Proving an angle and radius - HELP!!!!!

[englishmajor]

I need some serious help!!! I don't even know where to begin on this question:
• Let x = r(cos u + i sin u)
• Let y = t(cos v + i sin v)

1. Show the steps to prove that the angle of xy is (u + v).
a. Provide each step of your proof.
2. Show the steps to prove that the radius of xy is rt.
a. Provide each step of your proof.

Thank you!!! I am soooooooooooooo confused

 

[soroban]

Hello, englishmajor!

\(\displaystyle \begin{array}{ccc}x &= & r(\cos u + i\sin u) \\ y &=& t(\cos v + i\sin v)\end{array}\)

\(\displaystyle \text{1. Prove that the angle of }xy\text{ is }(u + v)\)

\(\displaystyle \text{We have: }\:xy \;=\;rt(\cos u + i\sin u)(\cos v + i\sin v)\)

. . . . . . . . . .\(\displaystyle =\;rt(\cos u\cos v + i\sin u\cos v + i\cos u\sin v + i^2\sin u\sin v)\)

. . . . . . . . . .\(\displaystyle =\;rt\bigg[\underbrace{(\cos u\cos v - \sin u\sin v)}_{\text{This is }\cos(u+v)} + i\underbrace{(\sin u\cos v + \cos u\sin v)}_{\text{This is }\sin(u+v)}\bigg]\)

. . . . . . . . . .\(\displaystyle =\;rt\bigg[\cos(u+v) + i\sin(u+v)\bigg]\)


\(\displaystyle \text{Therefore, the angle of }xy\text{ is: }\:u+v\)

\(\displaystyle \text{2. Prove that the radius of }xy\text{ is }rt\)

\(\displaystyle \text{We want the }magnitude\text{ of }xy\)

\(\displaystyle \text{We know that: }\:xy \;=\;rt\bigg[\cos(u+v) + i\sin(u+v)\bigg]\)

\(\displaystyle \text{Then: }\;|xy| \;=\;|rt|\,\underbrace{\sqrt{\left[\cos(u+v)\right]^2 + \left[\sin(u+v)\right]^2}} _{\text{This is 1}}\)

\(\displaystyle \text{Therefore: }\;|xy| \:=\:rt\)

 

[englishmajor]

Thank you soooooo much!! I'm going over this problem and the entire chapter so that I can really master this!! I do so appreciate your help.