Products, quotients and powers in cis form Problem set B Worked solutions

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 \(zw\).

   • \(|zw| = 3 \times 8 = 24\)
   • \(\arg(zw) = \frac{\pi}{5} + \frac{2\pi}{5} = \frac{3\pi}{5}\)
   • \(zw = 24\,\text{cis}\left(\frac{3\pi}{5}\right)\)

2. (b) \(z = 6\,\text{cis}\left(\frac{5\pi}{6}\right)\) and \(w = 2\,\text{cis}\left(\pi\right)\). Find \(zw\).

   • \(|zw| = 6 \times 2 = 12\)
   • \(\arg(zw) = \frac{5\pi}{6} + \pi = \frac{11\pi}{6}\)
   • \(zw = 12\,\text{cis}\left(\frac{11\pi}{6}\right)\)

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 \(zw\).

   • \(|zw| = 6 \times 1.5 = 9\)
   • \(\arg(zw) = \frac{7\pi}{4} + \frac{5\pi}{6} = \frac{31\pi}{12}\)
   • \(\frac{31\pi}{12} - 2\pi = \frac{7\pi}{12}\)
   • \(zw = 9\,\text{cis}\left(\frac{7\pi}{12}\right)\)

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 \(\dfrac{z}{w}\).

   • \(\left|\frac{z}{w}\right| = \frac{20}{4} = 5\)
   • \(\arg\left(\frac{z}{w}\right) = \frac{2\pi}{3} - \frac{\pi}{6} = \frac{\pi}{2}\)
   • \(\frac{z}{w} = 5\,\text{cis}\left(\frac{\pi}{2}\right)\)

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

   • \(\left|\frac{z}{w}\right| = \frac{9}{6} = 1.5\)
   • \(\arg\left(\frac{z}{w}\right) = \frac{\pi}{6} - \frac{3\pi}{4} = -\frac{7\pi}{12}\)
   • \(-\frac{7\pi}{12} + 2\pi = \frac{17\pi}{12}\)
   • \(\frac{z}{w} = 1.5\,\text{cis}\left(\frac{17\pi}{12}\right)\)

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 \(z^{5}\).

   • \(|z^5| = 3^5 = 243\)
   • \(\arg(z^5) = 5 \times \frac{\pi}{10} = \frac{\pi}{2}\)
   • \(z^5 = 243\,\text{cis}\left(\frac{\pi}{2}\right)\)

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

   • \(|z^5| = 2^5 = 32\)
   • \(\arg(z^5) = 5 \times \frac{7\pi}{6} = \frac{35\pi}{6}\)
   • \(\frac{35\pi}{6} - 4\pi = \frac{11\pi}{6}\)
   • \(z^5 = 32\,\text{cis}\left(\frac{11\pi}{6}\right)\)

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.

   • \(|z^2| = 8^2 = 64,\quad \arg(z^2) = 2 \times \frac{\pi}{4} = \frac{\pi}{2}\)
   • \(|w^3| = 2^3 = 8,\quad \arg(w^3) = 3 \times \frac{\pi}{12} = \frac{\pi}{4}\)
   • \(\left|\frac{z^2}{w^3}\right| = \frac{64}{8} = 8,\quad \arg = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\)
   • \(\frac{z^2}{w^3} = 8\,\text{cis}\left(\frac{\pi}{4}\right) = 4 \sqrt{2}+4 \sqrt{2}i\)

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.

   • \(|z^3| = 2^3 = 8,\quad \arg(z^3) = 3 \times \frac{\pi}{6} = \frac{\pi}{2}\)
   • \(|w^2| = 2^2 = 4,\quad \arg(w^2) = 2 \times \frac{\pi}{12} = \frac{\pi}{6}\)
   • \(\left|\frac{z^3}{w^2}\right| = \frac{8}{4} = 2,\quad \arg = \frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}\)
   • \(\frac{z^3}{w^2} = 2\,\text{cis}\left(\frac{\pi}{3}\right) = 1+\sqrt{3}i\)

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 \(r\,\text{cis}\,\theta\).

   • \(|DS| = 0.5 \times 8 = 4\)
   • \(\arg(DS) = -\frac{\pi}{3} + \frac{\pi}{6} = -\frac{\pi}{6}\)
   • \(-\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}\)
   • \(DS = 4\,\text{cis}\left(\frac{11\pi}{6}\right)\)

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.

   • \(|D^3| = 0.5^3 = 0.125,\quad \arg(D^3) = 3 \times \left(-\frac{\pi}{3}\right) = -\pi\)
   • \(|SD^3| = 0.125 \times 8 = 1\)
   • \(\arg = \frac{\pi}{6} + \left(-\pi\right) = -\frac{5\pi}{6}\)
   • \(-\frac{5\pi}{6} + 2\pi = \frac{7\pi}{6}\)
   • \(SD^3 = \text{cis}\left(\frac{7\pi}{6}\right)\)

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 \(z^n\).

   • \(\arg(z^n) = \frac{n\pi}{6}\)
   • Real and negative: \(\arg(z^n) = \pi\), \(3\pi\), … so \(\frac{n\pi}{6} = \pi\) and \(n = 6\)
   • \(|z^6| = 3^6 = 729, \text{ so } z^6 = 729\,\text{cis}\,\pi = -729\)

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 \(r\,\text{cis}\,\theta\).

   • \(|zw|\times\left|\frac{z}{w}\right| = |z|^2 = 18 \times 2 = 36, \text{ so } |z| = 6 \text{ and } |w| = 18 \div 6 = 3\)
   • \(\arg z + \arg w = \frac{13\pi}{12}, \quad \arg z - \arg w = \frac{7\pi}{12}\)
   • \(\text{Add: } 2\arg z = \frac{20\pi}{12}, \text{ so } \arg z = \frac{5\pi}{6}. \text{ Then } \arg w = \frac{13\pi}{12} - \frac{10\pi}{12} = \frac{\pi}{4}\)