Math 9 Quiz Review Solution Key

9.6-9.7 Quiz Review Name: Hour: HPC Plot the point that has the given polar coordinates. Label each point! 1. (3, 5π/4) 2. (-4, 2π/3) 3. (1, 7π/4) 4. (3, -13π/6) 5. (-2, 5π/6) 6. (-1, -7π) 7. (-3, -5π/6) 8. (-2, -π/3) 9. (4, 4π/3) 10. (-1, -3π/4) 11. (-2, 6π) 12. (3, -7π/6) Determine which point in the figure, P, Q, R, or S has the given polar coordinate. 13. (2, -π/6) S 14. (2, π/4) R 15. (-2, -5π/6) P 16. (-2, -11π/6) R 17. (-2, -11π/6) R 18. (2, 11π/6) P 19. (2, 7π/6) S Find three other polar coordinate representations of the point where -2π < θ < 2π. 20. (-5, -π/6) (5, 5π/6) (-5, 11π/6) (5, -7π/6) 21. (8, 2π/3) (-8, -π/3) (-8, 5π/3) (8, -4π/3) 22. (10, π/4) (-10, 5π/4) (-10, -3π/4) (10, -7π/4) 23. (-3, -2π/3) (3, π/3) (-3, 4π/3) (3, -5π/3) 24. (-2, -π/2) (2, π/2) (-2, 3π/2) (2, -3π/2) 25. (-1, -5π/6) (1, π/6) (-1, 7π/6) (1, -11π/6) Find the corresponding rectangular coordinates for the given polar coordinate. 26. (4, 2π/3) x = 4 cos(2π/3) = 4(-1/2) = -2 y = 4 sin(2π/3) = 4(√3/2) = 2√3 (-2, 2√3) 27. (3, -7π/6) x = 3 cos(-7π/6) = 3(-√3/2) = -3√3/2 y = 3 sin(-7π/6) = 3(1/2) = 3/2 (-3√3/2, 3/2) 28. (6, 17π/6) x = 6 cos(17π/6) = 6(0) = 0 y = 6 sin(17π/6) = 6(-1/2) = -3 (0, -3) 29. (-4, 13π/6) x = -4 cos(13π/6) = -4(√3/2) = -2√3 y = -4 sin(13π/6) = -4(1/2) = -2 (-2√3, -2) 30. (3, 8π) x = 3 cos(8π) = 3(1) = 3 y = 3 sin(8π) = 3(0) = 0 (3, 0) 31. (6.42, 3.59) (Calculator!) x = 6.42 cos(3.59) ≈ -5.79 y = 6.42 sin(3.59) ≈ -2.78 (-5.79, -2.78) Find the corresponding polar coordinates for the given rectangular coordinate where 0 ≤ θ < 2π. 32. (-5, -5) r = √((-5)² + (-5)²) = 5√2 θ = arctan(-5/-5) = π/4 + π = 5π/4 (5√2, 5π/4) 33. (-√3, 1) r = √((-√3)² + 1²) = 2 θ = arctan(1/-√3) = -π/6 + π = 5π/6 (2, 5π/6) 34. (4, -4√3) r = √(4² + (-4√3)²) = 8 θ = arctan(-4√3/4) = -π/3 + 2π = 5π/3 (8, 5π/3) 35. (0, 8) r = √(0² + 8²) = 8 θ = arctan(8/0) = π/2 (8, π/2) 36. (-1, 0) r = √((-1)² + 0²) = 1 θ = arctan(0/-1) = 0 + π = π (1, π) 37. (-12, 5) r = √((-12)² + 5²) = 13 θ = arctan(5/-12) ≈ -0.395 (13, 2.75) Convert the rectangular equation to polar form. 38. x = 5 r cosθ = 5 r = 5 secθ 39. 4y + x - 2 = 0 4r sinθ + r cosθ = 2 r(4sinθ + cosθ) = 2 r = 2 / (4sinθ + cosθ) 40. x² + y² + 4y = 0 r² + 4r sinθ = 0 r(r + 4sinθ) = 0 r = -4sinθ 41. x² + y² + 8x = 0 r² + 8r cosθ = 0 r(r + 8cosθ) = 0 r = -8cosθ Convert the polar equation to rectangular form. 42. θ = 3π/4 tan⁻¹(y/x) = 3π/4 y/x = tan(3π/4) y = -x 43. r = 3sinθ x² + y² = 3y x² + y² - 3y = 0 44. r = 4 r² = 16 x² + y² = 16