Question:

a)Given that the complex numbers $\displaystyle w_{1} $ and $\displaystyle w_{2} $ are the roots of the equation $\displaystyle z^2-5-12i=0 $, express $\displaystyle w_{1} $ and $\displaystyle w_{2} $ in the form of $\displaystyle a +ib $ where a and b are real

b)i)Indicate the point sets in an argand diagram corresponding to the sets of the complex numbers

$\displaystyle A= \left \{z:\left | z \right | =3, z \epsilon \mathbb{C} \right \}$

$\displaystyle B=\left \{z:\left | z \right | =2, z \epsilon \mathbb{C} \right \} $

ii)Shade the region corresponding to the values of z for which the inequalities

$\displaystyle 2 <\left | z \right |<3 $ and $\displaystyle 30^{\circ}<arg(z)<60^{\circ} $

are simultaneously satisfied.

My attempt:

a)

$\displaystyle z^2-5-12i=0 $

$\displaystyle z^2= 5+12i $

]$\displaystyle z= \pm (5+12i)^{(0.5)}$

note that $\displaystyle (3+2i)^2 = 5+12i $

thus $\displaystyle z = \pm (3+2i) $

$\displaystyle w_{1}= 3+2i $

$\displaystyle w_{2}= -(3+2i) $

b)i)

$\displaystyle A={z:\left | z \right | =3, z \epsilon \mathbb{C}} $

$\displaystyle B={z:\left | z \right | =2, z \epsilon \mathbb{C}} $

ii)$\displaystyle 2 <\left | z \right |<3 $ and $\displaystyle 30^{\circ}<arg(z)<60^{\circ} $

I shaded the region I thought would be the right answer in purple

The part I am finding difficult is part b. Please help