Polar and exponential form: mixed practice Problem set B

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

1. (a) \(-12 i\)
2. (b) \(- 4 \sqrt{3} - 4 i\)
3. (c) \(7 - 7 i\)

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

1. (a) \(|z| = 15,\ \arg z = -\frac{\pi}{2}\)
2. (b) \(|z| = 10,\ \arg z = \frac{\pi}{6}\)
3. (c) \(|z| = 3 \sqrt{2},\ \arg z = -\frac{3\pi}{4}\)

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

1. (a) \((5\,\mathrm{cis}\left(\frac{3\pi}{4}\right))(2\,\mathrm{cis}\left(\frac{5\pi}{4}\right))\)
2. (b) \(\dfrac{18\,\mathrm{cis}\left(\frac{\pi}{3}\right)}{6\,\mathrm{cis}\left(\frac{5\pi}{6}\right)}\)

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

1. (a) \(4e^{\frac{2\pi i}{5}} \times 3e^{\frac{\pi i}{10}}\)
2. (b) \(20e^{\frac{\pi i}{2}} \div 5e^{\frac{3\pi i}{4}}\)

Work without a calculator.

1. (a) Write in the form \(x + iy\): \(\left(e^{\frac{\pi i}{4}}\right)^{6}\)
2. (b) Simplify \(\mathrm{cis}\,0.45 \times \mathrm{cis}\,2.1\)

The diagram shows the point that represents \(z = 3 + i\) 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 \(x + iy\).

\(z = -5 - 5 i\).

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 \(x + iy\).

\(z = 5 \sqrt{3} - 5 i\) and \(w = - 4 \sqrt{3} - 4 i\). 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) \((2\,\mathrm{cis}\left(\frac{\pi}{3}\right))(3\,\mathrm{cis}\left(\frac{\pi}{3}\right))\)
2. (b) \(3\,\mathrm{cis}\left(\frac{\pi}{6}\right) + \mathrm{cis}\left(\pi\right)\)

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) \(\left(2\,\mathrm{cis}\left(\frac{3\pi}{7}\right)\right)^{3}\left(5\,\mathrm{cis}\left(\frac{\pi}{14}\right)\right)^{2}\)
2. (b) \(\dfrac{\left(\mathrm{cis}\left(\frac{5\pi}{12}\right)\right)^{4}}{\left(3\,\mathrm{cis}\left(\frac{\pi}{10}\right)\right)^{2}}\)

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 \(z^{5}\) in Cartesian form.
3. (c) \(w = 1 - i\) and \(u = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\). Find \(w^{6}u^{5}\) 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 \(z = 3\,\mathrm{cis}\left(\frac{5\pi}{16}\right)\).
2. (b) Find the smallest positive integer \(n\) for which \(w^n = i\), where \(w = \mathrm{cis}\left(\frac{3\pi}{10}\right)\).

A floating platform rises and falls with two swells. At time \(t\) minutes the height of the platform, in metres, is \(H(t) = 3.5\sin(2t + 1) + 2\sin(2t - 0.8)\). Write the total as a single sinusoid \(A\sin(2t + B)\), 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 \(V_1 = 8\cos(30t)\) and \(V_2 = 5\cos(30t - 2)\).

1. (a) On the axes provided, sketch the graphs of \(V_1\) and \(V_2\) for \(0 \le t \le 0.4\). 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 \(A\cos(30t + c)\), where \(A > 0\) and \(-\pi < c \le \pi\).

The numbers of trout \(T\) and bass \(B\) in a lake vary through the year. At time \(t\) days after 1 January, \(T = 120 + 25\sin\left(\frac{2\pi}{365}t\right)\) and \(B = 90 + 18\sin\left(\frac{2\pi}{365}t + 2.3\right)\). Write the total number of fish, \(N = T + B\), in the form \(C + A\sin(bt + c)\).

The complex numbers \(z\) and \(w\) have arguments between \(0\) and \(\pi\). Given that \(zw = 18 i\) and \(\dfrac{z}{w} = \sqrt{3} + i\), 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 \(\cos\left(\frac{11\pi}{12}\right)\).

Let \(z = 9 + 5e^{2.5i}\).

1. (a) Find the modulus and argument of \(9 + 5e^{2.5i}\).
2. (b) Hence write \(9\sin\theta + 5\sin(\theta + 2.5)\) in the form \(A\sin(\theta + \alpha)\), where \(A > 0\) and \(-\pi < \alpha \le \pi\).