Write the complex number in polar form with argument ? between 0 and 2p. 9 radical 3 - 9i

Question

Write the complex number in polar form with argument θ between 0 and 2π. 9 radical 3 − 9i

Answer 1

Given a complex number z = 9√3 - 9i,  let z= x + yi  then x= 9√3, y = -9

writing in polar form z = r(cosθ + i sinθ)

where r =  |z| = √(x² + y²) = 18

         θ = tan^-1(y/x) = tan^-1(-1/√3) = - π/6, or 11π/6 where π = pi

∴  z = 18(cos(11π/6) + i sin(11π/6))

Answer 2

Here the polar form of a complex number a+bj has a constant of square root of sum of squares of a and b and a theta which is inverse tangent of b/a. So here theta is inverse tan of 9 radical 3/ 9 or pi/3 of a typical 30-60-99 degree triangle. Its magnitconstant stand) is twice of nine or 18.

 

Hope that helps,

Deanna