Find the cube roots of 4 Write the result in trigonometric f

Find the cube roots of -4. Write the result in trigonometric form

Solution

z^3 = -4

z= (-4)^1/3

In polar form : z^3 = re^(i*theta) = 4e^(i*pi) = 4(cospi +isinpi)

Z = [4(cos(pi+2kpi) + isin(pi+2kpi)) ]^1/3

k=0 z1 = 4^1/3[cospi/3 +isinpi/3]

k = 1 ; z2 = 4^1/3[ cos(4pi/3 +isin4pi/3 ]

k = 2 ; z3 = 4^1/3[cos5pi/3 + isin5pi/3 ]