Math C

The Complex Plane: Practice Problems

In polar form, multiplication by a complex number is a rotation combined with a scaling. Practise De Moivre's theorem and the rotation formula.

Basic, Standard: Grade 12 · Term 1 / Advanced: Grade 12 · Exam prep

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Absolute value, argument and polar form

For $z = a + bi$ , $|z| = \sqrt{a^2 + b^2}$ . Writing $z = r(\cos\theta + i\sin\theta)$ with $r = |z|$ is the polar form, and $\theta$ is the argument.

Products and quotients

|z_1z_2| = |z_1||z_2|,\quad \arg z_1z_2 = \arg z_1 + \arg z_2,\qquad \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|},\quad \arg\frac{z_1}{z_2} = \arg z_1 - \arg z_2

De Moivre's theorem and n-th roots

De Moivre's theorem

(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

To solve $z^n = w$ , put $z = r(\cos\theta + i\sin\theta)$ and compare absolute values and arguments, allowing the argument to differ by $2k\pi$ . Taking $k = 0, 1, \ldots, n - 1$ gives the $n$ solutions.

Rotations and geometry

Rotating $\beta$ about $\alpha$ by $\theta$ gives $\gamma$ with

\gamma - \alpha = (\beta - \alpha)(\cos\theta + i\sin\theta).

In polar form, $\dfrac{\gamma - \alpha}{\beta - \alpha}$ has absolute value $\dfrac{\mathrm{AC}}{\mathrm{AB}}$ and argument $\angle \mathrm{BAC}$ , which reveals the shape of the triangle.

Loci

Q1BasicGr 12 · Term 1The complex plane

For the points $\mathrm{A}(2 - 4i)$ and $\mathrm{B}(-5 + 3i)$ , find the following.

(1)The distance AB
(2)The complex number of the point dividing AB internally in the ratio $2 : 1$

Answer

(1) $\mathrm{AB} = 7\sqrt{2}$
(2) $-\frac{8}{3} + \frac{2}{3}i$

Solution

The distance is
$|(-5 + 3i) - (2 - 4i)| = |-7 + 7i| = 7\sqrt{2}$
The division point is
$\frac{(2 - 4i) + 2(-5 + 3i)}{3} = -\frac{8}{3} + \frac{2}{3}i$

Q2StandardGr 12 · Term 1The complex plane

For $\mathrm{A}(1)$ and $\mathrm{B}(-2)$ , find all complex numbers $\mathrm{C}$ such that $\triangle \mathrm{ABC}$ is equilateral.

Answer

-\frac{1}{2} - \frac{3\sqrt{3}}{2}\,i,\ -\frac{1}{2} + \frac{3\sqrt{3}}{2}\,i

Solution

C is B rotated about A by $\pm \frac{\pi}{3}$ :
$\gamma = (1) + (-3)\left(\frac{1}{2} \pm \frac{\sqrt{3}}{2}i\right)$
Computing,
$\gamma = -\frac{1}{2} - \frac{3\sqrt{3}}{2}\,i,\ -\frac{1}{2} + \frac{3\sqrt{3}}{2}\,i$

Q3AdvancedExam prepThe complex plane

Let $z = \sqrt{3} + i$ . Find the smallest natural number $n$ for which $z^n$ is a positive real number, and the value of $z^n$ for that $n$ .

(1)The value of $n$
(2)The value of $z^n$

Answer

(1) $n = 12$
(2) $4096$

Solution

In polar form,
$z = 2\left(\cos \frac{\pi}{6} + i\sin \frac{\pi}{6}\right)$
By De Moivre,
$z^n = 2^n\left(\cos\frac{n\pi}{6} + i\sin\frac{n\pi}{6}\right)$
It is a positive real when the sine is 0 and the cosine is positive.
$n = 12,\qquad z^{12} = 4096$

Q1BasicGr 12 · Term 1The complex plane

For the points $\mathrm{A}(3)$ and $\mathrm{B}(4 - 2i)$ , find the following.

(1)The distance AB
(2)The complex number of the point dividing AB internally in the ratio $1 : 2$

Q2StandardGr 12 · Term 1The complex plane

Evaluate.

(-1 + i)^{9}

Q3AdvancedExam prepThe complex plane

In the complex plane, the points $z$ with $|z - (1 - 2i)| = 2|z + 2 + 2i|$ form a circle. Find its center and radius.

(1)The center
(2)The radius

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The Complex Plane: Practice Problems

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