How do I find smallest theta value, largest theta value, and a value in between

Question

Evaluate the cube root of z when z = 125 cis(255°).

 (Let 0 ≤ 𝜃 < 360°.)

 

Find

(Smallest  𝜃-value)

(In between value)

(Largest  𝜃-value)

Answer 1

Finding roots and performing multiplication and exponentiation on complex numbers in polar form is a breeze! Polar form lends itself to precisely these calculations.

To find the 3 cube roots of a complex number z = rcisΘ, we take the cube root of r and we divide Θ by 3. This gives the first of the 3 cube roots. From there, we remember that roots of complex numbers are evenly spaced around the origin in the complex plane. Thus, cube roots share an r value, and have angles that are 360/3 = 120° apart from one another.

Thus, ³√(125cis255°) = 5cis85° or 5cis205° or 5cis325°

All 3 of these distinct complex numbers will cube to give 125cis255°

Answer 2

z^1/3 = 125^1/3(cis (225° / 3))

125^1/3(cis ((225° + 360°) / 3))

125^1/3(cis ((225° + 720°) / 3))