Products, quotients and powers in cis form Problem set B

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

1. (a) \(z = 3\,\text{cis}\left(\frac{\pi}{5}\right)\) and \(w = 8\,\text{cis}\left(\frac{2\pi}{5}\right)\). Find \(zw\).
2. (b) \(z = 6\,\text{cis}\left(\frac{5\pi}{6}\right)\) and \(w = 2\,\text{cis}\left(\pi\right)\). Find \(zw\).
3. (c) \(z = 6\,\text{cis}\left(\frac{7\pi}{4}\right)\) and \(w = 1.5\,\text{cis}\left(\frac{5\pi}{6}\right)\). Find \(zw\).

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

1. (a) \(z = 20\,\text{cis}\left(\frac{2\pi}{3}\right)\) and \(w = 4\,\text{cis}\left(\frac{\pi}{6}\right)\). Find \(\dfrac{z}{w}\).
2. (b) \(z = 9\,\text{cis}\left(\frac{\pi}{6}\right)\) and \(w = 6\,\text{cis}\left(\frac{3\pi}{4}\right)\). Find \(\dfrac{z}{w}\).

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

1. (a) \(z = 3\,\text{cis}\left(\frac{\pi}{10}\right)\). Find \(z^{5}\).
2. (b) \(z = 2\,\text{cis}\left(\frac{7\pi}{6}\right)\). Find \(z^{5}\).

Work by hand. Give exact values.

1. (a) \(z = 8\,\text{cis}\left(\frac{\pi}{4}\right)\) and \(w = 2\,\text{cis}\left(\frac{\pi}{12}\right)\). Write \(\dfrac{z^2}{w^3}\) in the form \(x + iy\), giving exact values.
2. (b) \(z = 2\,\text{cis}\left(\frac{\pi}{6}\right)\) and \(w = 2\,\text{cis}\left(\frac{\pi}{12}\right)\). Write \(\dfrac{z^3}{w^2}\) in the form \(x + iy\), giving exact values.

An audio signal is stored as the complex number \(S = 8\,\text{cis}\left(\frac{\pi}{6}\right)\), where the modulus is the loudness and the argument is the phase in radians. A delay unit multiplies the signal by \(D = 0.5\,\text{cis}\left(-\frac{\pi}{3}\right)\). Give each phase in the range \(0 \le \theta < 2\pi\).

1. (a) Write the signal that leaves one delay unit, \(DS\), in the form \(r\,\text{cis}\,\theta\).
2. (b) The signal passes through three identical delay units in a row. Write the output \(SD^3\) in the form \(r\,\text{cis}\,\theta\) and state its loudness and phase.

Answer each part.

1. (a) \(z = 3\,\text{cis}\left(\frac{\pi}{6}\right)\). Find the smallest positive integer \(n\) for which \(z^n\) is real and negative, and write down \(z^n\).
2. (b) \(zw = 18\,\text{cis}\left(\frac{13\pi}{12}\right)\) and \(\dfrac{z}{w} = 2\,\text{cis}\left(\frac{7\pi}{12}\right)\). Find \(z\) and \(w\) in the form \(r\,\text{cis}\,\theta\).