Chapter 6 Practice Problems
6.1 Solving quadratic equations
Find the solutions of the following quadratic equations.
\(z^2 -4z + 7 = 0\)
\(z^2 + 8 = 0\)
\(z^2 + 24z + 27 = 0\)
\(3z^2 -z + 5 = 0\)
6.2 Argand diagrams
Draw Argand diagrams to represent the following complex numbers.
\(z = 2 + 3i\)
\(z = -3 + 4i\)
\(z = \frac{1}{2} - 3i\)
\(z = -4 - i\)
\(z = 2\)
6.3 Addition
Express the solutions to the following sums in the form \(a + ib\) where \(a\) and \(b\) are real.
\((1 - 2i) + (-2 + 3i)\)
\((4 - 3i) - (6 - 5i)\)
\(3(1 - 2i) + 4(3 + 3i)\)
6.4 Multiplication
Express the solutions to the following products in the form \(a + ib\) where \(a\) and \(b\) are real.
\(i(5 - i)\)
\((2 + 3i)(4 - 5i)\)
\((4 - 9i)(5 - 3i)\)
\((1 - i)(7 + 3i)\)
\((1 + i)(3 - 4i)(2 + 3i)\)
6.5 Complex conjugate and division
Express the complex conjugate \(\overline{z}\) in the form \(a + ib\) where \(a\) and \(b\) are real.
\(z = 7i\)
\(z = 2 - 3i\)
\(z = 6i - 7\)
Using the complex conjugate, express the following quotients in the form \(a + ib\) where \(a\) and \(b\) are real.
\(\frac{3 + 2i}{1 - 5i}\)
\(\frac{2 + 3i}{8 - i}\)
\(\frac{(2 - i)(2 + 5i)}{1 + i}\)
\(\frac{1}{4 - 2i} + \frac{1}{3 + i}\)
\(\frac{1 - i}{2 + i} + \frac{4 - i}{1 - i}\)
\(\frac{11i}{2 + 3i}\)
6.6 Polar co-ordinates
Find the modulus \(|z|\) and principal value of the argument \(-\pi<\text{Arg}(z)\leq\pi\) for each of the following.
\(z = 1 + i\)
\(z = \sqrt{3} - i\)
\(z = -4 - 4\sqrt{3}i\)
\(z = -4 + 4i\)
\(z = -\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})\)
Write the following complex numbers in polar form, using the modulus \(\left|z\right|\) and the principal value of the argument \(-\pi<\text{Arg}(z)\leq\pi\), in each case.
\(z = 2 - \sqrt{12}i\)
\(z = -9\)
\(z = 7 + 7i\)
\(z = -94 + 94i\)
\(z = -3 -3\sqrt{3}i\)
Write the following complex numbers in rectangular form i.e. \(z = a + bi\) where \(a, b\in\mathbb{R}\).
\(z = 2\text{cis}\left(\frac{\pi}{6}\right)\)
\(z = 5\text{cis}\left(-\frac{\pi}{3}\right)\)
\(z = 3\sqrt{2}\text{cis}\left(-\frac{3\pi}{4}\right)\)
\(z = 4\sqrt{3}\text{cis}\left(\frac{5\pi}{6}\right)\)
6.7 de Moivre’s theorem
Let \(z_1 = -2 + 2i\) and \(z_2 = 1 - \sqrt{3}i\). Convert \(z_1\) and \(z_2\) to polar form. Then use de Moivre’s theorem along with the rules for multiplication and division in polar form to find the following, leaving your answers in polar form \(z = r\text{cis}\left(\text{Arg}(z)\right)\). You may have to add or subtract multiples of \(2\pi\) from the argument you initially calculate, in order to have \(-\pi < \text{Arg}(z) \leq \pi\).
\(z_1z_2\)
\(z_1^3\)
\(\frac{1}{z_1^3}\)
\(z_1^3z_2^5\)
\(\frac{z_1^8}{z_2^6}\)
Use the polar form and de Moivre’s theorem to simplify the following.
\(z = \frac{\left(2 + 2i\right)^5}{1 - i}\)
\(z = \frac{\left(1 + \sqrt{3}i\right)^2}{\left(2 + 2i\right)^3}\)
\(z = \left(1 + i\right)^{10} + \left(1 - i\right)^{10}\)
\(z = \left(-\sqrt{3} + i\right)^{-5}\)
\(z = \frac{\left(\cos\left(\theta\right) + i\sin\left(\theta\right)\right)^4}{\left(\sin\left(\theta\right) + i\cos\left(\theta\right)\right)^2}\)
6.8 Applications to trigonometric identities
Use de Moivre’s theorem to express \(\sin(3\theta)\) and \(\cos(3\theta)\) in terms of \(\sin(\theta)\) and \(\cos(\theta)\).
Use de Moivre’s theorem to show that \[\begin{aligned} \cos^2(\theta) &= \frac{1}{2}\left(\cos(2\theta) + 1\right),\\ \sin(2\theta) &= 2\sin(\theta)\cos(\theta). \end{aligned}\]
Express \(\sin^5(\theta)\) in terms of sines of integer multiples of \(\theta\). Hence calculate \(\int\sin^5(\theta)\text{d}\theta\).
Express \(\cos^2(\theta)\sin^4(\theta)\) as linear combinations of cosines of integer multiples of \(\theta\).
Use your answer to the last question to compute \(\int_{0}^{\frac{\pi}{6}}\cos^2(\theta)\sin^4(\theta)\text{d}\theta\).
6.9 Roots
Find the following roots and give your answers in the form \(a + bi\) where \(a, b \in \mathbb{R}\).
The square roots of \(4i\).
The square roots of \(2 + 2\sqrt{3}i\).
The cube roots of \(-27\).
The cube roots of \(8i\).
The fourth roots of \(-8 - 8\sqrt{3}i\).
The sixth roots of \(729\).