question 11-22

744 Chapter 11 Analytic Geometry In Exercises 11-16, convert the rectangular coordinates 48. a. Find a complete graph of r = 1 - 3 sin 26 b. Predict what the graph of r = 1 3 sin 30 wi to polar coordinates. look like . Then check your prediction with a calculator. 17. (3V/3, -3) 12. (2V/3 , -2) 13. ( 2, 4 ) C. Predict what the graph of r = 1 - 3 sin 40) with 14. (3 , - 2) 15. (-5 , 2.5 ) 16. (-6.2, -3) look like. Then check your prediction with a calculator. In Exercises 17-22, sketch the graph of the equation without using a calculator. 49. If a and b are constants such that ab * 0, show that the graph of r = a sin 0 + b cos 0 is a circle. 17. 1 = 4 19. 0 = - 3 Hint: Multiply both sides by r and convert to 18. r = -1 rectangular coordinates. 20. 0 = 5 TT 22. 0 = -4 50. Critical Thinking Prove that the coordinate 6 21. 0 = 1 conversion formulas are valid when r 0, 31. r = sin 30 32. r = sin 40 s > 0, and 0 > B, then the triangle with vertices (r, 0), (s, B), (0, 0) has an angle of 0 - B, whose 33. 12 = 4 cos 20 34. 12 = sin 20 sides have lengths r and s. Use the Law of Cosines. 35. r = 2 + 4 cos 0 36. r = 1 + 2 cos 0 52. Critical Thinking Explain why the following symmetry tests for the graphs of polar equations 37. r = sin 0 + cos 0 38. r = 4 cos 0 + 4 sin 0 are valid. a. If replacing 0 by -0 produces an equivalent 39. r = sin ? 40. r = 4 tan 0 equation, then the graph is symmetric with respect to the line 0 = 0 (the x-axis). 41. r = sin 0 tan 0 (cissoid) b. If replacing 0 by T - 0 produces an equivalent equation, then the graph is symmetric with 42. r = 4 + 2 sec0 (conchoid) respect to the line 0 = TT/2 (the y-axis). c. If replacing r by -r produces an equivalent 43. r = e (logarithmic spiral) equation, then the graph is symmetric with respect to the pole (origin). 44. 12 = 45. 1 = (0 > 0) 46. 12 = 0 47. a. Find a complete graph of r = 1 - 2 sin 30. b. Predict what the graph of r = 1 - 2 sin 40 will look like. Then check your prediction with a calculator. c. Predict what the graph of r = 1 - 2 sin 50 will look like. Then check your prediction with a calculator