Euler's Theorem Converting from Trignometric to Exponential Form

1. Aug 27, 2008

jaylwood

r(cos u + i sin u)

t(cos v + i sin v)

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

2. Aug 27, 2008

tiny-tim

Welcome to PF!

Hi jaylwood! Welcome to PF!

cos u + i sin u = eiu

(I don't understand why you're not recognising that? )

3. Aug 27, 2008

jaylwood

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. Aug 27, 2008

tiny-tim

ah … so that's the problem!

ok … x = r eiu, y = t eiv

so multiply them, and you get xy = … ?

5. Aug 27, 2008

jaylwood

rt eiu eiv What do i do to simplify that? Or reconvert it back to trignometric form?

6. Aug 27, 2008

tiny-tim

oh come on …

eiu eiv = … ?

7. Aug 27, 2008

jaylwood

ei(u+v)

8. Aug 28, 2008

tiny-tim

amplitude …

(just got up … :zzz:)

That's right!

So the amplitude of xy is … ?

9. Aug 28, 2008

jaylwood

u+v ? but what happens to the rt?

Last edited: Aug 28, 2008
10. Aug 28, 2008

tiny-tim

Yes!

(it's that easy )

Any other problems?

11. Aug 28, 2008

jaylwood

what happens to the rt?

12. Aug 28, 2008

tiny-tim

They're just ordinary numbers.

Treat them as usual …

xy = rt ei(u+v)

13. Aug 28, 2008

jaylwood

So what would be my final answer?

14. Aug 28, 2008

tiny-tim

Well, the question was …
… so the answer is that the amplitude of their sum is u + v, which is the sum of their amplitudes!

(which is why you didn't need to bother with x and t at the end )

15. Aug 28, 2008

jaylwood

Thank you so much.