In Exercises polarpointgraphfirst - polarpointgraphlast, plot the point given in polar coordinates in the \(xy\)-plane and then give three different expressions for the point such that (a) \(r < 0\) and \(0 \leq \theta \leq 2\pi \), (b) \(r > 0\) and \(\theta \leq 0\) (c) \(r > 0\) and \(\theta \geq 2\pi \)
\(\left ( 2, \frac {\pi }{3} \right )\)

[Picture]

(a) \(\left ( -2, \frac {4\pi }{3} \right )\)

(b) \(\left ( 2, -\frac {5\pi }{3} \right )\)

(c) \(\left ( 2, \frac {7\pi }{3} \right )\)

\(\left ( 5, \frac {7\pi }{4} \right )\)

[Picture]

(a) \(\left ( -5, \frac {3\pi }{4} \right )\)

(b) \(\left ( 5, -\frac {\pi }{4} \right )\)

(c) \(\left ( 5, \frac {15\pi }{4} \right )\)

\(\left ( \frac {1}{3}, \frac {3\pi }{2} \right )\)

[Picture]

(a) \(\left ( -\frac {1}{3}, \frac {\pi }{2} \right )\)

(b) \(\left ( \frac {1}{3}, -\frac {\pi }{2} \right )\)

(c) \(\left ( \frac {1}{3}, \frac {7\pi }{2} \right )\)

\(\left ( \frac {5}{2}, \frac {5\pi }{6} \right )\)

[Picture]

(a) \(\left ( -\frac {5}{2}, \frac {11\pi }{6} \right )\)

(b) \(\left ( \frac {5}{2}, -\frac {7\pi }{6} \right )\)

(c) \(\left ( \frac {5}{2}, \frac {17\pi }{6} \right )\)

\(\left ( 12, -\frac {7\pi }{6} \right )\)

[Picture]

(a) \(\left ( -12, \frac {11\pi }{6} \right )\)

(b) \(\left ( 12, -\frac {19\pi }{6} \right )\)

(c) \(\left (12, \frac {17\pi }{6} \right )\)

\(\left ( 3, -\frac {5\pi }{4} \right )\)

[Picture]

(a) \(\left ( -3, \frac {7\pi }{4} \right )\)

(b) \(\left ( 3, -\frac {13\pi }{4} \right )\)

(c) \(\left (3, \frac {11\pi }{4} \right )\)

\(\left ( 2\sqrt {2}, -\pi \right )\)

[Picture]

(a) \(\left ( -2 \sqrt {2}, 0 \right )\)

(b) \(\left ( 2 \sqrt {2}, -3\pi \right )\)

(c) \(\left (2\sqrt {2}, 3\pi \right )\)

\(\left ( \frac {7}{2}, -\frac {13\pi }{6} \right )\)

[Picture]

(a) \(\left ( -\frac {7}{2}, \frac {5\pi }{6} \right )\)

(b) \(\left ( \frac {7}{2}, - \frac {\pi }{6} \right )\)

(c) \(\left (\frac {7}{2}, \frac {23\pi }{6} \right )\)

\(\left ( -20, 3\pi \right )\)

[Picture]

(a) \(\left ( -20, \pi \right )\)

(b) \(\left ( 20, - 2\pi \right )\)

(c) \(\left (20,4\pi \right )\)

\(\left ( -4, \frac {5\pi }{4} \right )\)

[Picture]

(a) \(\left ( -4, \frac {5\pi }{4} \right )\)

(b) \(\left ( 4, - \frac {7\pi }{4} \right )\)

(c) \(\left (4, \frac {9\pi }{4} \right )\)

\(\left ( -1, \frac {2\pi }{3} \right )\)

[Picture]

(a) \(\left ( -1, \frac {2\pi }{3} \right )\)

(b) \(\left ( 1, - \frac {\pi }{3} \right )\)

(c) \(\left (1, \frac {11\pi }{3} \right )\)

\(\left ( -3, \frac {\pi }{2} \right )\)

[Picture]

(a) \(\left ( -3, \frac {\pi }{2} \right )\)

(b) \(\left ( 3, - \frac {\pi }{2} \right )\)

(c) \(\left (3, \frac {7\pi }{2} \right )\)

