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Section 3-6 : Polar Coordinates

  1. For the point with polar coordinates \(\displaystyle \left( { - 9,\frac{{3\pi }}{7}} \right)\) determine three different sets of coordinates for the same point all of which have angles different from \(\displaystyle - \frac{{2\pi }}{3}\) and are in the range \( - 2\pi \le \theta \le 2\pi \).
  2. For the point with polar coordinates \(\displaystyle \left( {7, - \frac{{2\pi }}{3}} \right)\) determine three different sets of coordinates for the same point all of which have angles different from \(\displaystyle \frac{{3\pi }}{7}\) and are in the range \( - 2\pi \le \theta \le 2\pi \).
  3. The polar coordinates of a point are \(\left( {14,\,\,2.48} \right)\). Determine the Cartesian coordinates for the point.
  4. The polar coordinates of a point are \(\left( {\displaystyle - \frac{3}{{10}},\, - 5.29} \right)\). Determine the Cartesian coordinates for the point.
  5. The Cartesian coordinate of a point are \(\left( { - 3,5} \right)\). Determine a set of polar coordinates for the point.
  6. The Cartesian coordinate of a point are \(\left( {4, - 7} \right)\). Determine a set of polar coordinates for the point.
  7. The Cartesian coordinate of a point are \(\left( { - 3, - 12} \right)\). Determine a set of polar coordinates for the point.

For problems 8 and 9 convert the given equation into an equation in terms of polar coordinates.

  1. \(7{x^2}y + 8y = 3 - 6{x^2} - 6{y^2}\)
  2. \(\displaystyle \frac{{7y}}{{{x^2} + {y^2} - 8x}} = 9 + {y^2}\)

For problems 10 – 13 convert the given equation into an equation in terms of Cartesian coordinates.

  1. \(\displaystyle r - \frac{{8\sin \theta }}{r} = 2\cos \theta \)
  2. \({r^3}\csc \theta = 5\cos \theta - 6\)
  3. \(8 - r = {r^2}\sin \left( {2\theta } \right)\)
  4. \(r = 2a\cos \theta + 2b\sin \theta \)

For problems 14 – 27 sketch the graph of the given polar equation.

  1. \( - 7 = r\sin \theta \)
  2. \(\displaystyle \theta = \frac{{5\pi }}{7}\)
  3. \(\displaystyle \theta = - \frac{{9\pi }}{5}\)
  4. \(r\cos \theta = 4\)
  5. \(r = 6\sin \theta \)
  6. \(r = 100\)
  7. \(r = 24\cos \theta \)
  8. \(r = - 15\sin \theta \)
  9. \(r = 4 + 12\cos \theta \)
  10. \(r = 7 - 7\sin \theta \)
  11. \(r = 1 + 3\sin \theta \)
  12. \(r = 5 - 4\cos \theta \)
  13. \(r = 8 + 3\sin \theta \)
  14. \(r = 1 - \cos \theta \)