1. 1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved Chapter 7 Applications of Trigonometric Functions

2. OBJECTIVES Polar Coordinates SECTION 7.7 1 2 Plot points using polar coordinates. Convert points between polar and rectangular forms. Convert equations between rectangular and polar forms. Graph polar equations. 3 4

3. 3 © 2011 Pearson Education, Inc. All rights reserved POLAR COORDINATES In a polar coordinate system, we draw a horizontal ray, called the polar axis, in the plane; its endpoint is called the pole. A point P in the plane is described by an ordered pair of numbers (r,  ), the polar coordinates of P.

4. 4 © 2011 Pearson Education, Inc. All rights reserved POLAR COORDINATES The figure below shows the point P(r,  ) in the polar coordinate system, where r is the “directed distance” from the pole O to the point P and θ is a directed angle from the polar axis to the line segment OP.

5. 5 © 2011 Pearson Education, Inc. All rights reserved POLAR COORDINATES The polar coordinates of a point are not unique. The polar coordinates (3, 60º), (3, 420º), and (3, −300º) all represent the same point. In general, if a point P has polar coordinates (r,  ), then for any integer n, (r,  + n · 360º) or (r,  + 2nπ) are also polar coordinates of P.

6. 6 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding Different Polar Coordinates a. Plot the point P with polar coordinates (3, 225º). Find another pair of polar coordinates of P for which the following is true. b. r < 0 and 0º <  < 360º c. r < 0 and –360º <  < 0º d. r > 0 and –360º <  < 0º

7. 7 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding Different Polar Coordinates Solution b. r < 0 and 0º <  < 360º a.

8. 8 © 2011 Pearson Education, Inc. All rights reserved Solution continued EXAMPLE 1 Finding Different Polar Coordinates c. r < 0 and –360º <  < 0º d. r > 0 and –360º <  < 0º

9. 9 © 2011 Pearson Education, Inc. All rights reserved POLAR AND RECTANGULAR COORDINATES Let the positive x-axis of the rectangular coordinate system serve as the polar axis and the origin as the pole for the polar coordinate system. Each point P has both polar coordinates (r,  ) and rectangular coordinates (x, y).

10. 10 © 2011 Pearson Education, Inc. All rights reserved RELATIONSHIPS BETWEEN POLAR AND RECTANGULAR COORDINATES

11. 11 © 2011 Pearson Education, Inc. All rights reserved CONVERTING FROM POLAR TO RECTANGULAR COORDINATES To convert the polar coordinates (r,  ) of a point to rectangular coordinates (x, y), use the equations

12. 12 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 2 Converting from Polar to Rectangular Coordinates Convert the polar coordinates of each point to rectangular coordinates. Solution The rectangular coordinates of (2, –30º) are 2

13. 13 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 2 Converting from Polar to Rectangular Coordinates Solution continued The rectangular coordinates of are

14. 14 © 2011 Pearson Education, Inc. All rights reserved CONVERTING FROM RECTANGULAR TO POLAR COORDINATES To convert the rectangular coordinates (x, y) of a point to polar coordinates, follows these steps: 1.Find the quadrant in which the given point (x, y) lies. 2.Useto find r. 3.Find  by usingand choose  so that it lies in the same quadrant as the point (x, y).

15. 15 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 3 Convert from Rectangular to Polar Coordinates Find polar coordinates (r,  ) of the point P with r > 0 and 0≤ θ < 2π, whose rectangular coordinates are (x, y) = 2.2. Solution 1. The point Plies in quadrant II with

16. 16 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 3 Convert from Rectangular to Polar Coordinates 3.3. Solution continued Choose because it lies in quadrant II. The polar coordinates of

17. 17 © 2011 Pearson Education, Inc. All rights reserved CONVERTING EQUATIONS BETWEEN RECTANGULAR AND POLAR FORMS An equation that has the rectangular coordinates x and y as variables is called a rectangular (or Cartesian) equation. An equation where the polar coordinates r and  are the variables is called a polar equation. Some examples of polar equations are

18. 18 © 2011 Pearson Education, Inc. All rights reserved To convert a rectangular equation to a polar equation, replace x with r cos  and y with r sin , and then simplify where possible. CONVERTING EQUATIONS BETWEEN RECTANGULAR AND POLAR FORMS

19. 19 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 5 Converting an Equation from Rectangular to Polar Form Convert the equation x 2 + y 2 – 3x + 4 = 0 to polar form. Solution The equation form of the given rectangular equation. is the polar

20. 20 © 2011 Pearson Education, Inc. All rights reserved CONVERTING AN EQUATION FROM POLAR TO RECTANGULAR FORM Converting an equation from polar to rectangular form frequently requires some ingenuity in order to use the substitutions

21. 21 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Convert each polar equation to a rectangular equation and identify its graph. Solution Circle: center (0, 0) radius = 3 units

22. 22 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Solution continued Line through the origin with a slope of 1

23. 23 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Solution continued Horizontal line with y-intercept = 1

24. 24 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Solution continued Circle: Center (1, 0) radius = 1

25. 25 © 2011 Pearson Education, Inc. All rights reserved THE GRAPH OF A POLAR EQUATION To graph polar equations, we plot points in polar coordinates. The graph of a polar equation is the set of all points P(r,  ) that have at least one polar coordinate representation that satisfies the equation. Make a table of several ordered pair solutions (r,  ) of the equation, plot the points, and join them with a smooth curve.

26. 26 © 2011 Pearson Education, Inc. All rights reserved TESTS FOR SYMMETRY IN POLAR COORDINATES Symmetry with respect to the polar axis Replace (r,  ) with (r, –  ) or (–r, π –  ).

27. 27 © 2011 Pearson Education, Inc. All rights reserved Symmetry with respect to the line Replace (r,  ) with (r, π –  ) or (–r, –  ). TESTS FOR SYMMETRY IN POLAR COORDINATES

28. 28 © 2011 Pearson Education, Inc. All rights reserved Symmetry with respect to the pole Replace (r,  ) with (r, π +  ) or (–r,  ). TESTS FOR SYMMETRY IN POLAR COORDINATES

29. 29 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 8 Sketching the Graph of a Polar Equation Sketch the graph of the polar equation Solution Because cos  is even, the graph is symmetric about the polar axis. Thus, compute values for 0 ≤  ≤ π. θ0π r = 2(1 + cos θ)4 2 + ≈ 3.73 3212 – ≈ 0.270

30. 30 © 2011 Pearson Education, Inc. All rights reserved Solution continued EXAMPLE 8 Sketching the Graph of a Polar Equation This type of curve is called a cardioid because it resembles a heart.

31. 31 © 2011 Pearson Education, Inc. All rights reserved The graph of an equation of the form is a limaçon. If b > a, the limaçon has a loop. If b = a, the limaçon is a cardioid. LIMA Ç ONS

32. 32 © 2011 Pearson Education, Inc. All rights reserved LIMA Ç ONS

33. 33 © 2011 Pearson Education, Inc. All rights reserved ROSE CURVES The graph of an equation of the form is a rose curve. If n is odd, the rose has n petals. If n is even, the rose has 2n petals.

34. 34 © 2011 Pearson Education, Inc. All rights reserved ROSE CURVES

35. 35 © 2011 Pearson Education, Inc. All rights reserved CIRCLES

36. 36 © 2011 Pearson Education, Inc. All rights reserved LEMNISCATES

37. 37 © 2011 Pearson Education, Inc. All rights reserved SPIRALS