\(\left ( -3, -\frac {11\pi }{6} \right )\)

[Picture]

(a) \(\left ( -3, \frac {\pi }{6} \right )\)

(b) \(\left ( 3, - \frac {5\pi }{6} \right )\)

(c) \(\left (3, \frac {19\pi }{6} \right )\)

\(\left ( -2.5, -\frac {\pi }{4} \right )\)

[Picture]

(a) \(\left ( -2.5, \frac {7\pi }{4} \right )\)

(b) \(\left ( 2.5, - \frac {5\pi }{4} \right )\)

(c) \(\left (2.5, \frac {11\pi }{4} \right )\)

\(\left ( -\sqrt {5}, -\frac {4\pi }{3} \right )\)

[Picture]

(a) \(\left ( -\sqrt {5}, \frac {2\pi }{3} \right )\)

(b) \(\left ( \sqrt {5}, - \frac {\pi }{3} \right )\)

(c) \(\left (\sqrt {5}, \frac {11\pi }{3} \right )\)

\(\left ( -\pi , -\pi \right )\)

[Picture]

(a) \(\left ( -\pi ,\pi \right )\)

(b) \(\left ( \pi , - 2\pi \right )\)

(c) \(\left (\pi , 2\pi \right )\)

In Exercises polartorectfirst - polartorectlast, convert the point from polar coordinates into rectangular coordinates.
\(\left ( 5, \frac {7\pi }{4} \right )\)

\(\left ( \frac {5\sqrt {2}}{2}, -\frac {5\sqrt {2}}{2} \right )\)
\(\left ( 2, \frac {\pi }{3} \right )\)

\(\left ( \answer {1}, \answer {\sqrt {3}} \right )\)

\(\left ( 11, -\frac {7\pi }{6} \right )\)

\(\left ( -\frac {11\sqrt {3}}{2}, \frac {11}{2} \right )\)
\(\left ( -20, 3\pi \right )\)

\(\left ( \answer {20}, \answer {0} \right )\)

\(\left ( \frac {3}{5}, \frac {\pi }{2} \right )\)

\(\left ( 0, \frac {3}{5} \right )\)
\(\left ( -4, \frac {5\pi }{6} \right )\)

\(\left ( \answer {2\sqrt {3}}, \answer {-2} \right )\)

\(\left ( 9, \frac {7\pi }{2} \right )\)

\(\left ( 0, -9 \right )\)
\(\left ( -5, -\frac {9\pi }{4} \right )\)

\(\left ( \answer {-\frac {5\sqrt {2}}{2}}, \answer {\frac {5\sqrt {2}}{2}} \right )\)

\(\left ( 42, \frac {13\pi }{6} \right )\)

\(\left ( 21\sqrt {3}, 21 \right )\)
\(\left ( -117, 117\pi \right )\)

\(\left (\answer {117}, \answer {0} \right )\)

\(\left ( 6, \arctan (2) \right )\)

\(\left ( \frac {6\sqrt {5}}{5}, \frac {12\sqrt {5}}{5} \right )\)
\(\left (10, \arctan (3) \right )\)

\(\left (\answer {\sqrt {10}}, \answer {3\sqrt {10}} \right )\)

\(\left ( -3, \arctan \left (\frac {4}{3}\right ) \right )\)

\(\left ( -\frac {9}{5}, -\frac {12}{5} \right )\)
\(\left ( 5, \arctan \left (-\frac {4}{3}\right ) \right )\)

\(\left ( \answer {3},\answer {-4} \right )\)

\(\left ( 2, \pi - \arctan \left (\frac {1}{2}\right ) \right )\)

\(\left ( -\frac {4\sqrt {5}}{5}, \frac {2\sqrt {5}}{5} \right )\)
\(\left ( -\frac {1}{2}, \pi - \arctan \left (5\right ) \right )\)

\(\left ( \answer {\frac {\sqrt {26}}{52}}, \answer {-\frac {5\sqrt {26}}{52}} \right )\)

\(\left ( -1, \pi + \arctan \left (\frac {3}{4}\right ) \right )\)

\(\left ( \frac {4}{5}, \frac {3}{5} \right )\)
\(\left ( \frac {2}{3}, \pi + \arctan \left (2\sqrt {2}\right ) \right )\)

