Question 8
Given $z=2\,\text{cis}\left(\dfrac{\pi}{3}\right)$, determine $z^3$.
That’s it. Step through the hints below to see why. In 2024, 72.9% of Queensland students chose (A).
Not quite. The modulus is cubed, not multiplied by 3. It’s the argument that gets multiplied by 3. In 2024, 2.6% of Queensland students chose (B).
Not quite. Two things need fixing: the modulus is cubed rather than tripled, and $\text{cis}(\pi)$ equals $-1$. In 2024, 4.5% of Queensland students chose (C).
Not quite. The modulus is right, but the argument becomes $\pi$, and $\text{cis}(\pi)=-1$. The point lands on the negative real axis. In 2024, 19.7% of Queensland students chose (D).
De Moivre’s theorem says that to raise a complex number in polar form to a power, you raise the modulus to that power and multiply the argument by it.
$$\big(r\,\text{cis}\,\theta\big)^n=r^n\,\text{cis}(n\theta)$$
Here $r=2$, $\theta=\frac{\pi}{3}$ and $n=3$.
$$z^3=2^3\,\text{cis}\left(3\times\frac{\pi}{3}\right)=8\,\text{cis}(\pi)$$
Convert back to Cartesian form. An argument of $\pi$ points along the negative real axis.
$$\text{cis}(\pi)=\cos\pi+i\sin\pi=-1+0i$$
Step through the powers on the left. Each time, the modulus doubles and the point turns another $\frac{\pi}{3}$ anticlockwise.
$$z^3=2^3\,\text{cis}\left(3\times\frac{\pi}{3}\right)=8\,\text{cis}(\pi)=8(-1)=-8$$
How Queensland students answered in 2024
From the QCAA Specialist Mathematics subject report for 2024. 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 2024 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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