Find the cubic roots of unity and
plot them on an Argand diagram.
Finding the cubic roots of unity is
like saying what are the solutions to the equation 𝑧 cubed equals one. To find them, we can use the
general formula for the 𝑛th roots of unity. That’s cos of two 𝜋𝑘 over 𝑛 plus
𝑖 sin of two 𝜋𝑘 over 𝑛 for integer values of 𝑘 between zero and 𝑛 minus
one. Since we’re finding the cubic roots
of unity, in this example, our value of 𝑛 is three, which means 𝑘 will take the
values zero, one, and two.
Let’s begin with the case when 𝑘
is equal to zero. This root is cos of zero plus 𝑖
sin of zero. Well, cos of zero is one. And sin of zero is zero. So the first root is one. And it makes complete sense if we
think about it that a solution to the equation 𝑧 cubed equals one would be one.
Next, we let 𝑘 be equal to
one. This root is cos of two 𝜋 by three
plus 𝑖 sin of two 𝜋 by three. And in exponential form, that’s 𝑒
to the two 𝜋 by three 𝑖. Finally, we let 𝑘 be equal to
two. The root here is cos four 𝜋 by
three plus 𝑖 sin of four 𝜋 by three. Notice though that the argument for
this root is outside of the range for the principal argument. We therefore subtract two 𝜋 from
four 𝜋 by three to get negative two 𝜋 by three. So our third and final root is cos
of negative two 𝜋 by three plus 𝑖 sin of negative two 𝜋 by three. Or in exponential form, that’s 𝑒
to the negative two 𝜋 by three 𝑖.
Now that we have the cubic roots of
unity, we need to plot them on an Argand diagram. There are two ways we could
approach this problem. We could convert each number to
algebraic form. That’s one, negative a half plus
root three over two 𝑖, and negative a half minus root three over two 𝑖. These are plotted on the Argand
diagram as shown. Alternatively, we could’ve used the
modulus and argument of each root.
Either way, let’s notice that the
points that represent these complex numbers are the vertices of an equilateral
triangle. This triangle is inscribed in a
unit circle whose centre is the origin. Actually, an interesting
geometrical property of the 𝑛th roots of unity is that, on an Argand diagram, they
are all evenly spaced about the unit circle whose centre is the origin. They form a regular 𝑛-gon. It has a vertex at the point whose
Cartesian coordinates are one, zero.