Polar and exponential form: mixed practice Problem set B One question at a time

Write each number in the form \(r\,\mathrm{cis}\,\theta\) with \(0 \le \theta < 2\pi\).

1. (a) \(-12 i\)

   • \(|z| = \sqrt{0^2 + (-12)^2} = 12\)
   • On the negative imaginary axis: \(\theta = \frac{3\pi}{2}\)
   • \(z = 12\,\mathrm{cis}\left(\frac{3\pi}{2}\right)\)

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

   • \(|z| = \sqrt{(- 4 \sqrt{3})^2 + (-4)^2} = 8\)
   • The point is in the third quadrant. The angle with the real axis is \(\frac{\pi}{6}\), so \(\theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}\)
   • \(z = 8\,\mathrm{cis}\left(\frac{7\pi}{6}\right)\)

3. (c) \(7 - 7 i\)

   • \(|z| = \sqrt{7^2 + (-7)^2} = 7 \sqrt{2}\)
   • The point is in the fourth quadrant. The angle with the real axis is \(\frac{\pi}{4}\), so \(\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)
   • \(z = 7 \sqrt{2}\,\mathrm{cis}\left(\frac{7\pi}{4}\right)\)

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

1. (a) \(|z| = 15,\ \arg z = -\frac{\pi}{2}\)

   • \(z = 15\left(\cos(-\frac{\pi}{2}) + i\sin(-\frac{\pi}{2})\right)\)
   • \(x = 15\cos(-\frac{\pi}{2}) = 0,\quad y = 15\sin(-\frac{\pi}{2}) = -15\)
   • \(z = -15 i\)

2. (b) \(|z| = 10,\ \arg z = \frac{\pi}{6}\)

   • \(z = 10\left(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6})\right)\)
   • \(x = 10\cos(\frac{\pi}{6}) = 5 \sqrt{3},\quad y = 10\sin(\frac{\pi}{6}) = 5\)
   • \(z = 5 \sqrt{3} + 5 i\)

3. (c) \(|z| = 3 \sqrt{2},\ \arg z = -\frac{3\pi}{4}\)

   • \(z = 3 \sqrt{2}\left(\cos(-\frac{3\pi}{4}) + i\sin(-\frac{3\pi}{4})\right)\)
   • \(x = 3 \sqrt{2}\cos(-\frac{3\pi}{4}) = -3,\quad y = 3 \sqrt{2}\sin(-\frac{3\pi}{4}) = -3\)
   • \(z = -3 - 3 i\)

Write each in the form \(r\,\mathrm{cis}\,\theta\) with \(0 < \theta \le 2\pi\). Work without a calculator.

1. (a) \((5\,\mathrm{cis}\left(\frac{3\pi}{4}\right))(2\,\mathrm{cis}\left(\frac{5\pi}{4}\right))\)

   • \(|z| = 5 \times 2 = 10\)
   • \(\arg = \frac{3\pi}{4} + \frac{5\pi}{4} = 2\pi\)
   • \(= 10\,\mathrm{cis}\left(2\pi\right)\)

2. (b) \(\dfrac{18\,\mathrm{cis}\left(\frac{\pi}{3}\right)}{6\,\mathrm{cis}\left(\frac{5\pi}{6}\right)}\)

   • \(|z| = 18 \div 6 = 3\)
   • \(\arg = \frac{\pi}{3} - \frac{5\pi}{6} = -\frac{\pi}{2}\) This is outside the interval, so add \(2\pi\): \(\frac{3\pi}{2}\)
   • \(= 3\,\mathrm{cis}\left(\frac{3\pi}{2}\right)\)

Write each in exponential form \(re^{i\theta}\) with \(-\pi < \theta \le \pi\).

