Polar and exponential form: mixed practice Problem set B Answers

1. (a) \(12\,\mathrm{cis}\left(\frac{3\pi}{2}\right)\)
2. (b) \(8\,\mathrm{cis}\left(\frac{7\pi}{6}\right)\)
3. (c) \(7 \sqrt{2}\,\mathrm{cis}\left(\frac{7\pi}{4}\right)\)

1. (a) \(-15 i\)
2. (b) \(5 \sqrt{3} + 5 i\)
3. (c) \(-3 - 3 i\)

1. (a) \(10\,\mathrm{cis}\left(2\pi\right)\)
2. (b) \(3\,\mathrm{cis}\left(\frac{3\pi}{2}\right)\)

1. (a) \(12e^{\frac{\pi i}{2}}\)
2. (b) \(4e^{-\frac{\pi i}{4}}\)

1. (a) \(-i\)
2. (b) \(\mathrm{cis}\,2.55\)

1. (a) \(z^2 = 8 + 6i\)
2. (b) \(iz = -1 + 3i\)

1. (a) \(5 \sqrt{2}\)
2. (b) \(-\frac{3\pi}{4}\)
3. (c) \(|z^2| = 50,\ \arg z^2 = \frac{\pi}{2}\)
4. (d) \(50 i\)

1. (a) \(-\frac{\pi}{6}\)
2. (b) \(-\frac{5\pi}{6}\)
3. (c) \(\pi\)

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

1. (a) \(-44.5 - 195i\)
2. (b) \(-0.0116 - 0.111i\)

1. (a) \(2\,\mathrm{cis}\left(\frac{5\pi}{6}\right)\)
2. (b) \(16 \sqrt{3} + 16 i\)
3. (c) \(-4 - 4 \sqrt{3} i\)

1. (a) \(n = 16\)
2. (b) \(n = 15\)

\(H(t) = 3.62\sin(2t + 0.431)\)

1. (a) \(V_1\) starts at 8, \(V_2\) starts at -2.08; both have period 0.209. \(V_1\) is greatest at t = 0, \(V_2\) at t = 0.0667.
2. (b) \(V = 7.46\cos(30t - 0.655)\)

\(N = 210 + 18.7\sin\left(\frac{2\pi}{365}t + 0.801\right)\)

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

1. (a) \(|z| = 2,\ \arg z = \frac{5\pi}{6},\ |w| = \sqrt{2},\ \arg w = -\frac{\pi}{4}\)
2. (b) \(\dfrac{w}{z} = \frac{\sqrt{2}}{2}\,\mathrm{cis}\left(\frac{11\pi}{12}\right) = \frac{- \sqrt{3} - 1 - \left(1 - \sqrt{3}\right) i}{4}\)
3. (c) \(\frac{- \sqrt{6} - \sqrt{2}}{4}\)

1. (a) \(|z| = 5.82,\ \arg z = 0.540\)
2. (b) \(5.82\sin(\theta + 0.540)\)