Exponential form Problem set A One question at a time

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 \(zw\).

   • \(zw = 5e^{i\frac{\pi}{9}} \times 3e^{i\frac{2\pi}{9}} = 15e^{i\left(\frac{\pi}{9} + \frac{2\pi}{9}\right)} = 15e^{i\frac{\pi}{3}}\)
   • \(zw = 15e^{i\frac{\pi}{3}}\)

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

   • \(zw = 2e^{i\frac{3\pi}{4}} \times 7e^{i\frac{5\pi}{6}} = 14e^{i\left(\frac{3\pi}{4} + \frac{5\pi}{6}\right)} = 14e^{i\frac{19\pi}{12}}\)
   • \(zw = 14e^{i\frac{19\pi}{12}}\)

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

   • \(zw = 4e^{i\frac{7\pi}{6}} \times 1.5e^{i\frac{5\pi}{4}} = 6e^{i\left(\frac{7\pi}{6} + \frac{5\pi}{4}\right)} = 6e^{i\frac{29\pi}{12}}\)
   • \(\frac{29\pi}{12} - 2\pi = \frac{5\pi}{12}\)
   • \(zw = 6e^{i\frac{5\pi}{12}}\)

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

   • \(\dfrac{z}{w} = \dfrac{24e^{i\frac{5\pi}{6}}}{6e^{i\frac{\pi}{3}}} = 4e^{i\left(\frac{5\pi}{6} - \frac{\pi}{3}\right)} = 4e^{i\frac{\pi}{2}}\)
   • \(\dfrac{z}{w} = 4e^{i\frac{\pi}{2}}\)

2. (b) \(z = 10e^{i\frac{\pi}{5}}\) and \(w = 4e^{i\frac{7\pi}{10}}\). Find \(\dfrac{z}{w}\).

   • \(\dfrac{z}{w} = \dfrac{10e^{i\frac{\pi}{5}}}{4e^{i\frac{7\pi}{10}}} = 2.5e^{i\left(\frac{\pi}{5} - \frac{7\pi}{10}\right)} = 2.5e^{i\left(-\frac{\pi}{2}\right)}\)
   • \(-\frac{\pi}{2} + 2\pi = \frac{3\pi}{2}\)
   • \(\dfrac{z}{w} = 2.5e^{i\frac{3\pi}{2}}\)

3. (c) \(z = 15e^{i\frac{3\pi}{2}}\) and \(w = 5e^{i\frac{\pi}{6}}\). Find \(\dfrac{z}{w}\).

   • \(\dfrac{z}{w} = \dfrac{15e^{i\frac{3\pi}{2}}}{5e^{i\frac{\pi}{6}}} = 3e^{i\left(\frac{3\pi}{2} - \frac{\pi}{6}\right)} = 3e^{i\frac{4\pi}{3}}\)
   • \(\dfrac{z}{w} = 3e^{i\frac{4\pi}{3}}\)

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

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

   • \(|z| = \sqrt{1^2 + 1^2} = \sqrt{2},\quad \arg z = \arctan 1 = \frac{\pi}{4}\)
   • \(|w| = \sqrt{3 + 1} = 2,\quad \arg w = \arctan\frac{1}{\sqrt{3}} = \frac{\pi}{6}\)
   • \(z = \sqrt{2}e^{i\frac{\pi}{4}},\quad w = 2e^{i\frac{\pi}{6}}\)

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

   • \(zw = \sqrt{2} \times 2\,e^{i\left(\frac{\pi}{4} + \frac{\pi}{6}\right)} = 2\sqrt{2}e^{i\frac{5\pi}{12}}\)
   • \(\dfrac{z}{w} = \dfrac{\sqrt{2}}{2}e^{i\left(\frac{\pi}{4} - \frac{\pi}{6}\right)} = \dfrac{\sqrt{2}}{2}e^{i\frac{\pi}{12}}\)

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

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

   • \(z^* = 3e^{-i\frac{5\pi}{4}}\): same modulus, opposite argument.
   • \(-\frac{5\pi}{4} + 2\pi = \frac{3\pi}{4}\), so \(z^* = 3e^{i\frac{3\pi}{4}}\).

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

   • \(zz^* = 3e^{i\frac{5\pi}{4}} \times 3e^{i\frac{3\pi}{4}} = 9e^{2\pi i} = 9\).
   • This is \(|z|^2 = 3^2\).

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

   • \(z^* = 3\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)\)
   • \(= 3\left(-\frac{\sqrt{2}}{2}\right) + 3\left(\frac{\sqrt{2}}{2}\right)i = -\frac{3\sqrt{2}}{2} + \frac{3\sqrt{2}}{2}i\)

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.

   • \(z_{\pi/2} = \frac{1}{2}\cdot\frac{\pi}{2}e^{i\frac{\pi}{2}} = 0.785i\)
   • \(z_{\pi} = \frac{1}{2}\pi e^{i\pi} = -1.57\)
   • \(z_{3\pi/2} = \frac{1}{2}\cdot\frac{3\pi}{2}e^{i\frac{3\pi}{2}} = -2.36i\)

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.

   • \(|z_\theta| = \frac{1}{2}\theta = 4\), so \(\theta = 8\).
   • \(z = 4e^{8i} = 4\cos 8 + 4\sin 8\,i = -0.582 + 3.96i\) (radian mode).

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.

   • \(I = \dfrac{24e^{i\frac{\pi}{4}}}{3e^{i\frac{\pi}{6}}} = 8e^{i\left(\frac{\pi}{4} - \frac{\pi}{6}\right)} = 8e^{i\frac{\pi}{12}}\)
   • \(I = 8e^{i\frac{\pi}{12}}\)

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.

   • \(I' = \dfrac{24e^{i\frac{\pi}{4}}}{4e^{i\frac{3\pi}{4}}} = 6e^{i\left(\frac{\pi}{4} - \frac{3\pi}{4}\right)} = 6e^{i\left(-\frac{\pi}{2}\right)}\)
   • \(-\frac{\pi}{2} + 2\pi = \frac{3\pi}{2}\)
   • \(I' = 6e^{i\frac{3\pi}{2}}\)

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.

   • \(\dfrac{1}{z} = \dfrac{1e^{i0}}{6e^{i\frac{\pi}{4}}} = \frac{1}{6}e^{i\left(0 - \frac{\pi}{4}\right)} = \frac{1}{6}e^{-i\frac{\pi}{4}}\)
   • \(-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}\)
   • \(\dfrac{1}{z} = \frac{1}{6}e^{i\frac{7\pi}{4}}\)

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

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