1. (a) \(4e^{\frac{2\pi i}{5}} \times 3e^{\frac{\pi i}{10}}\)

   • \(|z| = 4 \times 3 = 12\)
   • \(\text{angle} = \frac{2\pi}{5} + \frac{\pi}{10} = \frac{\pi}{2}\)
   • \(= 12e^{\frac{\pi i}{2}}\)

2. (b) \(20e^{\frac{\pi i}{2}} \div 5e^{\frac{3\pi i}{4}}\)

   • \(|z| = 20 \div 5 = 4\)
   • \(\text{angle} = \frac{\pi}{2} - \frac{3\pi}{4} = -\frac{\pi}{4}\)
   • \(= 4e^{-\frac{\pi i}{4}}\)

Work without a calculator.

1. (a) Write in the form \(x + iy\): \(\left(e^{\frac{\pi i}{4}}\right)^{6}\)

   • \(\left(e^{\frac{\pi i}{4}}\right)^{6} = e^{\frac{3\pi i}{2}} = \cos(\frac{3\pi}{2}) + i\sin(\frac{3\pi}{2})\)
   • \(= -i\)

2. (b) Simplify \(\mathrm{cis}\,0.45 \times \mathrm{cis}\,2.1\)

   • \(\mathrm{cis}\,0.45 \times \mathrm{cis}\,2.1 = \mathrm{cis}\left(0.45 + 2.1\right) = \mathrm{cis}\,2.55\)

The diagram shows the point that represents \(z = 3 + i\) on an Argand diagram.

1. (a) Mark and label \(z^2\) on a copy of the diagram. Write \(z^2\) in the form \(x + iy\).

   • \(z^2 = (3 + i)^2 = 9 + 6i + 1i^2 = 8 + 6i\)
   • Plot the point \((8, 6)\). Its modulus is \(|z|^2 = 10\) and its argument is twice the argument of z.

2. (b) Mark and label \(iz\). Write \(iz\) in the form \(x + iy\).

   • \(iz = i(3 + i) = 3i + 1i^2 = -1 + 3i\)
   • Plot the point \((-1, 3)\). Multiplying by \(i\) rotates z through \(\frac{\pi}{2}\) anticlockwise about the origin.

\(z = -5 - 5 i\).

1. (a) Find \(|z|\).

   • \(|z| = \sqrt{(-5)^2 + (-5)^2} = 5 \sqrt{2}\)

2. (b) Find \(\arg z\), with \(-\pi < \arg z \le \pi\).

   • The point is in the third quadrant. The angle with the real axis is \(\frac{\pi}{4}\), so \(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\)
   • For \(-\pi < \theta \le \pi\), subtract \(2\pi\): \(\arg z = -\frac{3\pi}{4}\)

3. (c) Hence write down the modulus and argument of \(z^2\), with \(-\pi < \arg z^2 \le \pi\).

   • \(|z^2| = |z|^2 = 50\)
   • \(\arg z^2 = 2 \times -\frac{3\pi}{4} = -\frac{3\pi}{2} = \frac{\pi}{2}\)

4. (d) Hence write \(z^2\) in the form \(x + iy\).

   • \(z^2 = 50\left(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})\right) = 50 i\)

\(z = 5 \sqrt{3} - 5 i\) and \(w = - 4 \sqrt{3} - 4 i\). Give each argument in the interval \(-\pi < \theta \le \pi\).

1. (a) Find \(\arg z\).

   • The point is in the fourth quadrant. The angle with the real axis is \(\frac{\pi}{6}\), so \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\)
   • For \(-\pi < \theta \le \pi\), subtract \(2\pi\): \(\arg z = -\frac{\pi}{6}\)

2. (b) Find \(\arg w\).

   • The point is in the third quadrant. The angle with the real axis is \(\frac{\pi}{6}\), so \(\theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}\)
   • For \(-\pi < \theta \le \pi\), subtract \(2\pi\): \(\arg w = -\frac{5\pi}{6}\)

3. (c) Find \(\arg(zw)\).

   • \(\arg(zw) = \arg z + \arg w = -\frac{\pi}{6} + -\frac{5\pi}{6} = -\pi\) This is not in \(-\pi < \theta \le \pi\), so add \(2\pi\): \(\pi\)

Write each in the form \(x + iy\). Do not use a calculator.

