Exponential form Problem set B Worked solutions

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

1. (a) \(z = 3e^{i\frac{2\pi}{5}}\) and \(w = 8e^{i\frac{\pi}{10}}\). Find \(zw\).

   • \(zw = 3e^{i\frac{2\pi}{5}} \times 8e^{i\frac{\pi}{10}} = 24e^{i\left(\frac{2\pi}{5} + \frac{\pi}{10}\right)} = 24e^{i\frac{\pi}{2}}\)
   • \(zw = 24e^{i\frac{\pi}{2}}\)

2. (b) \(z = 6e^{i\frac{2\pi}{3}}\) and \(w = 2e^{i\frac{\pi}{6}}\). Find \(zw\).

   • \(zw = 6e^{i\frac{2\pi}{3}} \times 2e^{i\frac{\pi}{6}} = 12e^{i\left(\frac{2\pi}{3} + \frac{\pi}{6}\right)} = 12e^{i\frac{5\pi}{6}}\)
   • \(zw = 12e^{i\frac{5\pi}{6}}\)

3. (c) \(z = 5e^{i\frac{5\pi}{3}}\) and \(w = 0.4e^{i\frac{3\pi}{4}}\). Find \(zw\).

   • \(zw = 5e^{i\frac{5\pi}{3}} \times 0.4e^{i\frac{3\pi}{4}} = 2e^{i\left(\frac{5\pi}{3} + \frac{3\pi}{4}\right)} = 2e^{i\frac{29\pi}{12}}\)
   • \(\frac{29\pi}{12} - 2\pi = \frac{5\pi}{12}\)
   • \(zw = 2e^{i\frac{5\pi}{12}}\)

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

1. (a) \(z = 36e^{i\frac{2\pi}{3}}\) and \(w = 9e^{i\frac{\pi}{4}}\). Find \(\dfrac{z}{w}\).

   • \(\dfrac{z}{w} = \dfrac{36e^{i\frac{2\pi}{3}}}{9e^{i\frac{\pi}{4}}} = 4e^{i\left(\frac{2\pi}{3} - \frac{\pi}{4}\right)} = 4e^{i\frac{5\pi}{12}}\)
   • \(\dfrac{z}{w} = 4e^{i\frac{5\pi}{12}}\)

2. (b) \(z = 14e^{i\frac{\pi}{3}}\) and \(w = 8e^{i\frac{5\pi}{6}}\). Find \(\dfrac{z}{w}\).

   • \(\dfrac{z}{w} = \dfrac{14e^{i\frac{\pi}{3}}}{8e^{i\frac{5\pi}{6}}} = 1.75e^{i\left(\frac{\pi}{3} - \frac{5\pi}{6}\right)} = 1.75e^{i\left(-\frac{\pi}{2}\right)}\)
   • \(-\frac{\pi}{2} + 2\pi = \frac{3\pi}{2}\)
   • \(\dfrac{z}{w} = 1.75e^{i\frac{3\pi}{2}}\)

3. (c) \(z = 21e^{i\frac{\pi}{2}}\) and \(w = 3e^{i\frac{3\pi}{4}}\). Find \(\dfrac{z}{w}\).

   • \(\dfrac{z}{w} = \dfrac{21e^{i\frac{\pi}{2}}}{3e^{i\frac{3\pi}{4}}} = 7e^{i\left(\frac{\pi}{2} - \frac{3\pi}{4}\right)} = 7e^{i\left(-\frac{\pi}{4}\right)}\)
   • \(-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}\)
   • \(\dfrac{z}{w} = 7e^{i\frac{7\pi}{4}}\)

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

1. (a) Write \(z\) and \(w\) in exponential form.

   • \(|z| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}. \text{ The point is in the second quadrant, so } \arg z = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\)
   • \(|w| = \sqrt{1 + 3} = 2,\quad \arg w = \arctan\sqrt{3} = \frac{\pi}{3}\)
   • \(z = \sqrt{2}e^{i\frac{3\pi}{4}},\quad w = 2e^{i\frac{\pi}{3}}\)

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

   • \(zw = \sqrt{2} \times 2\,e^{i\left(\frac{3\pi}{4} + \frac{\pi}{3}\right)} = 2\sqrt{2}e^{i\frac{13\pi}{12}}\)
   • \(\dfrac{z}{w} = \dfrac{\sqrt{2}}{2}e^{i\left(\frac{3\pi}{4} - \frac{\pi}{3}\right)} = \dfrac{\sqrt{2}}{2}e^{i\frac{5\pi}{12}}\)

\(z = 5e^{i\frac{7\pi}{6}}\). Give each argument in the range \(0 \le \theta < 2\pi\).

