Can someone help me here?
"The product of 4(cos30 + i sin30) and 3(cos90 + i sin90) is equal to..."
The answer is 6(-1 + i rad(3)) apparently but I don't know where that came from.
I can never remember how to do these for some reason...
Do you mean?
$\displaystyle 4(\cos(30) + i \sin(30)) \times 3(\cos(90) + i \sin(90))$
$\displaystyle 4\left(\frac{\sqrt{3}}{2} + i \frac{1}{2}\right) \times 3( 0+ i\times 1)$
$\displaystyle 12\left(\frac{\sqrt{3}}{2} + i \frac{1}{2}\right) \times i$
$\displaystyle 12\left(\frac{\sqrt{3}}{2}i + i^2 \frac{1}{2}\right) $
$\displaystyle 12\left(\frac{\sqrt{3}}{2}i -\frac{1}{2}\right) $
$\displaystyle 6\sqrt{3}i -6 $
That is surely the hard way! The whole point of the "polar form", $\displaystyle r cis(\theta)$ (also written $\displaystyle r(cos(\theta)+ i sin(\theta)$ and even $\displaystyle re^{i\theta}$), is that
$\displaystyle (r_1 cis(\theta_1))(r_2 cis(\theta_2))= (r_1r_2) cis(\theta_1+ \theta_2)$.
Here, (4 cis(30))(3 cis(90))= (12) cis(120)= 12(cos(120)+ i sin(120)). Since the numbers were given in "cis" form, I would leave the answer in that form. If you like, $\displaystyle cos(120)= -\frac{1}{2}$ and $\displaystyle sin(120)= \frac{\sqrt{3}}{2}$ so $\displaystyle 12 cis(120)= -6+ 6i\sqrt{3}$