\(\left ( \answer {-\frac {2}{9}}, \answer {-\frac {4\sqrt {2}}{9}} \right )\)

\(\left ( \pi , \arctan (\pi ) \right )\)

\(\left ( \frac {\pi }{\sqrt {1+\pi ^2}}, \frac {\pi ^2}{\sqrt {1+\pi ^2}} \right )\)
\(\left ( 13, \arctan \left ( \frac {12}{5} \right ) \right )\)

\(\left ( \answer {5}, \answer {12} \right )\)

In Exercises recttopolarfirst - recttopolarlast, convert the given point from rectangular coordinates to polar coordinates. Choose \(r \geq 0\) and \(0 \leq \theta < 2\pi \). Remember it helps to plot the point first to help choose the correct value for \(\theta \).
\((0, 5)\)

\(\left ( 5, \frac {\pi }{2} \right )\)
\((3, \sqrt {3})\)

\(\left ( \answer {2\sqrt {3}}, \answer {\frac {\pi }{6}} \right )\)

\((7, -7)\)

\(\left ( 7\sqrt {2}, \frac {7\pi }{4} \right )\)
\((-3, -\sqrt {3})\)

\(\left ( \answer {2\sqrt {3}}, \answer {\frac {7\pi }{6}} \right )\)

\((-3, 0)\)

\(\left ( 3, \pi \right )\)
\(\left ( -\sqrt {2}, \sqrt {2} \right )\)

\(\left ( \answer {2}, \answer {\frac {3\pi }{4}} \right )\)

\(\left ( -4,-4\sqrt {3} \right )\)

\(\left ( 8, \frac {4\pi }{3} \right )\)
\(\left ( \frac {\sqrt {3}}{4}, -\frac {1}{4} \right )\)

\(\left ( \answer {\frac {1}{2}}, \answer {\frac {11\pi }{6}} \right )\)

\(\left ( -\frac {3}{10}, -\frac {3\sqrt {3}}{10} \right )\)

\(\left ( \frac {3}{5}, \frac {4\pi }{3} \right )\)
\(\left ( -\sqrt {5}, -\sqrt {5} \right )\)

\(\left ( \answer {\sqrt {10}}, \answer {\frac {5\pi }{4}} \right )\)

\((6,8)\)

\(\left ( 10, \arctan \left (\frac {4}{3}\right ) \right )\)
\((\sqrt {5},2\sqrt {5})\)

\(\left ( \answer {5}, \answer {\arctan \left (2\right )} \right )\)

\((-8,1)\)

\(\left ( \sqrt {65}, \pi - \arctan \left (\frac {1}{8}\right )\right )\)
\((-2\sqrt {10}, 6\sqrt {10})\)

\((\answer {20}, \answer {\pi - \arctan (3)})\)

\(\left (-5, -12 \right )\)

\(\left (13, \pi + \arctan \left (\frac {12}{5}\right ) \right )\)
\(\left (-\frac {\sqrt {5}}{15}, -\frac {2\sqrt {5}}{15} \right )\)

\(\left (\answer {\frac {1}{3}}, \answer {\pi + \arctan \left (2\right )} \right )\)

\(\left (24, -7 \right )\)

\(\left (25, 2\pi - \arctan \left (\frac {7}{24}\right ) \right )\)
\(\left (12, -9\right )\)

\(\left ( \answer {15}, \answer {2\pi - \arctan \left (\frac {3}{4} \right )} \right )\)

\(\left (\frac {\sqrt {2}}{4}, \frac {\sqrt {6}}{4}\right )\)

\(\left (\frac {\sqrt {2}}{2}, \frac {\pi }{3}\right )\)
\(\left (-\frac {\sqrt {65}}{5}, \frac {2\sqrt {65}}{5}\right )\)

\(\left (\answer {\sqrt {13}}, \answer {\pi - \arctan (2)} \right )\)

In Exercises equrecttopolarfirst - equrecttopolarlast, convert the equation from rectangular coordinates into polar coordinates. Solve for \(r\) in all but solvethetaone through solvethetafour. In Exercises solvethetaone - solvethetafour, solve for \(\theta \)

\(x = 6\)

\(r = 6\sec (\theta )\)
\(x = -3\)

