Let $z_1 = \frac{\sqrt{6} - i\sqrt{2}}{2}$ and $z_2 = 1 - i$. Write $z_1$ and $z_2$ in

Question

Let $z_1 = \frac{\sqrt{6} - i\sqrt{2}}{2}$ and $z_2 = 1 - i$. Write $z_1$ and $z_2$ in the form $r(\cos\theta + i\sin\theta)$ where $r > 0$ and $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$. Show that $\frac{z_1}{z_2} = \cos\frac{\pi}{12} + i\sin\frac{\pi}{12}$. Find the value of $\frac{z_1}{z_2}$ in the form $a + bi$, where $a$ and $b$ are to be determined exactly in surd form. Hence, find the exact values of $\cos\frac{\pi}{12}$ and $\sin\frac{\pi}{12}$.

Accepted Answer

(a) $\sqrt \rm{cis} \left( -\frac{\pi}{6}\right)$; $\sqrt \rm{cis} \left( -\frac{\pi}{4}\right)$; (c) $\frac{\sqrt{6}+\sqrt{2}}{4} + \frac{\sqrt{6}-\sqrt{2}}{4}i $; (d) $\frac{\sqrt{6}+\sqrt{2}}{4}, \frac{\sqrt{6}-\sqrt{2}}{4} $.

Properties of argument

Properties of argument