Pre-Calc Unit 15: Polar Assignment Sheet May 4 th to May 31 st, 2012

Transcription

1 Pre-Calc Unit 15: Polar Assignment Sheet May 4 th to May 31 st, 2012 Date Objective/ Topic Assignment Did it Friday Polar Discover y Activity Day 1 pp. 2-3 May 4 th Monday Polar Discover y Activity Day 2 pp. 4-5 May 7 th Tuesday Converting between Polar and pp May 8 th Rectangular systems. Notes pp. 6-8 Wednesday May 9 th Converting day 2 Text pp odd Thursday May10th Physics EOC field test Precalc work day QUIZ converting Friday Graphing Polar Equations pp May 11 th Notes p. 11 Monday Writing Equations from Graphs p. 15 May 14 th Notes p. 14 Tuesday Unit 15 Review Worksheet pp May 15 th Wednesday Unit 15 wrap up study May 16 th Thursday Unit 15 Test Print out review May 17 th Friday May 18 th Work on Polar Project Work on project or review Monday May 21 st Work on Polar Project Optional 4 th term test Work on project or review Tuesday May 22 nd Review Work on Polar Project Work on project or review Wednesday May 23 rd Review Work on review Study Thursday May 24 th Review Work on review Study Friday May 25 th 1 st exam seniors 6 th exam everyone Work on review Study Tuesday May 29 th 2 nd and 4 th exams everyone 5 th, 7 th period classes underclassmen 7 th exam seniors Study Wednesday 3 rd and 5 th exams everyone Study May 30 th Thursday 1 st and 7 th exams underclassmen Have a great May 31 st summer!!! Page 1

2 POLAR GRAPHS DISCOVERY ACTIVITY Put your graphing calculator in POLAR mode and RADIAN mode. Graph the following equation on your calculator, sketch the graphs on this sheet, and answer the questions. 1. r = 2cosθ 2. r = 3 cosθ 3. r = 3cosθ 4. r = 2sinθ 5. r = 3sinθ 6. r = 3 sinθ 7. What is similar about the graphs of #1-3? 8. How are they different? 9. What is similar about the graphs of #4-6? 10. How are they different? Page 2

3 11. r = 2 + 2cosθ 12. r = 1+ 2cosθ 13. r = 2 + cosθ 14. r = 2 + 2sinθ 15. r = 1+ 2sinθ 16. r = 2 + sinθ 17. What is similar about the graphs of #11-13? 18. How are they different? 19. What is similar about the graphs of #14-16? 20. How are they different? Page 3

4 POLAR GRAPHS DISCOVERY ACTIVITY II Put your graphing calculator in POLAR mode and RADIAN mode. Graph the following equation on your calculator, sketch the graphs on this sheet, and answer the questions. 1. r = 2cos3θ 2. r = 3 cos5θ 3. r = 4 cos7θ 4. r = 2sin3θ 5. r = 3sin5θ 6. r = 4 sin7θ 7. How does the coefficient affect the graphs? 8. How does the coefficient of the θ affect the graphs? Page 4

5 9. r = 3 cos2θ 10. r = 2cos4θ 11. r = 4 cos6θ 12. r = 3 sin2θ 13. r = 2sin4θ 14. r = 4 sin6θ 15. How does the coefficient affect the graphs? 16. How does the coefficient of the θ affect the graphs? Page 5

6 Polar Coordinates Notes The Polar Coordinate System is an alternative to the Cartesian system of rectangular coordinates for locating points in a plane. It consists of a fixed point O, called the pole or origin and a fixed ray OA, called the polar axis with O as its initial point. The polar coordinates of a fixed point P in the polar coordinate system consist of an ordered pair (r, θ). The directed distance from the pole to P is R, and the measure of the angle from the polar axis to OP is θ. P (r, θ) O A Both r and θ can be either positive or negative. When r is positive, the polar distance is measured from O along the terminal side of the angle θ, and when r is negative, it is measured from O on the opposite the terminal side of θ. When θ is positive, the polar angle is obtained by rotating OP counterclockwise from the polar axis, and when θ is negative, the rotation is clockwise. rθ- plane is a plane where polar coordinates (r, θ) are used to identify its points. Examples. Graph: 1) P ( 5, 60 ) 2) Q ( 5, -60 ) 3) W ( -5, 60 ) 4) V ( -5, -60 ) 5) A ( 3 150º) 6) B (-3, -150º) Rotations of θ and θ + 2nπ or θ n produce the same angle so there are infinitely many ways to represent the same angle. Examples: 1) Plot the point P (2, 45 ) and find 3 other 2) Plot the point P (1, π) and find 3 other polar representations of the point. polar representations of the point. Page 6

7 Polar Equation: an equation with polar coordinates Polar Graph: a graph of the set of all points (r, θ) that satisfy a given polar equation. The two most basic polar equations are: r = c a circle of radius c θ = a line through the origin that forms an angle θ with the polar axis Examples. 1) Sketch r = 3. 2) Sketch r = 2. 3) Sketch θ = 30. 4) Sketch θ = 45. If you superimpose a Rectangular Coordinate system over a Polar Coordinate system: y r = x + y so r = 2 x + y 2 P(x, y) x cos θ = so x = r cosθ r x y sin θ = so y = r sinθ r polar axis tan θ = y x Convert from Rectangular to Polar Coordinates. 1) ( 3, 3) 2) (2, 2 3 ) 3) (0, -2) 4) ( 4 3, 4) Convert from Polar to Rectangular Coordinates. 1) (-2, π) 2) (3, 135 ) 3) ( -5, 240 ) 4) (4, 6 π ) Page 7

