Question

Find z if |z| = 4 and arg(z) = `(5pi)/6`.

Answer 1

Given that: |z| = 4 and arg(z) = `(5pi)/6`

⇒ θ = `(5pi)/6`

|z| = 4

⇒ r = 4

So Polar form of z = `r[cos theta + i  sin theta]`

= `4[cos  (5pi)/6 + i  sin  (5pi)/6]`

= `4[cos (pi - pi/6) + i  sin(pi - pi/6)]`

= `4[- cos  pi/6 + i  sin  pi/6]`

= `4[(-sqrt(3))/2 + i  1/2]`

= `-2sqrt(3) + 2i`

Hence z = `-2sqrt(3) + 2i`.