Comprehensive Notes on Complex Numbers and Polar Coordinates
- 17a. 8(\cos 240° + i \sin 240°)
- Convert to rectangular form: 8(-\frac{1}{2} - i\frac{\sqrt{3}}{2}) = -4 - 4i\sqrt{3}
- 17b. 6(\cos \frac{11\pi}{7} + i \sin \frac{11\pi}{7})
- This is already in trigonometric form. To convert, evaluate the cosine and sine and distribute.
- 17c. 10(\cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3})
- Convert to rectangular form: 10(\frac{1}{2} - i\frac{\sqrt{3}}{2}) = 5 - 5i\sqrt{3}
- 17d. 6(\cos 2\pi + i \sin 2\pi)
- Convert to rectangular form: 6(1 + i(0)) = 6
- 18a. [2(\cos 27° + i \sin 27°)][5(\cos 103° + i \sin 103°)]
- Multiply magnitudes and add angles: 2 * 5 [\cos(27° + 103°) + i \sin(27° + 103°)] = 10(\cos 130° + i \sin 130°)
- 18b. [6(\cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3})][5(\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3})]
- Multiply magnitudes and add angles: 6 * 5 [\cos(\frac{5\pi}{3} + \frac{4\pi}{3}) + i \sin(\frac{5\pi}{3} + \frac{4\pi}{3})] = 30(\cos \frac{9\pi}{3} + i \sin \frac{9\pi}{3}) = 30(\cos 3\pi + i \sin 3\pi)
- 18c. \frac{22(\cos 500° + i \sin 500°)}{2(\cos 120° + i \sin 120°)}
- Divide magnitudes and subtract angles: \frac{22}{2} [\cos(500° - 120°) + i \sin(500° - 120°)] = 11(\cos 380° + i \sin 380°)
- Simplify angle: 11(\cos (380° - 360°) + i \sin (380° - 360°)) = 11(\cos 20° + i \sin 20°)
- 18d. \frac{10(\cos 70° + i \sin 70°)}{2(\cos 330° + i \sin 330°)}
- Divide magnitudes and subtract angles: \frac{10}{2} [\cos(70° - 330°) + i \sin(70° - 330°)] = 5(\cos (-260°) + i \sin (-260°))
- Simplify angle: 5(\cos (-260° + 360°) + i \sin (-260° + 360°)) = 5(\cos 100° + i \sin 100°)
DeMoivre's Theorem
- 19a. (-2 + 3i)^6
- Convert to polar form: r = \sqrt{(-2)^2 + 3^2} = \sqrt{13}, \theta = \arctan(\frac{3}{-2}) + \pi
- Apply DeMoivre's Theorem: (\sqrt{13})^6 [\cos(6\theta) + i \sin(6\theta)]
- Convert back to standard form after calculating the values.
- 19b. [3(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4})]^4
- Apply DeMoivre's Theorem: 3^4 [\cos(4 * \frac{5\pi}{4}) + i \sin(4 * \frac{5\pi}{4})] = 81(\cos 5\pi + i \sin 5\pi)
- Simplify: 81(\cos \pi + i \sin \pi) = 81(-1 + 0i) = -81
- 19c. [5(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})]^3
- Apply DeMoivre's Theorem: 5^3 [\cos(3 * \frac{3\pi}{2}) + i \sin(3 * \frac{3\pi}{2})] = 125(\cos \frac{9\pi}{2} + i \sin \frac{9\pi}{2})
- Simplify: 125(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) = 125(0 + i) = 125i
- 19d. (4 - 4i)^3
- Convert to polar form: r = \sqrt{4^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2}, \theta = \arctan(\frac{-4}{4}) = -\frac{\pi}{4}
- Apply DeMoivre's Theorem: (4\sqrt{2})^3 [\cos(3 * -\frac{\pi}{4}) + i \sin(3 * -\frac{\pi}{4})] = (4\sqrt{2})^3 [\cos(-\frac{3\pi}{4}) + i \sin(-\frac{3\pi}{4}) = 128 \sqrt{2} [\cos(-\frac{3\pi}{4}) + i \sin(-\frac{3\pi}{4})]
Polar Coordinate Representations
- 20. A = (5,$\frac{\pi}{3}$)
- Plot the point. Find three additional representations:
- Add 2\pi to the angle: (5, \frac{\pi}{3} + 2\pi) = (5, \frac{7\pi}{3})
- Add \pi to the angle and negate r: (-5, \frac{\pi}{3} + \pi) = (-5, \frac{4\pi}{3})
- Subtract 2\pi from the original angle: (5, \frac{\pi}{3} - 2\pi) = (5, -\frac{5\pi}{3})
- 21. B = (-2,$\frac{3\pi}{4}$)
- Plot the point. Find three additional representations:
- Add 2\pi to the angle: (-2, \frac{3\pi}{4} + 2\pi) = (-2, \frac{11\pi}{4})
- Add \pi to the angle and negate r: (2, \frac{3\pi}{4} + \pi) = (2, \frac{7\pi}{4})
