Modulus-argument form Problem set A

Write each number in modulus-argument form \(r\,\text{cis}\,\theta\), with \(0 \le \theta < 2\pi\). Sketch each point first and give exact values.

1. (a) \(1 + \sqrt{3}i\)
2. (b) \(-4 + 4i\)
3. (c) \(- \sqrt{3} - i\)
4. (d) \(3 - 3i\)

Write down \(r\) and \(\theta\) for each number, to 3 significant figures, with \(-\pi < \theta \le \pi\). Use the calculator in radian mode.

1. (a) \(5 - 12i\)
2. (b) \(-7 + 2i\)

Write each complex number in the form \(x + iy\). Give exact values for the first two and 3 significant figures for the third.

1. (a) \(8\,\text{cis}\,\dfrac{\pi}{3}\)
2. (b) \(6\,\text{cis}\,\dfrac{5\pi}{6}\)
3. (c) \(7\,\text{cis}\,2.30\)

Write the conjugate of each number in the form \(r\,\text{cis}\,\theta\) with \(-\pi < \theta \le \pi\). Sketch \(z\) and \(z^*\) for the first.

1. (a) \(6\,\text{cis}\left(\dfrac{2\pi}{9}\right)\)
2. (b) \(4\,\text{cis}\left(\dfrac{5\pi}{6}\right)\)
3. (c) \(3\,\text{cis}\left(\dfrac{4\pi}{3}\right)\)

Take the port as the origin of an Argand diagram, with east as the real axis and north as the imaginary axis. Distances are in km. A ship is 8 km from the port on a bearing of 060 degrees.

1. (a) Find the argument of the ship's position, in radians and measured anticlockwise from east.
2. (b) Write the ship's position as a complex number in the form \(x + iy\), to 3 significant figures.
3. (c) A lighthouse is at \(5 - 12i\). Find its distance from the port and its argument, to 3 significant figures, with \(-\pi < \theta \le \pi\).

A student writes \(z = -2\,\text{cis}\,\dfrac{\pi}{6}\). The modulus of a complex number cannot be negative. Write \(z\) in the form \(r\,\text{cis}\,\theta\) with \(r > 0\) and \(0 \le \theta < 2\pi\).

Write \(z = 3\left(\cos\dfrac{\pi}{5} - i\sin\dfrac{\pi}{5}\right)\) in the form \(r\,\text{cis}\,\theta\).

The complex number \(z\) has modulus 6 and argument \(\dfrac{2\pi}{3}\).

1. (a) Write \(z\) in the form \(x + iy\).
2. (b) Write \(z^*\) in the form \(x + iy\) and in the form \(r\,\text{cis}\,\theta\) with \(-\pi < \theta \le \pi\).
3. (c) Find the distance between the points that represent \(z\) and \(z^*\).