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

2. Copyright © 2011 Pearson Education, Inc. Slide 10.6-2 Chapter 10: Applications of Trigonometry and Vectors 10.1The Law of Sines 10.2The Law of Cosines and Area Formulas 10.3Vectors and Their Applications 10.4Trigonometric (Polar) Form of Complex Numbers 10.5Powers and Roots of Complex Numbers 10.6Polar Equations and Graphs 10.7More Parametric Equations

3. Copyright © 2011 Pearson Education, Inc. Slide 10.6-3 10.6Polar Equations and Graphs The Polar Coordinate System –Based on A point called the pole, and 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 10.6Rectangular and Polar Coordinates Rectangular and Polar Coordinates If a point has rectangular coordinates (x, y) and polar coordinates (r,  ), then these coordinates are related as follows.

5. Copyright © 2011 Pearson Education, Inc. Slide 10.6-5 10.6Plotting Points with Polar Coordinates ExamplePlot each point by hand in the polar coordinate system. Then determine the rectangular coordinates of each point. Solution (a)In this case, r = 2 and  = 30º, so the point P is located 2 units from the origin in the positive direction on a ray, making a 30º angle with the polar axis.

6. Copyright © 2011 Pearson Education, Inc. Slide 10.6-6 10.6Plotting Points with Polar Coordinates (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

7. Copyright © 2011 Pearson Education, Inc. Slide 10.6-7 10.6Plotting Points with Polar Coordinates (c) Since  is negative, the angle is measured in the clockwise direction. The rectangular coordinates

8. Copyright © 2011 Pearson Education, Inc. Slide 10.6-8 10.6Giving 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 10.6Giving Alternative Forms for Coordinates of a Point (b)(–1, 1) lies in quadrant II. Since one possible value for  is 135 º. Also, Therefore, two pairs of polar coordinates are (Any angle coterminal with 135 º could have been used.)

10. Copyright © 2011 Pearson Education, Inc. Slide 10.6-10 10.6Graphs of Polar Equations Equations such as r = 3 sin , r = 2 + cos , or r = , are examples of polar equations where r and  are the variables. The simplest equation for many types of curves turns out to be a polar equation. Evaluate r in terms of  until a pattern appears. Technology Note When graphing in polar coordinates, make sure the calculator is in polar mode.

11. Copyright © 2011 Pearson Education, Inc. Slide 10.6-11 10.6Graphing a Polar Equation (Cardioid) ExampleGraph r = 1 + cos . Analytic SolutionFind some ordered pairs until a pattern is found.  r = 1 + cos  0º0º 2180 º 0 30 º 1.9225 º 0.3 60 º 1.5255 º 0.7 90 º 1270 º 1 105 º 0.7300 º 1.5 135 º 0.3330 º 1.9 The curve has been graphed on a polar grid.

12. Copyright © 2011 Pearson Education, Inc. Slide 10.6-12 10.6Graphing a Polar Equation (Cardioid) Graphing Calculator Solution Choose degree mode and graph it for  in the interval [0 º, 360 º ].

13. Copyright © 2011 Pearson Education, Inc. Slide 10.6-13 10.6Graphing a Polar Equation (Lemniscate) ExampleGraph r 2 = cos 2 . SolutionComplete a table of ordered pairs.  0º0º ±1 30 º ±0.7 45 º 0 135 º 0 150 º ±0.7 180 º ±1 Values of  for 45 º <  < 135 º are not included because corresponding values of cos 2  are negative and do not have real square roots.

14. Copyright © 2011 Pearson Education, Inc. Slide 10.6-14 10.6Classifying Polar Equations

15. Copyright © 2011 Pearson Education, Inc. Slide 10.6-15 10.6Converting 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.

16. Copyright © 2011 Pearson Education, Inc. Slide 10.6-16 10.6Converting a Polar Equation to a Rectangular One Square both sides. Rectangular equation

17. Copyright © 2011 Pearson Education, Inc. Slide 10.6-17 10.6Converting 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