Modulus-argument form Problem set A Worked solutions

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) \(1 + \sqrt{3}i\)

   • \(r = \sqrt{(1)^2 + (\sqrt{3})^2} = 2\)
   • \(\tan\alpha = \dfrac{\sqrt{3}}{1}\), so \(\alpha = \dfrac{\pi}{3}\). The point is in the first quadrant.
   • \(\theta = \alpha = \dfrac{\pi}{3}\)

2. (b) \(-4 + 4i\)

   • \(r = \sqrt{(-4)^2 + (4)^2} = 4 \sqrt{2}\)
   • \(\tan\alpha = \dfrac{4}{4}\), so \(\alpha = \dfrac{\pi}{4}\). The point is in the second quadrant.
   • \(\theta = \pi - \alpha = \dfrac{3\pi}{4}\)

3. (c) \(- \sqrt{3} - i\)

   • \(r = \sqrt{(- \sqrt{3})^2 + (-1)^2} = 2\)
   • \(\tan\alpha = \dfrac{1}{\sqrt{3}}\), so \(\alpha = \dfrac{\pi}{6}\). The point is in the third quadrant.
   • \(\theta = \pi + \alpha = \dfrac{7\pi}{6}\)

4. (d) \(3 - 3i\)

   • \(r = \sqrt{(3)^2 + (-3)^2} = 3 \sqrt{2}\)
   • \(\tan\alpha = \dfrac{3}{3}\), so \(\alpha = \dfrac{\pi}{4}\). The point is in the fourth quadrant.
   • \(\theta = 2\pi - \alpha = \dfrac{7\pi}{4}\)

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) \(5 - 12i\)

   • \(r = \sqrt{(5)^2 + (-12)^2} = 13\)
   • \(\alpha = \tan^{-1}\left(\dfrac{12}{5}\right) = 1.18\). The point is in the fourth quadrant.
   • \(\theta = -\alpha = -1.18\)

2. (b) \(-7 + 2i\)

   • \(r = \sqrt{(-7)^2 + (2)^2} = 7.28\)
   • \(\alpha = \tan^{-1}\left(\dfrac{2}{7}\right) = 0.278\). The point is in the second quadrant.
   • \(\theta = \pi - \alpha = 2.86\)

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) \(8\,\text{cis}\,\dfrac{\pi}{3}\)

   • \(x = 8\cos \dfrac{\pi}{3} = 4\)
   • \(y = 8\sin \dfrac{\pi}{3} = 4 \sqrt{3}\)
   • \(z = 4 + 4 \sqrt{3}i\)

2. (b) \(6\,\text{cis}\,\dfrac{5\pi}{6}\)

   • \(x = 6\cos \dfrac{5\pi}{6} = - 3 \sqrt{3}\)
   • \(y = 6\sin \dfrac{5\pi}{6} = 3\)
   • \(z = - 3 \sqrt{3} + 3i\)

3. (c) \(7\,\text{cis}\,2.30\)

   • From the calculator (radian mode): \(x = 7\cos(2.30) = -4.66\)
   • \(y = 7\sin(2.30) = 5.22\)
   • \(z = -4.66 + 5.22i\)

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) \(6\,\text{cis}\left(\dfrac{2\pi}{9}\right)\)

   • \(z^* = 6\,\text{cis}\left(-\dfrac{2\pi}{9}\right)\)
   • The argument is already in \(-\pi < \theta \le \pi\).

2. (b) \(4\,\text{cis}\left(\dfrac{5\pi}{6}\right)\)

   • \(z^* = 4\,\text{cis}\left(-\dfrac{5\pi}{6}\right)\)
   • The argument is already in \(-\pi < \theta \le \pi\).

3. (c) \(3\,\text{cis}\left(\dfrac{4\pi}{3}\right)\)

   • \(z^* = 3\,\text{cis}\left(-\dfrac{4\pi}{3}\right)\)
   • \(-\dfrac{4\pi}{3} + 2\pi = \dfrac{2\pi}{3}\), so \(z^* = 3\,\text{cis}\left(\dfrac{2\pi}{3}\right)\) with \(-\pi < \theta \le \pi\).

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.

   • A bearing is measured clockwise from north. The argument is measured anticlockwise from east.
   • \(90^\circ - 60^\circ = 30^\circ = \dfrac{\pi}{6}\) radians

2. (b) Write the ship's position as a complex number in the form \(x + iy\), to 3 significant figures.

   • \(x = 8\cos\dfrac{\pi}{6} = 6.93\)
   • \(y = 8\sin\dfrac{\pi}{6} = 4\)
   • The ship is at \(6.93 + 4i\).

3. (c) A lighthouse is at \(5 - 12i\). Find its distance from the port and its argument, to 3 significant figures, with \(-\pi < \theta \le \pi\).

   • \(r = \sqrt{5^2 + (-12)^2} = 13\)
   • The point is in the fourth quadrant: \(\theta = -\tan^{-1}\left(\dfrac{12}{5}\right) = -1.18\)

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 \(0 \le \theta < 2\pi\).

• \(-1\) times a number reflects it through the origin: the modulus stays 2 and the angle turns through \(\pi\).
• \(\dfrac{\pi}{6} + \pi = \dfrac{7\pi}{6}\)
• \(z = 2\,\text{cis}\,\dfrac{7\pi}{6}\)

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

• \(\cos(-\theta) = \cos\theta\) and \(\sin(-\theta) = -\sin\theta\).
• So \(\cos\dfrac{\pi}{5} - i\sin\dfrac{\pi}{5} = \cos\left(-\dfrac{\pi}{5}\right) + i\sin\left(-\dfrac{\pi}{5}\right)\).
• \(z = 3\,\text{cis}\left(-\dfrac{\pi}{5}\right)\). This is the conjugate of \(3\,\text{cis}\,\dfrac{\pi}{5}\).

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

1. (a) Write \(z\) in the form \(x + iy\).

   • \(x = 6\cos\dfrac{2\pi}{3} = -3\), \(y = 6\sin\dfrac{2\pi}{3} = 3\sqrt{3}\)
   • \(z = -3 + 3\sqrt{3}\,i\)

2. (b) Write \(z^*\) in the form \(x + iy\) and in the form \(r\,\text{cis}\,\theta\) with \(-\pi < \theta \le \pi\).

   • Change the sign of the imaginary part: \(z^* = -3 - 3\sqrt{3}\,i\).
   • In polar form the argument changes sign: \(z^* = 6\,\text{cis}\left(-\dfrac{2\pi}{3}\right)\).

3. (c) Find the distance between the points that represent \(z\) and \(z^*\).

   • The points are the same distance above and below the real axis.
   • Distance \(= 2 \times 3\sqrt{3} = 6\sqrt{3} = 10.4\)