Solutions to Equation z^3 = w

Question

Use w = 8( cos(pi/2) + i sin(pi/2) ) to find the three solutions to the equation z^3 = w,

Express your solutions in the form c + di, where c and d are real numbers.

Answer 1

QUESTION: Solutions to Equation z^3 = w
Use w = 8( cos(pi/2) + i sin(pi/2) ) to find the three solutions to the equation z^3 = w,
Express your solutions in the form c + di, where c and d are real numbers.

Let w = re^(it) = r{cos(t) + i.sin(t)}

Then w = r{cos(t + 2n.pi) + i.sin(t + 2n.pi)}

z^3 = re^(it)

z = r^(1/3).e(it/3) = r^(1/3){cos[(t + 2n.pi)/3] + i.sin[(t + 2n.pi)/3]}

putting r = 8 and t = pi/2,

z = 2{cos[(pi/2 + 2n.pi)/3] + i.sin[(pi/2 + 2n.pi)/3]}

z = 2{cos[(1 + 4n).pi/6] + i.sin[(1 + 4n).pi/6]}

Setting n = 0,1,2

z1 = 2{cos[pi/6] + i.sin[pi/6]}

z2 = 2{cos[5pi/6] + i.sin[5pi/6]}

z3 = 2{cos[9pi/6] + i.sin[9pi/6]}

 

Comments

thank you so much!