Need help with complex numbers and locus

Question

part1. Let z=rcisθ where 0<θ<pi. show that Arg(z^2)=2θ.

 

part2. sketch the region defined by {z: Arg(z^2)>pi/2}

in the answers it shows region pi/4 to pi/2 and region 5pi/8 to 3pi/2 should be shaded which I dont understand why these regions must be shaded.

Thanks for the help in advance

Answer 1

part 1: z = r(cos θ + i sinθ)

z² = r²(cos²θ+ 2isinθcosθ - sin²θ⁾ = r²[cos²θ - sin²θ) + i (2sinθcosθ)] = r²[cos(2θ) + i sin(2θ)].

 

Is this sufficient help?