Problem

The complex numbers $z_1$ and $z_2$ are given by

$z_1 = 3 + 5i \quad \text{and} \quad z_2 = -2 + 6i$

(a) Show $z_1$ and $z_2$ on a single Argand diagram.

(b) Without using your calculator and showing all stages of your working,

(i) determine the value of $|z_1|$

(ii) express $\dfrac{z_1}{z_2}$ in the form $a + bi$, where $a$ and $b$ are fully simplified fractions.

(c) Hence determine the value of $\arg\dfrac{z_1}{z_2}$

Give your answer in radians to $2$ decimal places.