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

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

1. (a) \(-6 i\)

   • \(|z| = \sqrt{0^2 + (-6)^2} = 6\)
   • On the negative imaginary axis: \(\theta = \frac{3\pi}{2}\)
   • \(z = 6\,\mathrm{cis}\left(\frac{3\pi}{2}\right)\)

2. (b) \(-3 - 3 i\)

   • \(|z| = \sqrt{(-3)^2 + (-3)^2} = 3 \sqrt{2}\)
   • The point is in the third quadrant. The angle with the real axis is \(\frac{\pi}{4}\), so \(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\)
   • \(z = 3 \sqrt{2}\,\mathrm{cis}\left(\frac{5\pi}{4}\right)\)

3. (c) \(2 - 2 \sqrt{3} i\)

   • \(|z| = \sqrt{2^2 + (- 2 \sqrt{3})^2} = 4\)
   • The point is in the fourth quadrant. The angle with the real axis is \(\frac{\pi}{3}\), so \(\theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}\)
   • \(z = 4\,\mathrm{cis}\left(\frac{5\pi}{3}\right)\)

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

1. (a) \(|z| = 9,\ \arg z = -\frac{\pi}{2}\)

   • \(z = 9\left(\cos(-\frac{\pi}{2}) + i\sin(-\frac{\pi}{2})\right)\)
   • \(x = 9\cos(-\frac{\pi}{2}) = 0,\quad y = 9\sin(-\frac{\pi}{2}) = -9\)
   • \(z = -9 i\)

2. (b) \(|z| = 6,\ \arg z = \frac{\pi}{3}\)

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

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

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

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

1. (a) \((3\,\mathrm{cis}\left(\frac{5\pi}{6}\right))(4\,\mathrm{cis}\left(\frac{4\pi}{3}\right))\)

   • \(|z| = 3 \times 4 = 12\)
   • \(\arg = \frac{5\pi}{6} + \frac{4\pi}{3} = \frac{13\pi}{6}\) This is outside the interval, so subtract \(2\pi\): \(\frac{\pi}{6}\)
   • \(= 12\,\mathrm{cis}\left(\frac{\pi}{6}\right)\)

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

   • \(|z| = 12 \div 3 = 4\)
   • \(\arg = \frac{\pi}{4} - \frac{2\pi}{3} = -\frac{5\pi}{12}\) This is outside the interval, so add \(2\pi\): \(\frac{19\pi}{12}\)
   • \(= 4\,\mathrm{cis}\left(\frac{19\pi}{12}\right)\)

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

1. (a) \(6e^{\frac{\pi i}{3}} \times 2e^{\frac{\pi i}{4}}\)

   • \(|z| = 6 \times 2 = 12\)
   • \(\text{angle} = \frac{\pi}{3} + \frac{\pi}{4} = \frac{7\pi}{12}\)
   • \(= 12e^{\frac{7\pi i}{12}}\)

2. (b) \(14e^{\frac{7\pi i}{6}} \div 2e^{\frac{\pi i}{3}}\)

   • \(|z| = 14 \div 2 = 7\)
   • \(\text{angle} = \frac{7\pi}{6} - \frac{\pi}{3} = \frac{5\pi}{6}\)
   • \(= 7e^{\frac{5\pi i}{6}}\)

Work without a calculator.

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

   • \(\left(e^{\frac{\pi i}{6}}\right)^{4} = e^{\frac{2\pi i}{3}} = \cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3})\)
   • \(= - \frac{1}{2} + \frac{\sqrt{3}}{2} i\)

2. (b) Simplify \(\mathrm{cis}\,0.9 \times \mathrm{cis}\,1.3\)

   • \(\mathrm{cis}\,0.9 \times \mathrm{cis}\,1.3 = \mathrm{cis}\left(0.9 + 1.3\right) = \mathrm{cis}\,2.2\)

The diagram shows the point that represents \(z = 2 + i\) on an Argand diagram.

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

   • \(z^2 = (2 + i)^2 = 4 + 4i + 1i^2 = 3 + 4i\)
   • Plot the point \((3, 4)\). Its modulus is \(|z|^2 = 5\) and its argument is twice the argument of z.

