Complex Numbers - Using Complex Numbers to Deduce Trigonometric Values

The complex number \(w\) has modulus \(\sqrt2\) and argument \(-3\pi/4\), and the complex number \(z\) has modulus \(2\) and argument \(-\pi/3\). Given that \(w=-1-i\) and \(z=1-\sqrt3\,i\).

1. Find the exact values of the modulus and argument of \(wz\).
2. By first expressing \(w\) and \(z\) in the form \(x+y\mathbf{i}\), find the exact real and imaginary part of \(wz\).
3. Hence show that \(\sin \frac{\pi }{12}=\frac{\sqrt{3}-1}{2\sqrt{2}}\) .