1. (a) Write \(z^*\) in exponential form.

   • \(z^* = 5e^{-i\frac{7\pi}{6}}\): same modulus, opposite argument.
   • \(-\frac{7\pi}{6} + 2\pi = \frac{5\pi}{6}\), so \(z^* = 5e^{i\frac{5\pi}{6}}\).

2. (b) Find \(zz^*\).

   • \(zz^* = 5e^{i\frac{7\pi}{6}} \times 5e^{i\frac{5\pi}{6}} = 25e^{2\pi i} = 25\).
   • This is \(|z|^2 = 5^2\).

3. (c) Write \(z^*\) in the form \(x + iy\), exact.

   • \(z^* = 5\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right)\)
   • \(= 5\left(-\frac{\sqrt{3}}{2}\right) + 5\left(\frac{1}{2}\right)i = -\frac{5\sqrt{3}}{2} + \frac{5}{2}i\)

A set of points is given by \(z_\theta = \frac{1}{4}\theta e^{i\theta}\), 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.

   • \(z_{\pi/2} = \frac{1}{4}\cdot\frac{\pi}{2}e^{i\frac{\pi}{2}} = 0.393i\)
   • \(z_{\pi} = \frac{1}{4}\pi e^{i\pi} = -0.785\)
   • \(z_{3\pi/2} = \frac{1}{4}\cdot\frac{3\pi}{2}e^{i\frac{3\pi}{2}} = -1.18i\)

2. (b) Find the value of \(\theta\) for which \(|z_\theta| = 3\), and write that point in the form \(x + iy\), to 3 significant figures.

   • \(|z_\theta| = \frac{1}{4}\theta = 3\), so \(\theta = 12\).
   • \(z = 3e^{12i} = 3\cos 12 + 3\sin 12\,i = 2.53 - 1.61i\) (radian mode).

A radar pulse \(T\) is sent towards a target. The echo is \(R = FT\), where the complex number \(F\) describes the target: its modulus is the amplitude factor and its argument is the phase shift in radians. The echo is \(R = 72e^{i\frac{5\pi}{6}}\). Give each phase in the range \(0 \le \theta < 2\pi\).

1. (a) The target has \(F = 9e^{i\frac{\pi}{4}}\). Write the pulse \(T = \dfrac{R}{F}\) in exponential form.

   • \(T = \dfrac{72e^{i\frac{5\pi}{6}}}{9e^{i\frac{\pi}{4}}} = 8e^{i\left(\frac{5\pi}{6} - \frac{\pi}{4}\right)} = 8e^{i\frac{7\pi}{12}}\)
   • \(T = 8e^{i\frac{7\pi}{12}}\)

2. (b) The same echo comes from a different target with \(F' = 12e^{i\frac{11\pi}{12}}\). Write the pulse \(T' = \dfrac{R}{F'}\) in exponential form.

   • \(T' = \dfrac{72e^{i\frac{5\pi}{6}}}{12e^{i\frac{11\pi}{12}}} = 6e^{i\left(\frac{5\pi}{6} - \frac{11\pi}{12}\right)} = 6e^{i\left(-\frac{\pi}{12}\right)}\)
   • \(-\frac{\pi}{12} + 2\pi = \frac{23\pi}{12}\)
   • \(T' = 6e^{i\frac{23\pi}{12}}\)

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

1. (a) \(z = 8e^{i\frac{\pi}{3}}\). Write \(\dfrac{1}{z}\) in exponential form.

   • \(\dfrac{1}{z} = \dfrac{1e^{i0}}{8e^{i\frac{\pi}{3}}} = 0.125e^{i\left(0 - \frac{\pi}{3}\right)} = 0.125e^{-i\frac{\pi}{3}}\)
   • \(-\frac{\pi}{3} + 2\pi = \frac{5\pi}{3}\)
   • \(\dfrac{1}{z} = 0.125e^{i\frac{5\pi}{3}}\)

2. (b) \(z = 7e^{i\frac{4\pi}{5}}\). Write \(\dfrac{z}{z^*}\) in exponential form.

   • \(z^* = 7e^{-i\frac{4\pi}{5}}\)
   • \(\dfrac{z}{z^*} = \dfrac{7}{7}e^{i\left(\frac{4\pi}{5} - \left(-\frac{4\pi}{5}\right)\right)} = e^{i\frac{8\pi}{5}}\)