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]

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!

cos u + i sin u = e^iu

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

 

[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 e^iu, y = t e^iv …

so multiply them, and you get xy = … ?

 

[jaylwood]

rt e^iu e^iv What do i do to simplify that? Or reconvert it back to trignometric form?

 

[tiny-tim]

rt e^iu e^iv What do i do to simplify that?

oh come on …

e^iu e^iv = … ?

 

[jaylwood]

e^i(u+v)

 

[tiny-tim]

amplitude …

e^i(u+v)

(just got up … :zzz:)

That's right!

So the amplitude of xy is … ?

 

[jaylwood]

u+v ? but what happens to the rt?

 

[tiny-tim]

u+v ?

Yes!

(it's that easy )

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 e^i(u+v)

 

[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!

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

 

[jaylwood]

Thank you so much.