3 The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by

$$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$

(a) Show that $z _ { 1 } = \mathrm { i }$.\\
(b) Show that $\left| z _ { 1 } \right| = \left| z _ { 2 } \right|$.\\
(c) Express both $z _ { 1 }$ and $z _ { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.\\
(d) Draw an Argand diagram to show the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 1 } + z _ { 2 }$.\\
(e) Use your Argand diagram to show that

$$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$

\hfill \mbox{\textit{AQA FP2  Q3}}