Use De Moivres theorem to change the given complex number to

Use De Moivre\'s theorem to change the given complex number to the form a + bi, where a and b are real numbers. (- Squareroot 3/2 - 1/2 i)^4

Solution

( -sqrt3/2 -i/2)^4

Polar form = r*e^i*theta

r = sqrt[ (-sqrt3/2)^2 + (1/2)^2 )] = 1

theta = tan^-1(-1/2 /-sqrt3/2) = pi +pi/3 = 4pi/3

( -sqrt3/2 -i/2)^4 = (re^i*theta)^4 = r^4e^i*4theta

= 1*e^i*16pi/3

= cos16pi/3 + isin16pi/3

= -0.5 -i*0.866