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Section 9.6 : Polar Coordinates

1. For the point with polar coordinates \(\displaystyle \left( {2,\frac{\pi }{7}} \right)\) determine three different sets of coordinates for the same point all of which have angles different from \(\displaystyle \frac{\pi }{7}\) and are in the range \( - 2\pi \le \theta \le 2\pi \). Solution
2. The polar coordinates of a point are \(\left( { - 5,0.23} \right)\). Determine the Cartesian coordinates for the point. Solution
3. The Cartesian coordinate of a point are \(\left( {2, - 6} \right)\). Determine a set of polar coordinates for the point. Solution
4. The Cartesian coordinate of a point are \(\left( { - 8,1} \right)\). Determine a set of polar coordinates for the point. Solution

For problems 5 and 6 convert the given equation into an equation in terms of polar coordinates.

1. \(\displaystyle \frac{{4x}}{{3{x^2} + 3{y^2}}} = 6 - xy\) Solution
2. \(\displaystyle {x^2} = \frac{{4x}}{y} - 3{y^2} + 2\) Solution

For problems 7 and 8 convert the given equation into an equation in terms of Cartesian coordinates.

1. \(6{r^3}\sin \theta = 4 - cos\theta \) Solution
2. \(\displaystyle \frac{2}{r} = \sin \theta - \sec \theta \) Solution

For problems 9 – 16 sketch the graph of the given polar equation.

1. \(\displaystyle \cos \theta = \frac{6}{r}\) Solution
2. \(\displaystyle \theta = - \frac{\pi }{3}\) Solution
3. \(r = - 14\cos \theta \) Solution
4. \(r = 7\) Solution
5. \(r = 9\sin \theta \) Solution
6. \(r = 8 + 8\cos \theta \) Solution
7. \(r = 5 - 2\sin \theta \) Solution
8. \(r = 4 - 9\sin \theta \) Solution