Powers and de Moivre's formula Problem set A One question at a time

Work by hand. Give exact values.

1. (a) \(z = 3e^{i\frac{\pi}{12}}\). Find \(z^{4}\) in the form \(x + iy\).

   • \(z^4 = \left(3e^{i\frac{\pi}{12}}\right)^4 = 3^{4}e^{i \times 4 \times \frac{\pi}{12}} = 81e^{i\frac{\pi}{3}}\)
   • \(z^4 = 81e^{i\frac{\pi}{3}} = 81\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) = \frac{81}{2}+\frac{81 \sqrt{3}}{2}i\)

2. (b) \(z = 2e^{i\frac{2\pi}{9}}\). Find \(z^{3}\) in the form \(x + iy\).

   • \(z^3 = \left(2e^{i\frac{2\pi}{9}}\right)^3 = 2^{3}e^{i \times 3 \times \frac{2\pi}{9}} = 8e^{i\frac{2\pi}{3}}\)
   • \(z^3 = 8e^{i\frac{2\pi}{3}} = 8\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) = -4+4 \sqrt{3}i\)

3. (c) \(z = \sqrt{2}e^{i\frac{5\pi}{8}}\). Find \(z^{4}\) in the form \(x + iy\).

   • \(z^4 = \left(\sqrt{2}e^{i\frac{5\pi}{8}}\right)^4 = (\sqrt{2})^{4}e^{i \times 4 \times \frac{5\pi}{8}} = 4e^{i\frac{5\pi}{2}}\)
   • \(\frac{5\pi}{2} - 2\pi = \frac{\pi}{2}\)
   • \(z^4 = 4e^{i\frac{\pi}{2}} = 4\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right) = 4i\)

Find the modulus and argument first, with \(0 \le \arg z < 2\pi\). Give exact values.

1. (a) \(z = \sqrt{3}+i\). Write \(z^{4}\) in the form \(x + iy\).

   • \(|z| = \sqrt{(\sqrt{3})^2 + (1)^2} = 2\)
   • \(\text{The point is in the first quadrant. The acute angle is } \arctan\frac{1}{\sqrt{3}} = \frac{\pi}{6}, \text{ so } \arg z = \frac{\pi}{6}\)
   • \(z = 2e^{i\frac{\pi}{6}}\)
   • \(z^4 = \left(2e^{i\frac{\pi}{6}}\right)^4 = 2^{4}e^{i \times 4 \times \frac{\pi}{6}} = 16e^{i\frac{2\pi}{3}}\)
   • \(z^4 = 16e^{i\frac{2\pi}{3}} = 16\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) = -8+8 \sqrt{3}i\)

2. (b) \(z = -1+i\). Write \(z^{6}\) in the form \(x + iy\).

   • \(|z| = \sqrt{(-1)^2 + (1)^2} = \sqrt{2}\)
   • \(\text{The point is in the second quadrant. The acute angle is } \arctan\frac{1}{1} = \frac{\pi}{4}, \text{ so } \arg z = \frac{3\pi}{4}\)
   • \(z = \sqrt{2}e^{i\frac{3\pi}{4}}\)
   • \(z^6 = \left(\sqrt{2}e^{i\frac{3\pi}{4}}\right)^6 = (\sqrt{2})^{6}e^{i \times 6 \times \frac{3\pi}{4}} = 8e^{i\frac{9\pi}{2}}\)
   • \(\frac{9\pi}{2} - 4\pi = \frac{\pi}{2}\)
   • \(z^6 = 8e^{i\frac{\pi}{2}} = 8\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right) = 8i\)

3. (c) \(z = 2-2i\). Write \(z^{4}\) in the form \(x + iy\).

   • \(|z| = \sqrt{2^2 + (-2)^2} = 2 \sqrt{2}\)
   • \(\text{The point is in the fourth quadrant. The acute angle is } \arctan\frac{2}{2} = \frac{\pi}{4}, \text{ so } \arg z = \frac{7\pi}{4}\)
   • \(z = 2 \sqrt{2}e^{i\frac{7\pi}{4}}\)
   • \(z^4 = \left(2 \sqrt{2}e^{i\frac{7\pi}{4}}\right)^4 = (2 \sqrt{2})^{4}e^{i \times 4 \times \frac{7\pi}{4}} = 64e^{i7\pi}\)
   • \(7\pi - 6\pi = \pi\)
   • \(z^4 = 64e^{i\pi} = 64\left(\cos\pi + i\sin\pi\right) = -64\)

A geometric sequence has first term \(u_1 = 20\) and common ratio \(r = 0.3 + 0.4i\).

1. (a) Find \(u_2\) and \(u_3\) in the form \(x + iy\).

   • \(u_2 = 20(0.3 + 0.4i) = 6 + 8i\)
   • \(u_3 = (6 + 8i)(0.3 + 0.4i) = -1.4 + 4.8i\)

2. (b) Show that \(|u_1|, |u_2|, |u_3|, \dots\) is also a geometric sequence, and write down its common ratio.

   • \(|u_1| = 20,\ |u_2| = 10,\ |u_3| = 5\).
   • Each modulus is 0.5 times the one before, since \(|u_{n+1}| = |u_n||r|\) and \(|r| = \sqrt{0.3^2 + 0.4^2} = 0.5\).

3. (c) Find the sum of the infinite sequence \(|u_1|, |u_2|, |u_3|, \dots\).

   • \(|r| < 1\), so \(S_\infty = \dfrac{20}{1 - 0.5} = 40\).

In a drawing program, a pen starts at the point \(1\) on the complex plane, measured in centimetres. At each step the pen's position is multiplied by \(m = 1.2e^{i\frac{\pi}{12}}\), so the pen spirals outwards. After \(n\) steps its position is \(m^n\).

1. (a) Find the position of the pen after 4 steps in the form \(x + iy\). Give \(x\) and \(y\) to 3 significant figures.

   • \(m^4 = 1.2^4e^{i \times 4 \times \frac{\pi}{12}} = 2.0736e^{i\frac{\pi}{3}}\)
   • \(x = 2.0736\cos\frac{\pi}{3} = 1.0368,\quad y = 2.0736\sin\frac{\pi}{3} = 1.7958…\)
   • \(m^4 \approx 1.04 + 1.80i\)

2. (b) Find the distance of the pen from the origin after 4 steps. Give your answer to 3 significant figures.

   • \(|m^4| = 1.2^4 = 2.0736\)
   • The distance from the origin is 2.07 cm to 3 significant figures.

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

1. (a) Write \(z\) in exponential form.

   • \(|z| = \sqrt{1 + 3} = 2,\quad \arg z = \arctan\sqrt{3} = \frac{\pi}{3}\)
   • \(z = 2e^{i\frac{\pi}{3}}\)

2. (b) Find the smallest positive integer \(n\) for which \(z^n\) is real and negative, and write down \(z^n\).

   • \(z^n = 2^ne^{i\frac{n\pi}{3}}\)
   • \(\text{Real and negative: } \frac{n\pi}{3} = \pi, 3\pi, \ldots, \text{ so the smallest is } n = 3\)
   • \(z^3 = 2^3e^{i\pi} = 8 \times (-1) = -8\)