Euler's Theorem Converting from Trignometric to Exponential Form

[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?

[tiny-tim]

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:)

[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.

[tiny-tim]

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:

[jaylwood]

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

[tiny-tim]

rt eiu eiv What do i do to simplify that?

oh come on …

eiu eiv = … ? :smile:

[jaylwood]

ei(u+v)

[tiny-tim]

ei(u+v)

(just got up … :zzz:)

That's right! :smile:

So the amplitude of xy is … ?

[jaylwood]

u+v ? but what happens to the rt?

[tiny-tim]

u+v ?

Yes! :smile:

(it's that easy :biggrin:)

Any other problems?

[jaylwood]

what happens to the rt?

[tiny-tim]

what happens to the rt?

They're just ordinary numbers.

Treat them as usual …

xy = rt ei(u+v) :smile:

[jaylwood]

So what would be my final answer?

[tiny-tim]

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:)

[jaylwood]

Thank you so much.