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

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

\(z = -3 + 3 i\).

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

   • \(|z| = \sqrt{(-3)^2 + 3^2} = 3 \sqrt{2}\)

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

   • The point is in the second quadrant. The angle with the real axis is \(\frac{\pi}{4}\), so \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\)

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

   • \(|z^2| = |z|^2 = 18\)
   • \(\arg z^2 = 2 \times \frac{3\pi}{4} = \frac{3\pi}{2} = -\frac{\pi}{2}\)

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

   • \(z^2 = 18\left(\cos(-\frac{\pi}{2}) + i\sin(-\frac{\pi}{2})\right) = -18 i\)

\(z = 3 + 3 \sqrt{3} i\) and \(w = -2 + 2 i\). Give each argument in the interval \(-\pi < \theta \le \pi\).

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

   • The point is in the first quadrant. The angle with the real axis is \(\frac{\pi}{3}\), so \(\theta = \frac{\pi}{3}\)

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

   • The point is in the second quadrant. The angle with the real axis is \(\frac{\pi}{4}\), so \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\)

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

   • \(\arg(zw) = \arg z + \arg w = \frac{\pi}{3} + \frac{3\pi}{4} = \frac{13\pi}{12}\) This is not in \(-\pi < \theta \le \pi\), so subtract \(2\pi\): \(-\frac{11\pi}{12}\)

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

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

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

2. (b) \(2\,\mathrm{cis}\left(\frac{\pi}{3}\right) + 4\,\mathrm{cis}\left(\frac{\pi}{2}\right)\)

   • \(2\,\mathrm{cis}\left(\frac{\pi}{3}\right) = 1 + \sqrt{3} i\)
   • \(4\,\mathrm{cis}\left(\frac{\pi}{2}\right) = 4 i\)
   • \(\text{Sum} = 1 + \left(\sqrt{3} + 4\right) i\)

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

   • \(|z| = 4^{2} \times 3^{3} = 432\)
   • \(\arg = 2 \times \frac{\pi}{5} + 3 \times \frac{\pi}{10} = \frac{7\pi}{10}\)
   • \(= 432\,\mathrm{cis}\left(\frac{7\pi}{10}\right)\)
   • From the calculator: \(432\cos(\frac{7\pi}{10}) = -254,\quad 432\sin(\frac{7\pi}{10}) = 349\)

2. (b) \(\dfrac{\left(6\,\mathrm{cis}\left(\frac{\pi}{3}\right)\right)^{3}}{\left(2\,\mathrm{cis}\left(\frac{\pi}{9}\right)\right)^{4}}\)

   • \(|z| = 6^{3} \div 2^{4} = \frac{27}{2}\)
   • \(\arg = 3 \times \frac{\pi}{3} - 4 \times \frac{\pi}{9} = \frac{5\pi}{9}\)
   • \(= 13.5\,\mathrm{cis}\left(\frac{5\pi}{9}\right)\)
   • From the calculator: \(13.5\cos(\frac{5\pi}{9}) = -2.34,\quad 13.5\sin(\frac{5\pi}{9}) = 13.3\)

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 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 \(2\pi\): \(\arg z = -\frac{\pi}{6}\)
   • \(z = 2\,\mathrm{cis}\left(-\frac{\pi}{6}\right)\)

2. (b) Hence find \(z^{8}\) in Cartesian form.

   • \(z^{8} = 2^{8}\,\mathrm{cis}\left(8 \times -\frac{\pi}{6}\right) = 256\,\mathrm{cis}\left(\frac{2\pi}{3}\right)\)
   • \(= 256\left(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3})\right) = -128 + 128 \sqrt{3} i\)

3. (c) \(w = -1 + i\) and \(u = \cos\frac{\pi}{9} + i\sin\frac{\pi}{9}\). Find \(w^{4}u^{6}\) in Cartesian form.

