Exponential form Problem set A

Give each argument in the range \(0 \le \theta < 2\pi\).

1. (a) \(z = 5e^{i\frac{\pi}{9}}\) and \(w = 3e^{i\frac{2\pi}{9}}\). Find \(zw\).
2. (b) \(z = 2e^{i\frac{3\pi}{4}}\) and \(w = 7e^{i\frac{5\pi}{6}}\). Find \(zw\).
3. (c) \(z = 4e^{i\frac{7\pi}{6}}\) and \(w = 1.5e^{i\frac{5\pi}{4}}\). Find \(zw\).

Give each argument in the range \(0 \le \theta < 2\pi\).

1. (a) \(z = 24e^{i\frac{5\pi}{6}}\) and \(w = 6e^{i\frac{\pi}{3}}\). Find \(\dfrac{z}{w}\).
2. (b) \(z = 10e^{i\frac{\pi}{5}}\) and \(w = 4e^{i\frac{7\pi}{10}}\). Find \(\dfrac{z}{w}\).
3. (c) \(z = 15e^{i\frac{3\pi}{2}}\) and \(w = 5e^{i\frac{\pi}{6}}\). Find \(\dfrac{z}{w}\).

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

1. (a) Write \(z\) and \(w\) in exponential form.
2. (b) Hence write \(zw\) and \(\dfrac{z}{w}\) in exponential form.

\(z = 3e^{i\frac{5\pi}{4}}\). Give each argument in the range \(0 \le \theta < 2\pi\).

1. (a) Write \(z^*\) in exponential form.
2. (b) Find \(zz^*\).
3. (c) Write \(z^*\) in the form \(x + iy\), exact.

A set of points is given by \(z_\theta = \frac{1}{2}\theta e^{i\theta}\), where \(\theta \ge 0\).

1. (a) Find \(z_\theta\) for \(\theta = \frac{\pi}{2},\ \pi,\ \frac{3\pi}{2}\) in the form \(x + iy\), to 3 significant figures, and plot the three points.
2. (b) Find the value of \(\theta\) for which \(|z_\theta| = 4\), and write that point in the form \(x + iy\), to 3 significant figures.

A voltage \(V = 24e^{i\frac{\pi}{4}}\) volts is applied to a circuit component. The component has impedance \(Z\) ohms and the current is \(I = \dfrac{V}{Z}\) amperes. Moduli are the amplitudes and arguments are the phases in radians. Give each phase in the range \(0 \le \theta < 2\pi\).

1. (a) The impedance is \(Z = 3e^{i\frac{\pi}{6}}\). Write \(I\) in exponential form.
2. (b) The same voltage is applied to a second component with impedance \(Z' = 4e^{i\frac{3\pi}{4}}\). Write the current \(I'\) in exponential form.

Answer each part. Give each argument in the range \(0 \le \theta < 2\pi\).

1. (a) \(z = 6e^{i\frac{\pi}{4}}\). Write \(\dfrac{1}{z}\) in exponential form.
2. (b) \(z = 5e^{i\frac{2\pi}{3}}\). Write \(\dfrac{z}{z^*}\) in exponential form.