Comprehensive Notes on Complex Numbers and Polar Coordinates

Complex Numbers in Standard Form

• 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: $6(1 + i(0)) = 6$

Operations in Trigonometric Form

• 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.