8 Convert the Polar Equations to Rectangular form. 1) r = 1 2) θ = 45 3) r = 5secθ 4) r = 4cscθ 5) r = 3sinθ 6) 6 r = 7) 2cosθ 3sinθ r = 2 2 cosθ Convert the rectangular Equations to Polar form. 1) 5 x + 7 y = 12 2) x = 11 3) y = ) x + y = 9 5) (y 2) 2 + x 2 = 4 Page 8

9 Polar Coordinates Homework Convert from Rectangular to Polar Coordinates then graph A ( 3, 3 3 ) B (4, 4 3 ) C (0, 5) D ( 3, 1 ) E (5, 5) Graph then, Convert from Polar to Rectangular Coordinates. F (1, π π ) G (6, 120 ) H ( 4, 270 ) I (2, ) J (3, π) 2 4 Give 3 additional coordinates for the points given. 1) ( 1, 45º) 2) ( 2, 210 ) Page 9

10 Convert the Polar Equations to Rectangular form 1) r = 3 2) θ = 30 3) r = 7secθ 4) r = 8cscθ 5) r = 2sinθ 6) 5 r = 7) 4cosθ 2sinθ 3 r = 1 sinθ Convert the rectangular Equations to Polar form. 1) 3 x 5y = 8 2) x = 4 3) y = ) x + y = 16 5) y 2 + (x 3) 2 = 9 Page 10

11 Notes: Graphing Polar Equations Circles: The form The form r = a cosθ and r = asinθ where a is the diameter of the graph r = a cosθ is symmetrical about the polar ( horizontal ) axis π r = asinθ is symmetrical about the line θ =. 2 Limacons: Cardiod: r = a ± bsinθ and r = a ± bcosθ a = b Heart Inner Loop: a < b Indentation: a > b one side also may appear flat Roses: r = a cosbθ and r = asin bθ If b is odd there are b petals. If b is even there are 2b petals. Make and Fill in the table and Graph each polar equation. a = the length of the petal 1. r = 4sin3θ 2. r = 2 + 2sinθ 3. r = 1 2cosθ 4. r = 3 + 2cosθ Page 11

12 Homework: Graphing Polar Equations For each equation, make a table then graph. 1. r = 4cos4θ 2. r = 4cos3θ 3. r = 5cosθ 4. r = 4sinθ 5. r = 6cos3θ 6. r = 5sin3θ Page 12

13 7. r = 3 3cosθ 8. r = 3 + 2cosθ 9. r = 2 3cosθ 10. r = 3 + 2sinθ 11. r = 2 4sinθ 12. r = 2 + 2sinθ 13. r = 1+ 4sinθ 14. r = 2 2cosθ Page 13

14 Notes: Writing Polar Equations 1) 2) 3) 4) 5) 6) 7) 8) 9) Page 14

15 Homework: Writing Polar Equations 1) 2) 3) 4) 5) 6) 7) 8) 9) Page 15

16 Pre Calc Review Polar Test 1. Graph each point on the Polar grid on page 18. A ( 4, π 2) B ( 3,5π 6) C ( 2,225 ) D ( 1, 300 ) E ( 5, 270 ) F ( 3,3π 4) G ( 4, 30 ) H ( 2,240 ) 2. Convert from Rectangular to Polar Coordinates. a. ( 5, 5) b. ( 3 3, 3) c. ( 4, 0) d. 1, Convert from Polar to Rectangular Coordinates. a. ( 3, 60 ) b. ( 2, π 2) c. ( 5,2π 3) d. (4, 210 ) 4. Give 3 additional coordinates for each of the points. a. ( 4, π 6) b. (3, 50 ) c. (2, 220 ) d. (5, 90 ) 5. Convert to Polar form. a. 2x + 4y = 7 b. x = 5 c. y = 3 d. (x 3) 2 + y 2 = 9 6. Convert to Rectangular form. a. r = 3cscθ b. r = 5cosθ c. r 5cosθ = 7sinθ d. r = 2 cosθ 3sinθ 7. Graph on a Polar grids on page 18. a. r = 2secθ b. r = 3cosθ c. r = 5sin3θ d. r = 4cos2θ e. r = 3 + 2cosθ f. r = 2 2sinθ g. r = 2 3sinθ Page 16

17 8. Matching: Use each letter twice. a. horizontal line b. vertical line c. oblique line d. circle with center (0, 0) e. circle: center on y-axis f. circle: center on x-axis g. limacon with inner loop h. limacon with indentation i. cardiod j. rose 1. r = 3secθ 2. r = 4cosθ 3. θ = r = 2sinθ 5. r = 6cscθ 6. r = 3 + 3sinθ 7. r = 3 8. r = 3 2sinθ 9. r = 7secθ 10. r = 2 3sinθ 11. r = 5cos3θ 12. r = 6cosθ 13. r = 4 + 2sinθ 14. r = 3cscθ 15. r = 4 + 5cosθ 16. r = r = 5 5cosθ 18. r = 3sin2θ r = 20. r = 6sinθ 2cosθ - 5sinθ 9. Write the polar equation. a) b) c) Page 17

18 Use for #1 8a. 8b. 8c. 8d. 8e. 8f. 8g. Page 18