\(r = -3\sec (\theta )\)
\(y = 7\)

\(r = 7\csc (\theta )\)
\(y = 0\)

\(\theta = 0\)

\(y = -x\)

\(\theta = \frac {3\pi }{4}\)
\(y = x\sqrt {3}\)

\(\theta = \frac {\pi }{3}\)
\(y = 2x\)

\(\theta = \arctan (2)\)
\(x^2 + y^2 = 25\)

\(r = 5\)
\(x^{2} + y^{2} = 117\)

\(r = \sqrt {117}\)
\(y = 4x - 19\)

\(r = \frac {19}{4\cos (\theta ) - \sin (\theta )}\)
\(x = 3y + 1\)

\(x = \frac {1}{\cos (\theta ) - 3\sin (\theta )}\)
\(y = -3x^{2}\)

\(r = \frac {-\sec (\theta )\tan (\theta )}{3}\)
\(4x = y^2\)

\(r = 4\csc (\theta )\cot (\theta )\)
\(x^2 + y^2 - 2y = 0\)

\(r=2\sin (\theta )\)
\(x^2 -4x + y^2 = 0\)

\(r = 4\cos (\theta )\)
\(x^2 + y^2 = x\)

\(r = \cos (\theta )\)
\(y^2 = 7y - x^2\)

\(r = 7\sin (\theta )\)
\((x+2)^2 + y^2 = 4\)

\(r= -4\cos (\theta )\)
\(x^{2} + (y - 3)^{2} = 9\)

\(r = 6\sin (\theta )\)
\(4x^2 + 4\left ( y - \frac {1}{2} \right )^2 = 1\)

\(r = \sin (\theta )\)
In Exercises equpolartorectfirst - equpolartorectlast, convert the equation from polar coordinates into rectangular coordinates.
\(r = 7\)

\(x^{2} + y^{2} = 49\)
\(r = -3\)

\(x^{2} + y^{2} = 9\)
\(r = \sqrt {2}\)

\(x^{2} + y^{2} = 2\)
\(\theta = \frac {\pi }{4}\)

\(y=x\)
\(\theta = \frac {2\pi }{3}\)

\(y = -\sqrt {3}x\)
\(\theta = \pi \)

\(y=0\)
\(\theta = \frac {3\pi }{2}\)

\(x=0\)
\(r = 4\cos (\theta )\)

\(x^2 + y^2 = 4x\) or \((x-2)^2 + y^2 = 4\)
\(5r = \cos (\theta )\)

\(5x^2 + 5y^2 = x\) or \(\left (x - \frac {1}{10}\right )^2+y^2 = \frac {1}{100}\)
\(r = 3\sin (\theta )\)

\(x^2 + y^2 = 3y\) or \(x^2 + \left (y - \frac {3}{2}\right )^2 = \frac {9}{4}\)
\(r = -2\sin (\theta )\)

\(x^2 + y^2 = -2y\) or \(x^2+(y+1)^2 = 1\)
\(r = 7\sec (\theta )\)

\(x=7\)
\(12r = \csc (\theta )\)

\(y = \frac {1}{12}\)
\(r = -2\sec (\theta )\)

\(x = -2\)
\(r = -\sqrt {5} \csc (\theta )\)

\(y= -\sqrt {5}\)
\(r = 2\sec (\theta )\tan (\theta )\)

\(x^2=2y\)
\(r = -\csc (\theta ) \cot (\theta )\)

\(y^2=-x\)
\(r^{2} = \sin (2\theta )\)

\(\left ( x^{2} + y^{2} \right )^{2} = 2xy\)
\(r = 1 - 2\cos (\theta )\)

\(\left ( x^{2} + 2x + y^{2} \right )^{2} = x^{2} + y^{2}\)
\(r = 1 + \sin (\theta )\)

\(\left ( x^{2} + y^{2} - y\right )^{2} = x^{2} + y^{2}\)
Convert the origin \((0,0)\) into polar coordinates in four different ways.

Any point of the form \((0, \theta )\) will work, e.g. \((0, \pi ), (0, -117), \left ( 0, \frac {23\pi }{4} \right ) \) and \((0, 0)\).
With the help of your classmates, use the Law of Cosines to develop a formula for the distance between two points in polar coordinates.