Exponential form Problem set B

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 \(zw\).
2. (b) \(z = 6e^{i\frac{2\pi}{3}}\) and \(w = 2e^{i\frac{\pi}{6}}\). Find \(zw\).
3. (c) \(z = 5e^{i\frac{5\pi}{3}}\) and \(w = 0.4e^{i\frac{3\pi}{4}}\). Find \(zw\).

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 \(\dfrac{z}{w}\).
2. (b) \(z = 14e^{i\frac{\pi}{3}}\) and \(w = 8e^{i\frac{5\pi}{6}}\). Find \(\dfrac{z}{w}\).
3. (c) \(z = 21e^{i\frac{\pi}{2}}\) and \(w = 3e^{i\frac{3\pi}{4}}\). Find \(\dfrac{z}{w}\).

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

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

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

1. (a) Write \(z^*\) in exponential form.
2. (b) Find \(zz^*\).
3. (c) Write \(z^*\) in the form \(x + iy\), exact.

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.
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.

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.
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.

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