- Subtract 2\pi from the original angle: (-2, \frac{3\pi}{4} - 2\pi) = (-2, -\frac{5\pi}{4})
- 22. C = (2,-$\frac{\pi}{4}$)
- Plot the point. Find three additional representations:
- Add 2\pi to the angle: (2, -\frac{\pi}{4} + 2\pi) = (2, \frac{7\pi}{4})
- Add \pi to the angle and negate r: (-2, -\frac{\pi}{4} + \pi) = (-2, \frac{3\pi}{4})
- Subtract 2\pi from the original angle: (2, -\frac{\pi}{4} - 2\pi) = (2, -\frac{9\pi}{4})
- 23. D = (-4,-$\frac{\pi}{3}$)
- Plot the point. Find three additional representations:
- Add 2\pi to the angle: (-4, -\frac{\pi}{3} + 2\pi) = (-4, \frac{5\pi}{3})
- Add \pi to the angle and negate r: (4, -\frac{\pi}{3} + \pi) = (4, \frac{2\pi}{3})
- Subtract 2\pi from the original angle: (-4, -\frac{\pi}{3} - 2\pi) = (-4, -\frac{7\pi}{3})
Rectangular to Polar Conversion
- 24. (-3, -3)
- r = \sqrt{(-3)^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2}
- \theta = \arctan(\frac{-3}{-3}) = \arctan(1)
- Since the point is in the third quadrant: \theta = \frac{5\pi}{4}
- Polar coordinates: (3\sqrt{2}, \frac{5\pi}{4})
- 25. (-\sqrt{2}, \sqrt{2})
- r = \sqrt{(-\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2
- \theta = \arctan(\frac{\sqrt{2}}{-\sqrt{2}}) = \arctan(-1)
- Since the point is in the second quadrant: \theta = \frac{3\pi}{4}
- Polar coordinates: (2, \frac{3\pi}{4})
- 26. (1, -\sqrt{3})
- r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
- \theta = \arctan(\frac{-\sqrt{3}}{1}) = \arctan(-\sqrt{3})
- Since the point is in the fourth quadrant: \theta = -\frac{\pi}{3}, equivalent to \frac{5\pi}{3}
- Polar coordinates: (2, \frac{5\pi}{3})
- 27. (5, 0)
- r = \sqrt{5^2 + 0^2} = \sqrt{25} = 5
- \theta = \arctan(\frac{0}{5}) = 0
- Polar coordinates: (5, 0)
Polar to Rectangular Conversion
- 28. (7,$\frac{5\pi}{6}$)
- x = r \cos \theta = 7 \cos(\frac{5\pi}{6}) = 7(-\frac{\sqrt{3}}{2}) = -\frac{7\sqrt{3}}{2}
- y = r \sin \theta = 7 \sin(\frac{5\pi}{6}) = 7(\frac{1}{2}) = \frac{7}{2}
- Rectangular coordinates: (- \frac{7\sqrt{3}}{2}, \frac{7}{2})
- 29. (5,$\frac{11\pi}{6}$)
- x = r \cos \theta = 5 \cos(\frac{11\pi}{6}) = 5(\frac{\sqrt{3}}{2}) = \frac{5\sqrt{3}}{2}
- y = r \sin \theta = 5 \sin(\frac{11\pi}{6}) = 5(-\frac{1}{2}) = -\frac{5}{2}
- Rectangular coordinates: (\frac{5\sqrt{3}}{2}, -\frac{5}{2})
- 30. (-3,$\pi$ )
- x = r \cos \theta = -3 \cos(\pi) = -3(-1) = 3
- y = r \sin \theta = -3 \sin(\pi) = -3(0) = 0
- Rectangular coordinates: (3, 0)
Calculator Conversions
- 31a. (9, 31°)
- x = 9 \cos(31°) \approx 7.71
- y = 9 \sin(31°) \approx 4.64
- Rectangular coordinates: (7.71, 4.64)
- 31b. (8, 173°)
- x = 8 \cos(173°) \approx -7.97
- y = 8 \sin(173°) \approx 0.96
- Rectangular coordinates: (-7.97, 0.96)
Calculator Conversions: Rectangular to Polar
- 32a. (-2, -4)
- r = \sqrt{(-2)^2 + (-4)^2} = \sqrt{20} \approx 4.47
- \theta = \arctan(\frac{-4}{-2}) = \arctan(2) \approx 1.11
- Since the point is in the third quadrant, add \pi: \theta \approx 1.11 + \pi \approx 4.25
- Polar coordinates: (4.47, 4.25)
- 32b. (4, 5)
- r = \sqrt{4^2 + 5^2} = \sqrt{41} \approx 6.40
- \theta = \arctan(\frac{5}{4}) \approx 0.896
- Polar coordinates: (6.40, 0.90)
Graphing Polar Functions
- 33. r = -4\sin(2\theta)
- Create a table of values for \theta and r to plot the graph. Look for symmetry and critical points.
- Show direction of the graph by indicating how r changes as \theta increases.
- Label at least four values of \theta on the curve to show orientation.