Question

What is the cube root of z^2020 if z = cos 11 pi/6 + I sin 7pi/ 6?​

Answer 1

Given: z = cos 11 pi/6 + I sin 7pi/ 6

To find: z^2020/3

Solution: first we will find the numerical value of z and for that, we need to solve cos 11pi/6 and sin 7pi/6

cos 11π/6 can have values either +√3/2 or -√3/2

since 11π/6 will be in the fourth quadrant

therefore, we can say that cos 11π/6 = +√3/2

similarly, in sin7π/6 ,7π/6 will be in the third quadrant

therefore we can say that sin 7π/6 = - 1/2

So z = +√3/2 - 1/2i

now we need to find z^ 2020/3

therefore, we can write it as

(1010√3)/3 - (1010/3) i

So, cube root of z^2020 will be (1010√3)/3 - (1010/3)i .