1. (a) \((2\,\mathrm{cis}\left(\frac{\pi}{3}\right))(3\,\mathrm{cis}\left(\frac{\pi}{3}\right))\)

   • \((2\,\mathrm{cis}\left(\frac{\pi}{3}\right))(3\,\mathrm{cis}\left(\frac{\pi}{3}\right)) = 6\,\mathrm{cis}\left(\frac{2\pi}{3}\right)\)
   • \(= -3 + 3 \sqrt{3} i\)

2. (b) \(3\,\mathrm{cis}\left(\frac{\pi}{6}\right) + \mathrm{cis}\left(\pi\right)\)

   • \(3\,\mathrm{cis}\left(\frac{\pi}{6}\right) = \frac{3 \sqrt{3}}{2} + \frac{3}{2} i\)
   • \(\mathrm{cis}\left(\pi\right) = -1\)
   • \(\text{Sum} = -1 + \frac{3 \sqrt{3}}{2} + \frac{3}{2} i\)

Write each in Cartesian form, to 3 significant figures. Combine the powers first, then use the calculator only to find the real and imaginary parts.

1. (a) \(\left(2\,\mathrm{cis}\left(\frac{3\pi}{7}\right)\right)^{3}\left(5\,\mathrm{cis}\left(\frac{\pi}{14}\right)\right)^{2}\)

   • \(|z| = 2^{3} \times 5^{2} = 200\)
   • \(\arg = 3 \times \frac{3\pi}{7} + 2 \times \frac{\pi}{14} = \frac{10\pi}{7}\)
   • \(= 200\,\mathrm{cis}\left(\frac{10\pi}{7}\right)\)
   • From the calculator: \(200\cos(\frac{10\pi}{7}) = -44.5,\quad 200\sin(\frac{10\pi}{7}) = -195\)

2. (b) \(\dfrac{\left(\mathrm{cis}\left(\frac{5\pi}{12}\right)\right)^{4}}{\left(3\,\mathrm{cis}\left(\frac{\pi}{10}\right)\right)^{2}}\)

   • \(|z| = 1^{4} \div 3^{2} = \frac{1}{9}\)
   • \(\arg = 4 \times \frac{5\pi}{12} - 2 \times \frac{\pi}{10} = \frac{22\pi}{15}\)
   • \(= 0.111\,\mathrm{cis}\left(\frac{22\pi}{15}\right)\)
   • From the calculator: \(0.111\cos(\frac{22\pi}{15}) = -0.0116,\quad 0.111\sin(\frac{22\pi}{15}) = -0.111\)

Work without a calculator.

1. (a) Write \(z = - \sqrt{3} + i\) in the form \(r\,\mathrm{cis}\,\theta\), with \(-\pi < \theta \le \pi\).

   • \(|z| = \sqrt{(- \sqrt{3})^2 + 1^2} = 2\)
   • The point is in the second quadrant. The angle with the real axis is \(\frac{\pi}{6}\), so \(\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\)
   • \(z = 2\,\mathrm{cis}\left(\frac{5\pi}{6}\right)\)

2. (b) Hence find \(z^{5}\) in Cartesian form.

   • \(z^{5} = 2^{5}\,\mathrm{cis}\left(5 \times \frac{5\pi}{6}\right) = 32\,\mathrm{cis}\left(\frac{\pi}{6}\right)\)
   • \(= 32\left(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6})\right) = 16 \sqrt{3} + 16 i\)

3. (c) \(w = 1 - i\) and \(u = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\). Find \(w^{6}u^{5}\) in Cartesian form.

   • \(|w| = \sqrt{2},\ \arg w = -\frac{\pi}{4}\)
   • \(w^{6}u^{5} = 8\,\mathrm{cis}\left(6 \times -\frac{\pi}{4} + 5 \times \frac{\pi}{6}\right) = 8\,\mathrm{cis}\left(-\frac{2\pi}{3}\right)\)
   • \(= 8\,\mathrm{cis}\left(-\frac{2\pi}{3}\right) = -4 - 4 \sqrt{3} i\)

Give each answer as a positive integer.

