1. Copyright © 2011 Pearson Education, Inc. Slide 10.6-1 Cartesian vs. Polar

2. Copyright © 2011 Pearson Education, Inc. Slide 10.6-2 Relationship… …using Pythagorean Theorem: r 2 = x 2 + y2

3. Copyright © 2011 Pearson Education, Inc. Slide 10.6-3 The Polar Coordinate System –Based on A point called the pole (often the origin) A ray called the polar axis, usually drawn in the direction of the positive x-axis. The ordered pair P(r,  ) gives the polar coordinates of point P.

4. Copyright © 2011 Pearson Education, Inc. Slide 10.6-4 Plotting Points with Polar Coordinates ExamplePlot the point by hand in the polar coordinate system. Then determine the rectangular coordinates of each point.

5. Copyright © 2011 Pearson Education, Inc. Slide 10.6-5 Negative r value… (b) Since r is –4, Q is 4 units in the negative direction from the pole on an extension of the ray. The rectangular coordinates

6. Copyright © 2011 Pearson Education, Inc. Slide 10.6-6 Negative q value … (c) Since  is negative, the angle is measured in the clockwise direction. The rectangular coordinates

7. Copyright © 2011 Pearson Education, Inc. Slide 10.6-7 Formulas Rectangular and Polar Coordinates If a point has rectangular coordinates (x, y) and polar coordinates (r,  ), then these coordinates are related as follows.

8. Copyright © 2011 Pearson Education, Inc. Slide 10.6-8 Giving Alternative Forms for Coordinates of a Point Example (a)Give three other pairs of polar coordinates for the point P(3, 140º). (b)Determine two pairs of polar coordinates for the point with rectangular coordinates (–1, 1). Solution (a)See the figure: (3, –220 º ), (–3, 320 º ), and (–3, –40 º ).

9. Copyright © 2011 Pearson Education, Inc. Slide 10.6-9 Classifying Polar Equations

10. Copyright © 2011 Pearson Education, Inc. Slide 10.6-10 Graphing a Polar Equation (Cardioid) Graphing Calculator Solution Under the mode key choose DEGREE and POL to graph a polar equation, graph it for  in the interval [0 º, 360 º ].

11. Copyright © 2011 Pearson Education, Inc. Slide 10.6-11 Converting a Polar Equation to a Rectangular One ExampleFor the polar equation (a)convert to a rectangular equation, (b)use a graphing calculator to graph the polar equation for 0    2 , and (c)use a graphing calculator to graph the rectangular equation. SolutionMultiply both sides by the denominator.

12. Copyright © 2011 Pearson Education, Inc. Slide 10.6-12 Converting a Polar Equation to a Rectangular One Square both sides. Rectangular equation

13. Copyright © 2011 Pearson Education, Inc. Slide 10.6-13 Converting a Polar Equation to a Rectangular One (b)The figure shows a graph with polar coordinates. (c)Solving x 2 = –8(y – 2) for y, we obtain