A question is asking me to "find the sixth roots of unity and represent them on an Argand diagram".
I don't need you to do the problem for me, I'd rather attempt it myself. However, I don't understand what it's asking me to do. What is unity? Is it $1$, i.e. $\cos(0+2\pi k)$ where $k$ is an integer?
Best Answer
Yes, unity represents $1$. So there are six complex roots of unity $z_i,$ such that $$z_i^6 = 1,\;\;\;1 \leq i \leq 6$$
From De Moivre's formula (valid for all real $x$ and integers $n$), we have
$$(\cos x + i \sin x)^n = \cos nx + i \sin nx.$$
Setting $x = 2π/n$ gives an $\color{blue}{\bf \text{nth root of unity}}$:
$$\left(\color{blue}{\bf \cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}}\right)^n = \cos 2\pi + i \sin 2\pi = 1,$$ and so for $k = 1, 2, ⋯ , n − 1,$
$$\left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^k= \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n} \neq 1$$