Question 3
If all roots of the equation $z^3=1$ are plotted on an Argand plane and then joined by straight lines, the shape formed is
That’s it. Step through the hints below to see why. In 2025, 75.5% of Queensland students chose (B).
Not quite. The roots are evenly spaced, $120^\circ$ apart around the circle, so every angle in the triangle is $60^\circ$. There is no right angle. In 2025, 4.9% of Queensland students chose (A).
Close. It does have two equal sides, but in fact all three sides are equal, $\sqrt3$ each. Equilateral is the precise answer. In 2025, 16.9% of Queensland students chose (C).
Not quite. A scalene triangle has no equal sides, but these three sides are all $\sqrt3$. In 2025, 2.3% of Queensland students chose (D).
Three equal sides and three $60^\circ$ angles make an equilateral triangle. This is the equation in the question.
With two roots, $1$ and $-1$, there is only a line segment.
The roots of $z^n=1$ always make a regular polygon with $n$ equal sides.
Use the approach from game plan 1.3. Expect $n$ roots, spaced $\frac{2\pi}{n}$ apart. Here $n=3$, so there are three roots, $\frac{2\pi}{3}$ or $120^\circ$ apart.
Start with $z_0$. For $w=r\,\text{cis}\,\theta$, the first root is $z_0=r^{1/n}\,\text{cis}\left(\frac{\theta}{n}\right)$. Here $w=1=1\,\text{cis}\,0$.
$$z_0=1^{1/3}\,\text{cis}\left(\tfrac{0}{3}\right)=1$$
Add $\frac{2\pi}{3}$ to the argument each time to get the others.
$$z_1=\text{cis}\left(\tfrac{2\pi}{3}\right)\qquad z_2=\text{cis}\left(\tfrac{4\pi}{3}\right)$$
Plot them. The roots sit on a circle of radius $r^{1/n}=1$ and form a regular $n$-gon. A regular polygon with three sides is an equilateral triangle.
If you want to check, every side is $\sqrt3$. The triangle is isosceles as well, but equilateral describes it exactly.
There are 3 roots, spaced $\frac{2\pi}{3}$ apart, starting at $z_0=1$.
$$z_0=1,\quad z_1=\text{cis}\left(\tfrac{2\pi}{3}\right),\quad z_2=\text{cis}\left(\tfrac{4\pi}{3}\right)$$
They lie on the unit circle and form a regular 3-gon, which is an equilateral triangle.
How Queensland students answered in 2025
From the QCAA Specialist Mathematics subject report for 2025. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2025 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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