Products, quotients and powers in cis form Problem set A

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

1. (a) \(z = 2\,\text{cis}\left(\frac{\pi}{7}\right)\) and \(w = 6\,\text{cis}\left(\frac{2\pi}{7}\right)\). Find \(zw\).
2. (b) \(z = 5\,\text{cis}\left(\frac{3\pi}{4}\right)\) and \(w = 3\,\text{cis}\left(\frac{\pi}{2}\right)\). Find \(zw\).
3. (c) \(z = 4\,\text{cis}\left(\frac{5\pi}{3}\right)\) and \(w = 2.5\,\text{cis}\left(\frac{2\pi}{3}\right)\). Find \(zw\).

Give each answer in the form \(r\,\text{cis}\,\theta\) with \(0 \le \theta < 2\pi\).

1. (a) \(z = 12\,\text{cis}\left(\frac{5\pi}{6}\right)\) and \(w = 3\,\text{cis}\left(\frac{\pi}{3}\right)\). Find \(\dfrac{z}{w}\).
2. (b) \(z = 7\,\text{cis}\left(\frac{\pi}{4}\right)\) and \(w = 2\,\text{cis}\left(\frac{2\pi}{3}\right)\). Find \(\dfrac{z}{w}\).

Give each answer in the form \(r\,\text{cis}\,\theta\) with \(0 \le \theta < 2\pi\).

1. (a) \(z = 2\,\text{cis}\left(\frac{\pi}{9}\right)\). Find \(z^{3}\).
2. (b) \(z = 3\,\text{cis}\left(\frac{5\pi}{6}\right)\). Find \(z^{4}\).

Work by hand. Give exact values.

1. (a) \(z = 9\,\text{cis}\left(\frac{\pi}{3}\right)\) and \(w = 3\,\text{cis}\left(\frac{\pi}{6}\right)\). Write \(\dfrac{z^2}{w^3}\) in the form \(x + iy\), giving exact values.
2. (b) \(z = 2\,\text{cis}\left(\frac{\pi}{4}\right)\) and \(w = 2\,\text{cis}\left(\frac{\pi}{8}\right)\). Write \(\dfrac{z^3}{w^2}\) in the form \(x + iy\), giving exact values.

An alternating voltage is stored as the complex number \(V = 12\,\text{cis}\left(\frac{\pi}{6}\right)\), in volts, with the phase in radians. A filter multiplies the signal by \(F = 0.5\,\text{cis}\left(-\frac{\pi}{4}\right)\). Give each phase in the range \(0 \le \theta < 2\pi\).

1. (a) Write the signal that leaves one filter, \(FV\), in the form \(r\,\text{cis}\,\theta\).
2. (b) The signal passes through two identical filters in a row. Write the output \(VF^2\) in the form \(r\,\text{cis}\,\theta\) and state its amplitude and phase.

Answer each part.

1. (a) \(z = 2\,\text{cis}\left(\frac{\pi}{8}\right)\). Find the smallest positive integer \(n\) for which \(z^n\) is real and negative, and write down \(z^n\).
2. (b) \(zw = 20\,\text{cis}\left(\frac{5\pi}{6}\right)\) and \(\dfrac{z}{w} = 5\,\text{cis}\left(\frac{\pi}{6}\right)\). Find \(z\) and \(w\) in the form \(r\,\text{cis}\,\theta\).