Modulus-argument form Problem set B

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

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) \(15 - 8i\)
2. (b) \(-5 + 3i\)

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) \(10\,\text{cis}\,\dfrac{\pi}{6}\)
2. (b) \(4\,\text{cis}\,\dfrac{3\pi}{4}\)
3. (c) \(9\,\text{cis}(-1.10)\)

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) \(9\,\text{cis}\left(\dfrac{\pi}{8}\right)\)
2. (b) \(2\,\text{cis}\left(-\dfrac{3\pi}{4}\right)\)
3. (c) \(6\,\text{cis}\left(\dfrac{5\pi}{4}\right)\)

Take the pilot as the origin of an Argand diagram, with east as the real axis and north as the imaginary axis. Distances are in metres. A drone is 50 m from the pilot on a bearing of 135 degrees.

1. (a) Find the argument of the drone's position, in radians and measured anticlockwise from east, with \(-\pi < \theta \le \pi\).
2. (b) Write the drone's position as a complex number in the form \(x + iy\), to 3 significant figures.
3. (c) A tree is at \(-8 + 6i\). Find its distance from the pilot and its argument, to 3 significant figures, with \(-\pi < \theta \le \pi\).

A student writes \(z = -5\,\text{cis}\,\dfrac{\pi}{4}\). 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 \(z = 4\left(\cos\dfrac{2\pi}{7} - i\sin\dfrac{2\pi}{7}\right)\) in the form \(r\,\text{cis}\,\theta\).

The complex number \(z\) has modulus 10 and argument \(\dfrac{5\pi}{6}\).

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 \(z^*\).