maths/ physics
posted by jon smith on .
Determine the two square roots of 3+j4 in both a)Cartesian form
b) Polar form
Any help with this question would appreciated

let the angle be Ø
then r = √(3^2+4^2) = 5
tanØ = 4/3, Ø = .9273
3 + 4j = 5(cos .9273 + jsin .9273)
by De Moivre's theorem
(3 + 4j)^(1/2) = √5(cos ((1/2).9273) + jsin ((1/2).9273))
= √5( cos .4636 + jsin .4636)
or 2 + j if expanded.
check:
if 2+j is the square root of 3+4j, then
(2+j)^2 should equal 3+4j
Left side = (2+j)^2
= 4 + 4j + j^2
= 4 + 4j  1
= 3 + 4j
so √(3+4j) = 2+j or √5(cos .4636 + jsin .4636)