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Paul
January 27, 2020

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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 $$\displaystyle \frac{{3\pi }}{7}$$ 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 $$\displaystyle - \frac{{2\pi }}{3}$$ 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. $$r = 1 - \cos \theta$$