The Scenario
An air traffic controller sees a plane on the radar screen. The plane's position can be reported in two ways: "the plane is 3 km east and 4 km north of the tower" (Cartesian coordinates), or "the plane is 5 km from the tower at a bearing of 053°" (distance and direction). Both describe the same point, but the second form directly tells us the distance and angle — two pieces of information that are often more useful in navigation.
The Core Question
If a complex number is represented by the point (x, y) on an Argand diagram, can we describe the same number using its distance from the origin and the angle it makes with the positive real axis? What would this form look like, and why might it be more useful for certain operations?
From Cartesian to Polar
For a complex number $z = x + \mathrm{i}y$ represented on an Argand diagram, the distance from the origin is the modulus $|z| = r = \sqrt{x^2 + y^2}$, and the angle measured anticlockwise from the positive real axis is the argument $\arg z = \theta$, where $\tan\theta = \dfrac{y}{x}$.
where $r = |z| \ge 0$ and $\theta = \arg z$, with $-\pi < \theta \le \pi$ (principal argument).
The exponential form is compact and simplifies multiplication and division.
The modulus of a product is the product of the moduli. The argument of a product is the sum of the arguments.
Task: In groups of 3–4, match each complex number in Cartesian form to its equivalent polar form and exponential form. You have 6 minutes.
Instructions:
- Your group will receive 6 cards showing complex numbers in Cartesian form $x + \mathrm{i}y$.
- Find the matching polar form $r(\cos\theta + \mathrm{i}\sin\theta)$ and exponential form $re^{\mathrm{i}\theta}$ for each.
- For each match, write down the modulus $r$ and principal argument $\theta$.
- The first group to correctly match all 6 sets wins.
Cards:
$1+\mathrm{i} = \sqrt{2}\left(\cos\frac{\pi}{4}+\mathrm{i}\sin\frac{\pi}{4}\right) = \sqrt{2}e^{\mathrm{i}\pi/4}$
$-1+\mathrm{i}\sqrt{3} = 2\left(\cos\frac{2\pi}{3}+\mathrm{i}\sin\frac{2\pi}{3}\right) = 2e^{\mathrm{i}2\pi/3}$
$\sqrt{3}+\mathrm{i} = 2\left(\cos\frac{\pi}{6}+\mathrm{i}\sin\frac{\pi}{6}\right) = 2e^{\mathrm{i}\pi/6}$
$-2\mathrm{i} = 2\left(\cos\left(-\frac{\pi}{2}\right)+\mathrm{i}\sin\left(-\frac{\pi}{2}\right)\right) = 2e^{-\mathrm{i}\pi/2}$
$-3 = 3\left(\cos\pi+\mathrm{i}\sin\pi\right) = 3e^{\mathrm{i}\pi}$
$-\sqrt{3}-\mathrm{i} = 2\left(\cos\left(-\frac{5\pi}{6}\right)+\mathrm{i}\sin\left(-\frac{5\pi}{6}\right)\right) = 2e^{-\mathrm{i}5\pi/6}$
Task: Work independently to convert each complex number into the requested form. Show your working clearly. You have 5 minutes.
- Express $z = -2 + 2\mathrm{i}$ in polar form $r(\cos\theta + \mathrm{i}\sin\theta)$.
- Express $z = 4\left(\cos\frac{2\pi}{3} + \mathrm{i}\sin\frac{2\pi}{3}\right)$ in Cartesian form $x + \mathrm{i}y$.
- Express $z = 3e^{-\mathrm{i}\pi/6}$ in Cartesian form $x + \mathrm{i}y$.
- Express $z = -4\sqrt{3} + 4\mathrm{i}$ in exponential form $re^{\mathrm{i}\theta}$.
$z = -2 + 2\mathrm{i}$. $r = \sqrt{(-2)^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$. $\tan\theta = \frac{2}{-2} = -1$. Since $x<0, y>0$, $\theta = \frac{3\pi}{4}$. So $z = 2\sqrt{2}\left(\cos\frac{3\pi}{4} + \mathrm{i}\sin\frac{3\pi}{4}\right)$.
$z = 4\left(\cos\frac{2\pi}{3} + \mathrm{i}\sin\frac{2\pi}{3}\right) = 4\left(-\frac{1}{2} + \mathrm{i}\frac{\sqrt{3}}{2}\right) = -2 + 2\sqrt{3}\mathrm{i}$.
$z = 3e^{-\mathrm{i}\pi/6} = 3\left(\cos\left(-\frac{\pi}{6}\right) + \mathrm{i}\sin\left(-\frac{\pi}{6}\right)\right) = 3\left(\frac{\sqrt{3}}{2} - \frac{\mathrm{i}}{2}\right) = \frac{3\sqrt{3}}{2} - \frac{3}{2}\mathrm{i}$.
$z = -4\sqrt{3} + 4\mathrm{i}$. $r = \sqrt{(-4\sqrt{3})^2 + 4^2} = \sqrt{48+16} = \sqrt{64} = 8$. $\tan\theta = \frac{4}{-4\sqrt{3}} = -\frac{1}{\sqrt{3}}$. Since $x<0, y>0$, $\theta = \frac{5\pi}{6}$. So $z = 8e^{\mathrm{i}5\pi/6}$.
Express $z = 1 + \sqrt{3}\,\mathrm{i}$ in polar form $r(\cos\theta + \mathrm{i}\sin\theta)$. Hence find $z^2$ in polar form.
$|z| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$.
