A complex number is given by $z = 3 + 4i$. Find the modulus and argument of z.
Question Q8
A complex number is given by $z = 3 + 4i$. Find the modulus and argument of z.
Mark Scheme
$z = 3 + 4i$
Modulus $|z| = sqrt{3^2 + 4^2} = sqrt{9+16} = sqrt{25} = 5$.
Argument is in Quadrant 1. $ an heta = frac{4}{3}$.
$ heta = arctan(frac{4}{3}) approx 0.927$ radians (or $53.1^circ$).
Given $z_1 = 2 \operatorname{cis} \frac{\pi}{3}$ and $z_2 = 4 \operatorname{cis} \frac{\pi}{6}$, find $z_1 z_2$ in the form $a+bi$.
Question Q9
Given $z_1 = 2 \operatorname{cis} \frac{\pi}{3}$ and $z_2 = 4 \operatorname{cis} \frac{\pi}{6}$, find $z_1 z_2$ in the form $a+bi$.
Mark Scheme
Using polar form multiplication: $z_1 z_2 = r_1 r_2 ext{cis}( heta_1 + heta_2)$.
$z_1 z_2 = (2)(4) ext{cis}(frac{pi}{3} + frac{pi}{6})$
$= 8 ext{cis}(frac{2pi}{6} + frac{pi}{6}) = 8 ext{cis}(frac{pi}{2})$
$= 8(cos frac{pi}{2} + i sin frac{pi}{2})$
$= 8(0 + i(1)) = 8i$.
Form $a+bi$ is $0 + 8i$.
