Pre-Calc Unit 14: Polar Assignment Sheet April 27 th to May 7 th 2015 Date Objective/ Topic Assignment Did it Monday Polar Discovery Activity pp. 4-5 April 27 th Tuesday April 28 th Converting between Polar and Rectangular systems. Notes pp. 6-8 Graphing Polar Equations pp. 9-10 Wednesday April 29 th Notes p. 11 Thursday Writing Equations from Graphs April 30 th Notes p. 14 Friday Unit 14 Review pp. 17-18 study May 1 st Monday Unit 14 test pp. 12-13 p. 15 review p.16 May 4 th Tuesday Work on Polar Project Work on project May 5 th Wednesday Work on Polar Project Work on project May 6 th Thursday Work on then Turn in Polar Project Print out last unit May 7 th Page 1
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 2cos 2. r 3cos 3. r 3cos 4. r 2sin 5. r 3sin 6. r 3sin 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
11. r 2 2cos 12. r 1 2cos 13. r 2 cos 14. r 2 2sin 15. r 1 2sin 16. r 2 sin 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
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 2cos 3 2. r 3cos5 3. r 4 cos 7 4. r 2sin3 5. r 3sin5 6. r 4 sin7 7. How does the coefficient affect the graphs? 8. How does the coefficient of the affect the graphs? Page 4
9. r 3cos 2 10. r 2cos 4 11. r 4 cos 6 12. r 3sin2 13. r 2sin4 14. r 4 sin6 15. How does the coefficient affect the graphs? 16. How does the coefficient of the affect the graphs? Page 5
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 + 360 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
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 2 2 2 x y so r = 2 x y 2 P(x, y) x cos so x r cos r x y sin so y r sin r polar axis tan 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 ) Page 7
Convert the Polar Equations to Rectangular form. Identify. 1) r = 1 2) θ = 45 3) r 5sec 4) r 4csc 5) r 3sin 6) 6 r 7) 2cos 3sin r 2 2 cos Convert the rectangular Equations to Polar form. Identify. 1) 5x 7y 12 2) x = 11 3) y = 6 2 2 4) x y 9 5) (y 2) 2 + x 2 = 4 Page 8
Polar Coordinates Homework April 28 th 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, ) 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
Convert the Polar Equations to Rectangular form. Identify. 1) r = 3 2) θ = 30 3) r 7sec 4) r 8csc 5) r 2sin 6) 5 r 7) 4cos 2sin 3 r 1 sin Convert the rectangular Equations to Polar form. Identify. 1) 3x 5y 8 2) x = 4 3) y = 9 2 2 4) x y 16 5) y 2 + (x 3) 2 = 9 Page 10
Notes: Graphing Polar Equations Circles: The form The form r acos and r asin where a is the diameter of the graph r acos is symmetrical about the polar ( horizontal ) axis r asin is symmetrical about the line. 2 Limacons: Cardiod: r a bsin and r a bcos a b Heart Inner Loop: a b Indentation: a b one side also may appear flat Roses: r acos b and r asin b If b is odd there are b petals. If b is even there are 2b petals. a = the length of the petal Make and Fill in a table then Graph each polar equation. Identify. 1. r 4sin3 2. r 2 2sin 3. r 1 2cos 4. r 3 2cos Page 11
Homework: Graphing Polar Equations April 29 th For each equation, make a table then graph. Identify. 1. r 4cos 4 2. r 4cos3 3. r 5cos 4. r 4sin 5. r 6cos3 6. r 5sin3 Page 12
7. r 3 3cos 8. r 3 2cos 9. r 2 3cos 10. r 3 2sin 11. r 2 4sin 12. r 2 2sin 13. r 1 4sin 14. r 2 2cos Page 13
Notes: Writing Polar Equations 1) 2) 3) 4) 5) 6) 7) 8) 9) Page 14
Homework: Writing Polar Equations April 30 th 1) 2) 3) 4) 5) 6) 7) 8) 9) Page 15
Review Polar Test Homework April 30 th 1. Graph each point on the Polar grid on page 18. A ( 4, 2) B ( 3,5 6) C ( 2,225 ) D ( 1, 300 ) E ( 5, 270 ) F ( 3,3 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, 4 3 4 3. Convert from Polar to Rectangular Coordinates. a. ( 3, 60 ) b. ( 2, 2) c. ( 5,2 3) d. (4, 210 ) 4. Give 3 additional coordinates for each of the points. a. ( 4, 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. 2 r cos 3sin 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
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 = 60 4. 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 = 7 17. r = 5 5cosθ 18. r = 3sin2θ 19. 3 r 20. r = 6sinθ 2cos - 5sin 9. Write the polar equation. a) b) c) Page 17
Use for #1 8a. 8b. 8c. 8d. 8e. 8f. 8g. Page 18