Number Systems: Problem Sheet 5

Here are some hints for Problem Sheet 5 given to people in office hours.

Question 2. If you are stuck, please first remind yourself of exponential form for complex numbers (there are some examples and exercises on the revision sheet) and look again at Problem 2.5.

If after doing all that you are still stuck, it might be because replacing the $8i$ in Problem 2.5 with $8$ somehow makes it less obvious how to get started. Let $w = \mathrm{e}^{i \theta}$. We need to solve

$r^3 \mathrm{e}^{3 i \theta} = 8.$

The modulus of the left-hand side is $r^3$ and the modulus of the right-hand side is $8$. Hence $r^3 = 8$ and so $r=2$.

The argument is the tricker bit. The argument of $r^3 \mathrm{e}^{3 i \theta}$ is $3 \theta$, and the argument of $8$ is $0$. But remember that we can add any multiple of $2 \pi$ and get another value of the argument. So the conclusion from comparing arguments is that

$3 \theta = 0 + 2n \pi$

for some $n \in \mathbb{Z}$. Now proceed as in Problem 2.5.

Finally, to convert your answers to exponential form you should use the known values of $\cos (2\pi/3)$ and $\sin (2\pi/3)$. You should know the values of cos and sin on the angles $0, \pi/6, \pi/3, \pi/4, \pi/2$: the values needed for this problem can be deduced from these.