   • \(|w| = \sqrt{2},\ \arg w = \frac{3\pi}{4}\)
   • \(w^{4}u^{6} = 4\,\mathrm{cis}\left(4 \times \frac{3\pi}{4} + 6 \times \frac{\pi}{9}\right) = 4\,\mathrm{cis}\left(\frac{11\pi}{3}\right)\)
   • \(= 4\,\mathrm{cis}\left(-\frac{\pi}{3}\right) = 2 - 2 \sqrt{3} i\)

Give each answer as a positive integer.

1. (a) Find the smallest positive integer \(n\) for which \(z^n\) is real, where \(z = 2\,\mathrm{cis}\left(\frac{4\pi}{15}\right)\).

   • \(z^n = 2^n\,\mathrm{cis}\left(n \times \frac{4\pi}{15}\right)\). This is real when the argument is a multiple of \(\pi\).
   • \(n \times \frac{4\pi}{15} = k\pi \Rightarrow n = \dfrac{15k}{4}\)
   • The smallest positive integer value of \(n\) is the first one for which \(k\) makes this a whole number: \(n = 15\)

2. (b) Find the smallest positive integer \(n\) for which \(w^n = i\), where \(w = \mathrm{cis}\left(\frac{3\pi}{14}\right)\).

   • \(|w^n| = 1,\ \arg w^n = n \times \frac{3\pi}{14}\). For \(w^n = i\) the argument must be \(\tfrac{\pi}{2} + 2k\pi\).
   • \(n \times \frac{3\pi}{14} = \tfrac{\pi}{2} + 2k\pi \Rightarrow n = \dfrac{14(1 + 4k)}{6}\)
   • Try \(k = 0, 1, 2, \ldots\) until \(n\) is a whole number: \(k = 0: n = \frac{7}{3}; k = 1: n = \frac{35}{3}; k = 2: n = 21\)
   • \(n = 21\)

A small boat is lifted by two sets of waves. At time \(t\) seconds the height of the boat, in metres, is \(H(t) = 1.2\sin(4t) + 0.8\sin(4t + 2.5)\). Write the total as a single sinusoid \(A\sin(4t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

• \(1.2\sin(4t) + 0.8\sin(4t + 2.5) = \mathrm{Im}\left(1.2e^{4ti} + 0.8e^{(4t + 2.5)i}\right)\)
• \(= \mathrm{Im}\left(e^{4ti}(1.2 + 0.8e^{2.5i})\right)\)
• From the calculator: \(1.2 + 0.8e^{2.5i} = 0.559 + 0.479i = 0.736e^{0.708i}\)
• \(= \mathrm{Im}\left(0.736e^{(4t + 0.708)i}\right) = 0.736\sin(4t + 0.708)\)

Two AC voltage sources are connected in a circuit. The voltages, in volts, at time \(t\) seconds are \(V_1 = 7\cos(25t)\) and \(V_2 = 10\cos(25t + 3)\).

1. (a) On the axes provided, sketch the graphs of \(V_1\) and \(V_2\) for \(0 \le t \le 0.5\). Show where each graph starts and where it first reaches its maximum.

   • \(V_1 = 7\cos(25t)\) has amplitude 7 and period \(\tfrac{2\pi}{25} = 0.251\). It starts at 7.
   • \(V_2 = 10\cos(25t + 3)\) has amplitude 10, the same period, and starts at \(10\cos(3) = -9.90\).
   • Each is greatest when its angle is a multiple of \(2\pi\): \(t = 0\) for \(V_1\), \(t = 0.131\) for \(V_2\).

2. (b) Find the total voltage \(V = V_1 + V_2\) in the form \(A\cos(25t + c)\), where \(A > 0\) and \(-\pi < c \le \pi\).

   • \(7\cos(25t) + 10\cos(25t + 3) = \mathrm{Re}\left(7e^{25ti} + 10e^{(25t + 3)i}\right)\)
   • \(= \mathrm{Re}\left(e^{25ti}(7 + 10e^{3i})\right)\)
   • From the calculator: \(7 + 10e^{3i} = -2.90 + 1.41i = 3.23e^{2.69i}\)
   • \(= \mathrm{Re}\left(3.23e^{(25t + 2.69)i}\right) = 3.23\cos(25t + 2.69)\)

The numbers of rabbits \(R\) and foxes \(F\) in a nature reserve vary through the year. At time \(t\) days after 1 January, \(R = 300 + 50\sin\left(\frac{2\pi}{365}t\right)\) and \(F = 40 + 15\sin\left(\frac{2\pi}{365}t + 1.2\right)\). Write the total number of animals, \(N = R + F\), in the form \(C + A\sin(bt + c)\).

