In need proof of all justifications on a moduli equation?
x=r(cos u + i sin u)
y=t(cos v + i sin v)
Hello vramirezThere are three vital things that you need to complete the proof:
- $\displaystyle i^2 = -1$ (1)
- $\displaystyle \cos(A+B) = \cos A\cos B -\sin A \sin B$ (2)
- $\displaystyle \sin(A+B) = \sin A\cos B + \cos A \sin B$ (3)
So, you simply multiply $\displaystyle x$ by $\displaystyle y$ by expanding the brackets, but leave the $\displaystyle r$ and $\displaystyle t$ outside:
$\displaystyle xy = r(\cos u +i\sin u).t(\cos v + i\sin v)$
$\displaystyle = rt(\cos u\cos v + \cos u.i\sin v + i\sin u \cos v + i^2\sin u \sin v)$
$\displaystyle = rt\Big(\cos u \cos v - \sin u \sin v + i(\sin u \cos v + \cos u \sin v)\Big)$, by re-arranging the order of the terms, and using (1)
$\displaystyle =rt\Big(\cos(u+v) + i\sin (u+v)\Big)$, using (2) and (3)
$\displaystyle =s(\cos w + i\sin w)$, where $\displaystyle s = rt$ and $\displaystyle w = u+v$
And this is a complex number whose modulus is $\displaystyle s$, and amplitude $\displaystyle w$, where $\displaystyle s = rt =$ product of the moduli of $\displaystyle x$ and $\displaystyle y$, and $\displaystyle w = u+v =$ sum of their amplitudes.
Grandad