ISM_T11_C10_E - Section 10.5 Polar Coordinates 10.5 POLAR...

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Section 10.5 Polar Coordinates 647 10.5 POLAR COORDINATES 1. a, e; b, g; c, h; d, f 2. a, f; b, h; c, g; d, e 3. (a) 2 2n and 2 (2n 1) , n an integer ˆ‰ ˆ ß± ² ß± ± 11 ## (b) ( 2n ) and ( (2n 1) ), n an integer ²#ß ± (c) 2 2n and 2 (2n 1) , n an integer ˆ ² 33 (d) ( (2n 1) ) and ( 2n ), n an integer ± ²#ß 4. (a) 3 2n and 3 2n , n an integer ˆ ² ß ± 44 5 (b) 3 2n and 3 2n , n an integer ˆ ²ß ± 5 (c) 3 2n and 3 2n , n an integer ˆ ß² ± ² ß ± 3 (d) 3 2n and 3 2n , n an integer ˆ ²ß² ± 3 5. (a) x r cos 3 cos 0 3, y r sin 3 sin 0 0 Cartesian coordinates are ( 0) œœ œ œœœ Ê $ ß )) (b) x r cos 3 cos 0 3, y r sin 3 sin 0 0 Cartesian coordinates are ( 0) ²œ ² ² œ Ê ² $ ß (c) x r cos 2 cos 1, y r sin 2 sin 3 Cartesian coordinates are 1 3 œ ² œ Ê ² ß 22 ÈÈ Š‹ (d) x r cos 2 cos 1, y r sin 2 sin 3 Cartesian coordinates are 1 3 œ œ Ê ß 77 (e) x r cos 3 cos 3, y r sin 3 sin 0 Cartesian coordinates are (3 0) Ê ß )1 (f) x r cos 2 cos 1, y r sin 2 sin 3 Cartesian coordinates are 1 3 œ Ê ß (g) x r cos 3 cos 2 3, y r sin 3 sin 2 0 Cartesian coordinates are ( 3 0) ² œ ² ² œ Ê ² ß (h) x r cos 2 cos 1, y r sin 2 sin 3 Cartesian coordinates are 1 3 ² ² œ ² ² ² œ Ê ² ß 6. (a) x 2 cos 1, y 2 sin 1 Cartesian coordinates are (1 1) Ê ß (b) x 1 cos 0 1, y 1 sin 0 0 Cartesian coordinates are (1 0) Ê ß (c) x 0 cos 0, y 0 sin 0 Cartesian coordinates are ( 0) Ê ! ß (d) x 2 cos 1, y 2 sin 1 Cartesian coordinates are ( 1 1) œ² œ² Ê ² ß² (e) x 3 cos , y 3 sin Cartesian coordinates are œ ß² 55 3 3 62 6 È È # (f) x 5 cos tan 3, y 5 sin tan 4 Cartesian coordinates are ( 4) Ê $ ß ±" ±" (g) x 1 cos 7 1, y 1 sin 7 0 Cartesian coordinates are (1 0) œ œ Ê ß (h) x 2 3 cos 3, y 2 3 sin 3 Cartesian coordinates are 3 3 ² Ê ² ß È È
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648 Chapter 10 Conic Sections and Polar Coordinates 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
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Section 10.5 Polar Coordinates 649 22. 23. r cos 2 x 2, vertical line through ( 0) 24. r sin 1 y 1, horizontal line through (0 1) )) œÊœ # ß œ ±Êœ ± ß ± 25. r sin 0 y 0, the x-axis 26. r cos 0 x 0, the y-axis 27. r 4 csc r r sin 4 y 4, a horizontal line through (0 4) œÊ œ Ê œ Ê œ ß 4 sin ) 28. r 3 sec r r cos 3 x 3, a vertical line through ( 3 0) œ± Ê œ Ê œ± Ê œ± ± ß ± 3 cos ) 29. r cos r sin 1 x y 1, line with slope m 1 and intercept b 1 ²œ Ê ² œ œ ± œ 30. r sin r cos y x, line with slope m 1 and intercept b 0 œ œ œ 31. r 1 x y 1, circle with center C ( 0) and radius 1 ## # œÊ ²œ œ ! ß 32. r 4r sin x y 4y x y 4y 4 4 x (y 2) 4, circle with center C (0 2) and radius 2 # # # # # ² œ Ê ² ± ² œ Ê ² ± œ œ ß ) 33. r r sin 2r cos 5 y 2x 5, line with slope m 2 and intercept b 5 ± œ Ê ± œ œ œ 5 sin 2 cos ± 34. r sin 2 2 2r sin cos 2 (r sin )(r cos ) 1 xy 1, hyperbola with focal axis y x ) ) ) œÊ œ œ 35. r cot csc r sin cos r sin r cos y x, parabola with vertex (0 0) œœ Ê œ Ê œ Ê œ ß ) ) ) ) ˆ‰ cos sin sin ) " # # which opens to the right 36. r 4 tan sec r 4 r cos 4 sin r cos 4r sin x 4y, parabola with œ Ê œ Ê œ Ê œ ) ) ) ) sin cos ) ) # # # vertex ( 0) which opens upward œ!
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ISM_T11_C10_E - Section 10.5 Polar Coordinates 10.5 POLAR...

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