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Honors Pre-Calculus Chapter 8 Polar Coordinates Mrs. Kendall Chapter 8 – Assignment Guide 3/18 M: QTR EXAM T: p. 590 9-37 eoo, 39, 45-51 o W: p. 590 57-71 o, p. 607 1-15 o, 31-59 eoo (calc & sketch) R: Review F: 8.1 - 8.2 QUIZ SPRING BREAK!!! :) Honors Pre-Calculus 8.1 Introduction to Polar Coordinates Learning Targets: Students will be able to graph polar coordinates and rewrite polar coordinates to rectangular and vice- versa. . Until now, we would plot points in a plane using the RECTANGULAR COORDINATE SYSTEM. This is where a set of ordered pairs (x, y) are found on the x-axis and y-axis which intersect at the origin (0, 0). Plot: A (2, -3) B (-4, -1) There are other types of coordinate systems out there. We are going to study: POLAR COORDINATE SYSTEM The polar coordinate system pairs r , whose coordinates are found at a distance r (radius) along the polar axis from fixed point (pole) at an angle of rotation . For each ordered pair r , , if: rotated axis Pole + - +r -r rotate counterclockwise rotate clockwise plot point along rotated axis plot point opposite of rotated axis Polar Axis For simplicity, relations between polar and rectangular: Pole = Origin Polar axis = positive x-axis Plot each point given in polar coordinates. 10. A (4, 270°) 14. B 5, 5 3 16. C (-3, 120°) 18. D 3, 3 4 Find 3 other ordered pairs that have the same location as the given point. (You may not rotate 360 degrees/2π.) 22. 3 4, 4 Converting between polar and rectangular: Find all of the equations that relate polar and rectangular coordinates. P Find the rectangular coordinates of each point. 34. (5, 300°) 38. 3 3, 4 52. 2, 2 3 Find the polar coordinates of each point. 50. (-3, 3) Honors Pre-Calculus 8.1 & 8.2 Polar Equations and Graphs Learning Targets: Students will be able to convert polar equations to rectangular equations and vice-versa. Students will also be able to graph different types of polar graphs giving a minimum of four points. r x r cos cos x r y r sin sin y r x2 y 2 r 2 r x2 y 2 tan y x y x 8.1 – continued Write each equation using POLAR coordinates. (for consistency, always solve for r or r 2 in terms of ) 58. x2 y 2 x 60. y2 2x 72. r Write each equation using RECTANGULAR coordinates. (for consistency, always set final equation = 0, in terms of x and y). 70. r4 3 3 cos 8.2 – Polar Equations and Graphs Circles r# center @ pole tangent to pole, center @ (0, a) tangent to pole center @ (a, 0) r 2a sin r 2a cos Lines through the pole horizontal: vertical: Other # r sin # r cos # Cardioid: ex: r 2 2cos Limaçon: …without an inner loop ex: r 3 2cos …with an inner loop ex: r 1 2cos Rose: ex: r 3cos 2 Lemniscate: When graphing these, you must put at least four point (at the quadrantal values). Example: ex: r 2 4sin 2 r 2 2cos Spiral: ex: r e 5 Honors Pre-Calculus Review 8.1 & 8.2 Polar Coordinates Learning Targets: Students will be able to graph polar coordinates and rewrite polar coordinates to rectangular and vice- versa. . Plot and label the point on the graph provided. Find 3 other ordered pairs that have the same location as the given point. 4 7 2 7 1. A 3, 2. B 2, 5. 6. 2, 3, 3 6 3 4 3 3. C 1, 4. D 5, 4 2 Convert the given polar coordinates into rectangular coordinates. 7. Convert the given coordinates into polar coordinates. 9. 5, 5 10. 0, 3 . 5 5, 6 8. 5 3, 3 Learning Targets: Students will be able to convert polar equations to rectangular equations and vice-versa. Students will also be able to graph different types of polar graphs giving a minimum of four points. Convert the given rectangular equation into a polar equation. x 2 y 2 4 x 11. 3x 5 y 2 12. Convert the given polar equation into a rectangular equation. 13. r 7 16. 14. Graph. (Be sure to plot all of the key points.) r 3 3sin 15. 5. 2, Answers 1-4 6. 3, D 4 5 3, 3, 3 3 3 5 3 5 , 2 2 C B 3 3 5 2, 2, 4 4 4 7. A 9. 5 2, 5 4 11. r 4cos 13. x 2 y 2 49 0 3 3 3 2 2 8. , 3 2 10. 3, 2 3cos 5sin 2 14. x y 2 8 x 0 12. r 15-16. r 8cos Graph Paper