PreCalculus
posted by Anne on .
Could someone help me with thiswe learned this today in school but I'm not getting it now that I'm home with my homeworkI have lots of these to do can someone help with this one and then maybe it'll clickthank you
Find (sqrt3/2 1/2i)^4 by using DeMoivre's Theorem
I have to convert to polar, use the theorem and put it back into rectangular form

tan(t) = y/x = (1/2) /(sqrt(3)/2) = 1/sqrt(3)
so, t = pi/6
r^2 = x^2 + y^2 = 3/4 + 1/4 = 1
r = 1
so, point z = (1,pi/6)
z^4 = 1^4 cis 4*(pi/6) = 4 cis 2pi/3
x = r cos 2pi/3 = 1 * 1/2
y = r sin 2pi/3 = 1 * sqrt(3)/2
z^4 = (1/2, sqrt(3)/2)
Just for confirmation,
(sqrt(3)/2  i/2)^4 =
9/16  4*3*sqrt(3)/(8*2)i + 6*3/4*1/4 i^2  4*sqrt(3)/2 * 1/8 i^3 + 1/16 i^4
= 9/16  3sqrt(3)/4 i  18/16 + 1/4 sqrt(3) i + 1/16
8/16  sqrt(3)/2 i