2020 DHS P1 Q3

The complex number \(z\) is such that \(\arg (z)=-\frac{\pi }{3}\) and the complex number \(w\) has modulus of \(2\) and an argument of \(\frac{\pi }{4}\). It is also known that \(\arg \,(w-z)=\frac{\pi }{3}\). Let \(W\) and \(Z\) be the points representing the complex numbers \(w\) and \(z\) respectively.

1. By using an Argand diagram with reference from the origin \(O\), show that angle \(OWZ=\frac{\pi }{12}\) radians.[2]
2. Hence or otherwise, find \(z\) and \(w\) in the form of \(a+b\mathrm{i}\) where \(a\) and \(b\) are real constants. Leave your answers correct to \(3\) significant figures.[3]