- The complex number \(z _ { 1 }\) is defined as
$$z _ { 1 } = \frac { \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right) ^ { 4 } } { \left( \cos \frac { \pi } { 3 } - i \sin \frac { \pi } { 3 } \right) ^ { 3 } }$$
- Without using your calculator show that
$$z _ { 1 } = \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$$
- Shade, on a single Argand diagram, the region \(R\) defined by
$$\left| z - z _ { 1 } \right| \leqslant 1 \quad \text { and } \quad 0 \leqslant \arg \left( z - z _ { 1 } \right) \leqslant \frac { 3 \pi } { 4 }$$
Given that the complex number \(z\) lies in \(R\)
- determine the smallest possible positive value of \(\arg z\)