• \(N = (300 + 40) + \left[50\sin\theta + 15\sin(\theta + 1.2)\right] \text{ where } \theta = \tfrac{2\pi}{365}t\)
• \(50\sin\theta + 15\sin(\theta + 1.2) = \mathrm{Im}\left(e^{\theta i}(50 + 15e^{1.2i})\right)\)
• From the calculator: \(50 + 15e^{1.2i} = 55.4 + 14.0i = 57.2e^{0.247i}\)
• \(N = 340 + 57.2\sin\left(\tfrac{2\pi}{365}t + 0.247\right)\)

The complex numbers \(z\) and \(w\) have arguments between \(0\) and \(\pi\). Given that \(zw = - 4 \sqrt{3} + 4 i\) and \(\dfrac{z}{w} = 2 i\), find the modulus and argument of \(z\) and of \(w\).

• \(|zw| = |z||w| = 8 \quad\text{and}\quad \left|\dfrac{z}{w}\right| = \dfrac{|z|}{|w|} = 2\)
• \(\text{Multiply: } |z|^2 = 8 \times 2 = 16 \Rightarrow |z| = 4,\ |w| = 2\)
• \(\arg(zw) = \arg z + \arg w = \frac{5\pi}{6} \quad\text{and}\quad \arg\left(\dfrac{z}{w}\right) = \arg z - \arg w = \frac{\pi}{2}\)
• \(\text{Add: } 2\arg z = \frac{4\pi}{3} \Rightarrow \arg z = \frac{2\pi}{3}.\ \text{Subtract: } 2\arg w = \frac{\pi}{3} \Rightarrow \arg w = \frac{\pi}{6}\)

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

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

   • \(|z| = 2, \arg z = \frac{\pi}{6}\)
   • \(|w| = \sqrt{2}, \arg w = \frac{3\pi}{4}\) (use the quadrant of each point).

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

   • \(\left|\dfrac{w}{z}\right| = \frac{\sqrt{2}}{2},\quad \arg = \frac{3\pi}{4} - \frac{\pi}{6} = \frac{7\pi}{12}\)
   • \(\dfrac{w}{z} = \dfrac{-1 + i}{\sqrt{3} + i} = \dfrac{(-1 + i)(\sqrt{3} - i)}{4} = \frac{1 - \sqrt{3} + \left(1 + \sqrt{3}\right) i}{4}\)

3. (c) Hence find the exact value of \(\cos\left(\frac{7\pi}{12}\right)\).

   • \(\text{Real part: } \frac{\sqrt{2}}{2}\cos\left(\frac{7\pi}{12}\right) = \frac{1}{4} - \frac{\sqrt{3}}{4}\)
   • \(\cos\left(\frac{7\pi}{12}\right) = \frac{- \sqrt{6} + \sqrt{2}}{4}\)

Let \(z = 6 + 4e^{1.5i}\).

1. (a) Find the modulus and argument of \(6 + 4e^{1.5i}\).

   • From the calculator: \(6 + 4e^{1.5i} = 6.28 + 3.99i\)
   • \(|6 + 4e^{1.5i}| = 7.44, \quad \arg = 0.566\)

2. (b) Hence write \(6\sin\theta + 4\sin(\theta + 1.5)\) in the form \(A\sin(\theta + \alpha)\), where \(A > 0\) and \(-\pi < \alpha \le \pi\).

   • \(6\sin\theta + 4\sin(\theta + 1.5) = \mathrm{Im}\left(e^{\theta i}(6 + 4e^{1.5i})\right)\)
   • \(= \mathrm{Im}\left(7.44e^{(\theta + 0.566)i}\right) = 7.44\sin(\theta + 0.566)\)