Polar and exponential form: mixed practice Problem set A Answers

1. (a) \(6\,\mathrm{cis}\left(\frac{3\pi}{2}\right)\)
2. (b) \(3 \sqrt{2}\,\mathrm{cis}\left(\frac{5\pi}{4}\right)\)
3. (c) \(4\,\mathrm{cis}\left(\frac{5\pi}{3}\right)\)

1. (a) \(-9 i\)
2. (b) \(3 + 3 \sqrt{3} i\)
3. (c) \(-2 + 2 i\)

1. (a) \(12\,\mathrm{cis}\left(\frac{\pi}{6}\right)\)
2. (b) \(4\,\mathrm{cis}\left(\frac{19\pi}{12}\right)\)

1. (a) \(12e^{\frac{7\pi i}{12}}\)
2. (b) \(7e^{\frac{5\pi i}{6}}\)

1. (a) \(- \frac{1}{2} + \frac{\sqrt{3}}{2} i\)
2. (b) \(\mathrm{cis}\,2.2\)

1. (a) \(z^2 = 3 + 4i\)
2. (b) \(iz = -1 + 2i\)

1. (a) \(3 \sqrt{2}\)
2. (b) \(\frac{3\pi}{4}\)
3. (c) \(|z^2| = 18,\ \arg z^2 = -\frac{\pi}{2}\)
4. (d) \(-18 i\)

1. (a) \(\frac{\pi}{3}\)
2. (b) \(\frac{3\pi}{4}\)
3. (c) \(-\frac{11\pi}{12}\)

1. (a) \(6 i\)
2. (b) \(1 + \left(\sqrt{3} + 4\right) i\)

1. (a) \(-254 + 349i\)
2. (b) \(-2.34 + 13.3i\)

1. (a) \(2\,\mathrm{cis}\left(-\frac{\pi}{6}\right)\)
2. (b) \(-128 + 128 \sqrt{3} i\)
3. (c) \(2 - 2 \sqrt{3} i\)

1. (a) \(n = 15\)
2. (b) \(n = 21\)

\(H(t) = 0.736\sin(4t + 0.708)\)

1. (a) \(V_1\) starts at 7, \(V_2\) starts at -9.90; both have period 0.251. \(V_1\) is greatest at t = 0, \(V_2\) at t = 0.131.
2. (b) \(V = 3.23\cos(25t + 2.69)\)

\(N = 340 + 57.2\sin\left(\frac{2\pi}{365}t + 0.247\right)\)

\(|z| = 4,\ \arg z = \frac{2\pi}{3},\ |w| = 2,\ \arg w = \frac{\pi}{6}\)

1. (a) \(|z| = 2,\ \arg z = \frac{\pi}{6},\ |w| = \sqrt{2},\ \arg w = \frac{3\pi}{4}\)
2. (b) \(\dfrac{w}{z} = \frac{\sqrt{2}}{2}\,\mathrm{cis}\left(\frac{7\pi}{12}\right) = \frac{1 - \sqrt{3} + \left(1 + \sqrt{3}\right) i}{4}\)
3. (c) \(\frac{- \sqrt{6} + \sqrt{2}}{4}\)

1. (a) \(|z| = 7.44,\ \arg z = 0.566\)
2. (b) \(7.44\sin(\theta + 0.566)\)