1. (a) Find the smallest positive integer \(n\) for which \(z^n\) is real, where \(z = 3\,\mathrm{cis}\left(\frac{5\pi}{16}\right)\).

   • \(z^n = 3^n\,\mathrm{cis}\left(n \times \frac{5\pi}{16}\right)\). This is real when the argument is a multiple of \(\pi\).
   • \(n \times \frac{5\pi}{16} = k\pi \Rightarrow n = \dfrac{16k}{5}\)
   • The smallest positive integer value of \(n\) is the first one for which \(k\) makes this a whole number: \(n = 16\)

2. (b) Find the smallest positive integer \(n\) for which \(w^n = i\), where \(w = \mathrm{cis}\left(\frac{3\pi}{10}\right)\).

   • \(|w^n| = 1,\ \arg w^n = n \times \frac{3\pi}{10}\). For \(w^n = i\) the argument must be \(\tfrac{\pi}{2} + 2k\pi\).
   • \(n \times \frac{3\pi}{10} = \tfrac{\pi}{2} + 2k\pi \Rightarrow n = \dfrac{10(1 + 4k)}{6}\)
   • Try \(k = 0, 1, 2, \ldots\) until \(n\) is a whole number: \(k = 0: n = \frac{5}{3}; k = 1: n = \frac{25}{3}; k = 2: n = 15\)
   • \(n = 15\)

A floating platform rises and falls with two swells. At time \(t\) minutes the height of the platform, in metres, is \(H(t) = 3.5\sin(2t + 1) + 2\sin(2t - 0.8)\). Write the total as a single sinusoid \(A\sin(2t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

• \(3.5\sin(2t + 1) + 2\sin(2t - 0.8) = \mathrm{Im}\left(3.5e^{(2t + 1)i} + 2e^{(2t - 0.8)i}\right)\)
• \(= \mathrm{Im}\left(e^{2ti}(3.5e^{1i} + 2e^{-0.8i})\right)\)
• From the calculator: \(3.5e^{1i} + 2e^{-0.8i} = 3.28 + 1.51i = 3.62e^{0.431i}\)
• \(= \mathrm{Im}\left(3.62e^{(2t + 0.431)i}\right) = 3.62\sin(2t + 0.431)\)

Two AC voltage sources are connected in a circuit. The voltages, in volts, at time \(t\) seconds are \(V_1 = 8\cos(30t)\) and \(V_2 = 5\cos(30t - 2)\).

1. (a) On the axes provided, sketch the graphs of \(V_1\) and \(V_2\) for \(0 \le t \le 0.4\). Show where each graph starts and where it first reaches its maximum.

   • \(V_1 = 8\cos(30t)\) has amplitude 8 and period \(\tfrac{2\pi}{30} = 0.209\). It starts at 8.
   • \(V_2 = 5\cos(30t - 2)\) has amplitude 5, the same period, and starts at \(5\cos(-2) = -2.08\).
   • Each is greatest when its angle is a multiple of \(2\pi\): \(t = 0\) for \(V_1\), \(t = 0.0667\) for \(V_2\).

2. (b) Find the total voltage \(V = V_1 + V_2\) in the form \(A\cos(30t + c)\), where \(A > 0\) and \(-\pi < c \le \pi\).

   • \(8\cos(30t) + 5\cos(30t - 2) = \mathrm{Re}\left(8e^{30ti} + 5e^{(30t - 2)i}\right)\)
   • \(= \mathrm{Re}\left(e^{30ti}(8 + 5e^{-2i})\right)\)
   • From the calculator: \(8 + 5e^{-2i} = 5.92 - 4.55i = 7.46e^{-0.655i}\)
   • \(= \mathrm{Re}\left(7.46e^{(30t - 0.655)i}\right) = 7.46\cos(30t - 0.655)\)

The numbers of trout \(T\) and bass \(B\) in a lake vary through the year. At time \(t\) days after 1 January, \(T = 120 + 25\sin\left(\frac{2\pi}{365}t\right)\) and \(B = 90 + 18\sin\left(\frac{2\pi}{365}t + 2.3\right)\). Write the total number of fish, \(N = T + B\), in the form \(C + A\sin(bt + c)\).

