How do you find #z, z^2, z^3, z^4# given #z=1/2(1+sqrt3i)#?

1 Answer
Aug 13, 2016

#z = cos(pi/3) + isin(pi/3)#

#z^2 = cos(2pi/3) + isin(2pi/3) = 1/2(-1 + sqrt(3)i)#

#z^3 = cos(3pi/3) + isin(3pi/3) = -1#

#z^4 = cos(4pi/3) + isin(4pi/3) = -1/2(1+sqrt(3)i)#

Explanation:

The easiest method is to use De Moivre's theorem. For complex number #z#

#z= r(costheta + isintheta)#

#z^n = r^n(cosntheta + isinntheta)#

So we want to convert our complex number to polar form. The modulus #r# of a complex number #a+bi# is given by

#r = sqrt(a^2+b^2)#

#r = sqrt((1/2)^2 + (sqrt(3)/2)^2) = sqrt(1/4 + 3/4) = 1#

The complex number will be in the first quadrant of an Argand diagram so the argument is given by:

#theta = tan^(-1)(b/a)#

#theta = tan^(-1)((sqrt(3)/2)/(1/2)) = tan^(-1)(sqrt(3)) = pi/3#

#z = cos(pi/3) + isin(pi/3)#

#z^2 = cos(2pi/3) + isin(2pi/3) = 1/2(-1 + sqrt(3)i)#

#z^3 = cos(3pi/3) + isin(3pi/3) = -1#

#z^4 = cos(4pi/3) + isin(4pi/3) = -1/2(1+sqrt(3)i)#