2018 AJC P1 Q8 (b)

The complex number \(u\) is given by \(u=\cos \theta +\mathrm{i}\sin \theta \), where \(0<\theta <\pi \) .

1. Express \(u-1\) in terms of \(\sin \frac{\theta }{2}\) and \(\cos \frac{\theta }{2}\). Hence, find \(\left| u-1 \right|\) and show \(\arg \left( u-1 \right)=\frac{\pi }{2}+\frac{\theta }{2}\).[4]
2. Given that \(v=-\sqrt{3}+\mathrm{i}\) and \({{\left( v^*u \right)}^{8}}\) is real and negative, find the smallest value of \(\theta \) in terms of \(\pi\).[2]