• \(N = (120 + 90) + \left[25\sin\theta + 18\sin(\theta + 2.3)\right] \text{ where } \theta = \tfrac{2\pi}{365}t\)
• \(25\sin\theta + 18\sin(\theta + 2.3) = \mathrm{Im}\left(e^{\theta i}(25 + 18e^{2.3i})\right)\)
• From the calculator: \(25 + 18e^{2.3i} = 13.0 + 13.4i = 18.7e^{0.801i}\)
• \(N = 210 + 18.7\sin\left(\tfrac{2\pi}{365}t + 0.801\right)\)

The complex numbers \(z\) and \(w\) have arguments between \(0\) and \(\pi\). Given that \(zw = 18 i\) and \(\dfrac{z}{w} = \sqrt{3} + i\), find the modulus and argument of \(z\) and of \(w\).

• \(|zw| = |z||w| = 18 \quad\text{and}\quad \left|\dfrac{z}{w}\right| = \dfrac{|z|}{|w|} = 2\)
• \(\text{Multiply: } |z|^2 = 18 \times 2 = 36 \Rightarrow |z| = 6,\ |w| = 3\)
• \(\arg(zw) = \arg z + \arg w = \frac{\pi}{2} \quad\text{and}\quad \arg\left(\dfrac{z}{w}\right) = \arg z - \arg w = \frac{\pi}{6}\)
• \(\text{Add: } 2\arg z = \frac{2\pi}{3} \Rightarrow \arg z = \frac{\pi}{3}.\ \text{Subtract: } 2\arg w = \frac{\pi}{3} \Rightarrow \arg w = \frac{\pi}{6}\)

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

1. (a) Find the modulus and argument of \(z\) and of \(w\), with \(-\pi < \arg \le \pi\).

   • \(|z| = 2, \arg z = \frac{5\pi}{6}\)
   • \(|w| = \sqrt{2}, \arg w = -\frac{\pi}{4}\) (use the quadrant of each point).

2. (b) Find \(\dfrac{w}{z}\) in modulus-argument form and in Cartesian form.

   • \(\left|\dfrac{w}{z}\right| = \frac{\sqrt{2}}{2},\quad \arg = -\frac{\pi}{4} - \frac{5\pi}{6} = -\frac{13\pi}{12} = \frac{11\pi}{12}\)
   • \(\dfrac{w}{z} = \dfrac{1 - i}{- \sqrt{3} + i} = \dfrac{(1 - i)(- \sqrt{3} - i)}{4} = \frac{- \sqrt{3} - 1 - \left(1 - \sqrt{3}\right) i}{4}\)

3. (c) Hence find the exact value of \(\cos\left(\frac{11\pi}{12}\right)\).

   • \(\text{Real part: } \frac{\sqrt{2}}{2}\cos\left(\frac{11\pi}{12}\right) = - \frac{\sqrt{3}}{4} - \frac{1}{4}\)
   • \(\cos\left(\frac{11\pi}{12}\right) = \frac{- \sqrt{6} - \sqrt{2}}{4}\)

Let \(z = 9 + 5e^{2.5i}\).

1. (a) Find the modulus and argument of \(9 + 5e^{2.5i}\).

   • From the calculator: \(9 + 5e^{2.5i} = 4.99 + 2.99i\)
   • \(|9 + 5e^{2.5i}| = 5.82, \quad \arg = 0.540\)

2. (b) Hence write \(9\sin\theta + 5\sin(\theta + 2.5)\) in the form \(A\sin(\theta + \alpha)\), where \(A > 0\) and \(-\pi < \alpha \le \pi\).

   • \(9\sin\theta + 5\sin(\theta + 2.5) = \mathrm{Im}\left(e^{\theta i}(9 + 5e^{2.5i})\right)\)
   • \(= \mathrm{Im}\left(5.82e^{(\theta + 0.540)i}\right) = 5.82\sin(\theta + 0.540)\)