • #1
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r(cos u + i sin u)


t(cos v + i sin v)

How do I convert these into exponential form using Euler's Theorem?
 

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  • #2
tiny-tim
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Welcome to PF!

r(cos u + i sin u)

t(cos v + i sin v)

How do I convert these into exponential form using Euler's Theorem?

Hi jaylwood! Welcome to PF! :smile:

cos u + i sin u = eiu :smile:

(I don't understand why you're not recognising that? :confused:)
 
  • #3
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okay here is the problem i have. Given x = r(cos u + i sin u) and y = t(cos v + i sin v)
Prove that the amplitude of (xy) is the sum of their amplitudes. I don't understand where to go with it.
 
  • #4
tiny-tim
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okay here is the problem i have. Given x = r(cos u + i sin u) and y = t(cos v + i sin v)
Prove that the amplitude of (xy) is the sum of their amplitudes. I don't understand where to go with it.

ah … so that's the problem!

ok … x = r eiu, y = t eiv

so multiply them, and you get xy = … ? :smile:
 
  • #5
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rt eiu eiv What do i do to simplify that? Or reconvert it back to trignometric form?
 
  • #6
tiny-tim
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rt eiu eiv What do i do to simplify that?

oh come on …

eiu eiv = … ? :smile:
 
  • #7
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ei(u+v)
 
  • #8
tiny-tim
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amplitude …

ei(u+v)

(just got up … :zzz:)

That's right! :smile:

So the amplitude of xy is … ?
 
  • #9
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u+v ? but what happens to the rt?
 
Last edited:
  • #10
tiny-tim
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  • #11
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what happens to the rt?
 
  • #12
tiny-tim
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what happens to the rt?

They're just ordinary numbers.

Treat them as usual …

xy = rt ei(u+v) :smile:
 
  • #13
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So what would be my final answer?
 
  • #14
tiny-tim
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So what would be my final answer?

Well, the question was …
Prove that the amplitude of (xy) is the sum of their amplitudes.
… so the answer is that the amplitude of their sum is u + v, which is the sum of their amplitudes! :smile:

(which is why you didn't need to bother with x and t at the end :wink:)
 
  • #15
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Thank you so much.
 

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