$\tan\theta = \dfrac{\sqrt{3}}{1} = \sqrt{3}$. Since $x>0$ and $y>0$, the point is in the first quadrant, so $\theta = \dfrac{\pi}{3}$.
$z = 2\left(\cos\dfrac{\pi}{3} + \mathrm{i}\sin\dfrac{\pi}{3}\right)$.
$|z^2| = |z|^2 = 2^2 = 4$.
$\arg(z^2) = 2\arg z = 2 \times \dfrac{\pi}{3} = \dfrac{2\pi}{3}$.
Therefore $z^2 = 4\left(\cos\dfrac{2\pi}{3} + \mathrm{i}\sin\dfrac{2\pi}{3}\right)$.
$z^2 = (1 + \sqrt{3}\mathrm{i})^2 = 1 + 2\sqrt{3}\mathrm{i} + 3\mathrm{i}^2 = 1 + 2\sqrt{3}\mathrm{i} - 3 = -2 + 2\sqrt{3}\mathrm{i}$.
$r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = 4$. $\tan\theta = \frac{2\sqrt{3}}{-2} = -\sqrt{3}$, and since $x<0, y>0$, $\theta = \frac{2\pi}{3}$. ✓
Write down the values of the modulus and the principal argument of each of these complex numbers.
(i) $8\left(\cos\dfrac{\pi}{5} + \mathrm{i}\sin\dfrac{\pi}{5}\right)$
(ii) $\dfrac{\cos 2.3 + \mathrm{i}\sin 2.3}{4}$
(iii) $4\left(\cos\dfrac{\pi}{3} - \mathrm{i}\sin\dfrac{\pi}{3}\right)$
$|z| = 8$, $\arg z = \dfrac{\pi}{5}$.
$|z| = \dfrac{1}{4}$, $\arg z = 2.3$ radians.
Rewrite as $4\left(\cos\left(-\dfrac{\pi}{3}\right) + \mathrm{i}\sin\left(-\dfrac{\pi}{3}\right)\right)$. So $|z| = 4$, $\arg z = -\dfrac{\pi}{3}$.
For each complex number, find the modulus and principal argument, and hence write the complex number in polar form. Give the argument in radians, either as a simple rational multiple of $\pi$ or correct to 3 decimal places.
(v) $1 + \mathrm{i}$
(vi) $-5 - 5\mathrm{i}$
(vii) $1 - \sqrt{3}\mathrm{i}$
(viii) $6\sqrt{3} + 6\mathrm{i}$
$|z| = \sqrt{1^2 + 1^2} = \sqrt{2}$. $\tan\theta = 1$, $x>0,y>0 \Rightarrow \theta = \frac{\pi}{4}$.
$z = \sqrt{2}\left(\cos\frac{\pi}{4} + \mathrm{i}\sin\frac{\pi}{4}\right)$.
$|z| = \sqrt{(-5)^2 + (-5)^2} = 5\sqrt{2}$. $\tan\theta = 1$, $x<0,y<0 \Rightarrow \theta = -\frac{3\pi}{4}$.
$z = 5\sqrt{2}\left(\cos\left(-\frac{3\pi}{4}\right) + \mathrm{i}\sin\left(-\frac{3\pi}{4}\right)\right)$.
$|z| = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1+3} = 2$. $\tan\theta = -\sqrt{3}$, $x>0,y<0 \Rightarrow \theta = -\frac{\pi}{3}$.
$z = 2\left(\cos\left(-\frac{\pi}{3}\right) + \mathrm{i}\sin\left(-\frac{\pi}{3}\right)\right)$.
$|z| = \sqrt{(6\sqrt{3})^2 + 6^2} = \sqrt{108+36} = \sqrt{144} = 12$. $\tan\theta = \frac{6}{6\sqrt{3}} = \frac{1}{\sqrt{3}}$, $x>0,y>0 \Rightarrow \theta = \frac{\pi}{6}$.
$z = 12\left(\cos\frac{\pi}{6} + \mathrm{i}\sin\frac{\pi}{6}\right)$.
Write each complex number with the given modulus and argument in the form $x + \mathrm{i}y$, giving surds in your answer where appropriate.
(i) $|z| = 2$, $\arg z = \dfrac{\pi}{2}$
(ii) $|z| = 3$, $\arg z = \dfrac{\pi}{3}$
(iii) $|z| = 7$, $\arg z = \dfrac{5\pi}{6}$
(iv) $|z| = 1$, $\arg z = -\dfrac{\pi}{4}$
$z = 2\left(\cos\frac{\pi}{2} + \mathrm{i}\sin\frac{\pi}{2}\right) = 2(0 + \mathrm{i}) = 0 + 2\mathrm{i}$.
$z = 3\left(\cos\frac{\pi}{3} + \mathrm{i}\sin\frac{\pi}{3}\right) = 3\left(\frac{1}{2} + \frac{\sqrt{3}}{2}\mathrm{i}\right) = \frac{3}{2} + \frac{3\sqrt{3}}{2}\mathrm{i}$.
$z = 7\left(\cos\frac{5\pi}{6} + \mathrm{i}\sin\frac{5\pi}{6}\right) = 7\left(-\frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}\right) = -\frac{7\sqrt{3}}{2} + \frac{7}{2}\mathrm{i}$.
$z = 1\left(\cos\left(-\frac{\pi}{4}\right) + \mathrm{i}\sin\left(-\frac{\pi}{4}\right)\right) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}\mathrm{i}$.
Exit Ticket
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