Polar and exponential form: mixed practice Problem set A

Write each number in the form \(r\,\mathrm{cis}\,\theta\) with \(0 \le \theta < 2\pi\).

1. (a) \(-6 i\)
2. (b) \(-3 - 3 i\)
3. (c) \(2 - 2 \sqrt{3} i\)

Write \(z\) in the form \(x + iy\).

1. (a) \(|z| = 9,\ \arg z = -\frac{\pi}{2}\)
2. (b) \(|z| = 6,\ \arg z = \frac{\pi}{3}\)
3. (c) \(|z| = 2 \sqrt{2},\ \arg z = \frac{3\pi}{4}\)

Write each in the form \(r\,\mathrm{cis}\,\theta\) with \(0 < \theta \le 2\pi\). Work without a calculator.

1. (a) \((3\,\mathrm{cis}\left(\frac{5\pi}{6}\right))(4\,\mathrm{cis}\left(\frac{4\pi}{3}\right))\)
2. (b) \(\dfrac{12\,\mathrm{cis}\left(\frac{\pi}{4}\right)}{3\,\mathrm{cis}\left(\frac{2\pi}{3}\right)}\)

Write each in exponential form \(re^{i\theta}\) with \(-\pi < \theta \le \pi\).

1. (a) \(6e^{\frac{\pi i}{3}} \times 2e^{\frac{\pi i}{4}}\)
2. (b) \(14e^{\frac{7\pi i}{6}} \div 2e^{\frac{\pi i}{3}}\)

Work without a calculator.

1. (a) Write in the form \(x + iy\): \(\left(e^{\frac{\pi i}{6}}\right)^{4}\)
2. (b) Simplify \(\mathrm{cis}\,0.9 \times \mathrm{cis}\,1.3\)

The diagram shows the point that represents \(z = 2 + i\) on an Argand diagram.

1. (a) Mark and label \(z^2\) on a copy of the diagram. Write \(z^2\) in the form \(x + iy\).
2. (b) Mark and label \(iz\). Write \(iz\) in the form \(x + iy\).

\(z = -3 + 3 i\).

1. (a) Find \(|z|\).
2. (b) Find \(\arg z\), with \(-\pi < \arg z \le \pi\).
3. (c) Hence write down the modulus and argument of \(z^2\), with \(-\pi < \arg z^2 \le \pi\).
4. (d) Hence write \(z^2\) in the form \(x + iy\).

\(z = 3 + 3 \sqrt{3} i\) and \(w = -2 + 2 i\). Give each argument in the interval \(-\pi < \theta \le \pi\).

1. (a) Find \(\arg z\).
2. (b) Find \(\arg w\).
3. (c) Find \(\arg(zw)\).

Write each in the form \(x + iy\). Do not use a calculator.

1. (a) \((3\,\mathrm{cis}\left(\frac{\pi}{6}\right))(2\,\mathrm{cis}\left(\frac{\pi}{3}\right))\)
2. (b) \(2\,\mathrm{cis}\left(\frac{\pi}{3}\right) + 4\,\mathrm{cis}\left(\frac{\pi}{2}\right)\)

Write each in Cartesian form, to 3 significant figures. Combine the powers first, then use the calculator only to find the real and imaginary parts.

1. (a) \(\left(4\,\mathrm{cis}\left(\frac{\pi}{5}\right)\right)^{2}\left(3\,\mathrm{cis}\left(\frac{\pi}{10}\right)\right)^{3}\)
2. (b) \(\dfrac{\left(6\,\mathrm{cis}\left(\frac{\pi}{3}\right)\right)^{3}}{\left(2\,\mathrm{cis}\left(\frac{\pi}{9}\right)\right)^{4}}\)

Work without a calculator.

1. (a) Write \(z = \sqrt{3} - i\) in the form \(r\,\mathrm{cis}\,\theta\), with \(-\pi < \theta \le \pi\).
2. (b) Hence find \(z^{8}\) in Cartesian form.
3. (c) \(w = -1 + i\) and \(u = \cos\frac{\pi}{9} + i\sin\frac{\pi}{9}\). Find \(w^{4}u^{6}\) in Cartesian form.

Give each answer as a positive integer.

1. (a) Find the smallest positive integer \(n\) for which \(z^n\) is real, where \(z = 2\,\mathrm{cis}\left(\frac{4\pi}{15}\right)\).
2. (b) Find the smallest positive integer \(n\) for which \(w^n = i\), where \(w = \mathrm{cis}\left(\frac{3\pi}{14}\right)\).

A small boat is lifted by two sets of waves. At time \(t\) seconds the height of the boat, in metres, is \(H(t) = 1.2\sin(4t) + 0.8\sin(4t + 2.5)\). Write the total as a single sinusoid \(A\sin(4t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

Two AC voltage sources are connected in a circuit. The voltages, in volts, at time \(t\) seconds are \(V_1 = 7\cos(25t)\) and \(V_2 = 10\cos(25t + 3)\).

1. (a) On the axes provided, sketch the graphs of \(V_1\) and \(V_2\) for \(0 \le t \le 0.5\). Show where each graph starts and where it first reaches its maximum.
2. (b) Find the total voltage \(V = V_1 + V_2\) in the form \(A\cos(25t + c)\), where \(A > 0\) and \(-\pi < c \le \pi\).

The numbers of rabbits \(R\) and foxes \(F\) in a nature reserve vary through the year. At time \(t\) days after 1 January, \(R = 300 + 50\sin\left(\frac{2\pi}{365}t\right)\) and \(F = 40 + 15\sin\left(\frac{2\pi}{365}t + 1.2\right)\). Write the total number of animals, \(N = R + F\), in the form \(C + A\sin(bt + c)\).

The complex numbers \(z\) and \(w\) have arguments between \(0\) and \(\pi\). Given that \(zw = - 4 \sqrt{3} + 4 i\) and \(\dfrac{z}{w} = 2 i\), find the modulus and argument of \(z\) and of \(w\).

\(z = \sqrt{3} + i\) and \(w = -1 + i\).

1. (a) Find the modulus and argument of \(z\) and of \(w\), with \(-\pi < \arg \le \pi\).
2. (b) Find \(\dfrac{w}{z}\) in modulus-argument form and in Cartesian form.
3. (c) Hence find the exact value of \(\cos\left(\frac{7\pi}{12}\right)\).

Let \(z = 6 + 4e^{1.5i}\).

1. (a) Find the modulus and argument of \(6 + 4e^{1.5i}\).
2. (b) Hence write \(6\sin\theta + 4\sin(\theta + 1.5)\) in the form \(A\sin(\theta + \alpha)\), where \(A > 0\) and \(-\pi < \alpha \le \pi\).