Polar Coordinates Packet 2012

Academic Pre-Calculus

Chapter 10 Topics in Analytic Geometry

10.7 Polar Coordinates 10.8 Graphs of Polar Equations

Name: __________________ Teacher: _____________ Period: _____

POLAR COORDINATES The position of a point P in the plane can be described using polar coordinates instead of rectangular coordinates.

Rectangular Coordinates Polar Coordinates

Origin (0, 0) Pole O

x-axis and y-axis Polar Axis (Positive x-axis)

P = (x, y)

x = horizontal distance from origin (independent variable)

y = vertical distance from origin (dependent variable)

P = (r, )

r = distance from the Pole (dependent variable)

= angle in standard position with the terminal side through P (independent variable)

1 Point = 1 Ordered Pair 1 Point = Many Polar Coordinates

O (Pole) Polar Axis

r

P (r, )

Plot the following points on the polar grid below:

1. A

3,3 , B

6,4 , C

2,2 , D

4

7,3 , E

6

5,5

2. Name point F three other ways where -2 2.

F

3

2,4 a) ________ b) ________ c) ________

3. Name point F two other ways where 2 4.

F

3

2,4 a) ________ b) ________

4. Name point G three other ways where -2 2.

G

4

3,5.2 a) ________ b) ________ c) ________

Converting Between Polar and Rectangular Coordinates I. Polar Form to Rectangular Form:

5.

35

,8 6.

2

3,2

7. (6,0 )

P

(r, ) (x, y)

x = _______________ y = _______________

II. Rectangular Form to Polar Form: 8. (1, -1) 9. (-3, 3) 10. (3.2, 0) 11. (0, -8)

P

(x, y) (r, )

r = _______________

tan _________

MUST PUT IN CORRECT QUADRANT!

Name ____________________________ Date__________ Period_____

WORKSHEET POLAR COORDINATES

NON-CALCULATOR Plot each point.

1. A

4,2 2. B

6

5,3

3. C

3,5 4. D

2,4

5. E 2,5 6. F

3

2,4

Name each point using polar coordinates four ways where -2 2.

7. G = ____________, ____________ ____________, ____________ 8. H = ____________, ____________ ____________, ____________ 9. I = ____________, ____________ ____________, ____________ 10. J = ____________, ____________ ____________, ____________

Name each point two ways using

polar coordinates where 2 4.

11. I = ____________, ____________ 12. J = ____________, ____________

I

G J

H

NON-CALCULATOR For each point (r, ), give rectangular coordinates (x, y).

13.

3

2,4 __________ 14.

2,3 __________

15.

6

11,10 __________ 16.

3

2,5 __________

For each point (x, y), give polar coordinates (r, ), where is in radians.

17. (-2, 2) __________ 18. (5, 0) __________

19. 2,2 __________ 20. 1,3 __________ 21. (0, 12) __________ 22. (-2, 0) __________

WARM UP

Plot and label each point:

1) 2

2,3

2) 4,

4

3)

52,

6

For each point (r, ), give rectangular coordinates (x, y)

4) 5

3,4

For each point (x, y), give polar coordinates (r, ), where is in radians.

5) 3, 1

GRAPHS OF SIMPLE POLAR EQUATIONS Graph each of the following polar equations. Then describe and sketch each graph in the box below.

Simple Polar Graphs

1. r = a ________________

2. = a ________________

3. rsin = a ______________

4. rcos = a ______________

5. r = asin _______________

6. r = acos _______________

1. r = 3 2. 4

3. rsin = 2 4. rcos = 1 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 5. r = 4sin 6. r = 4cos

Name __________________________ Date_________ Period_______

WORKSHEET SIMPLE POLAR GRAPHS

Non-Calculator Identify and graph each polar equation.

1. r = 4 _________________ 2. 3

2 _________________

3. sin3r _________________ 4. 4sinr _________________

5. cosr _________________ 6. 4

3 _________________

7. sin4r _________________ 8. r = -5 _________________

9. 2cosr _________________ 10. cos2r _________________

Write the equation for each graph.

11. ________________________ 12. _________________________

13. ________________________ 14. _________________________

15. ________________________ 16. _________________________

Plot each point and write three other polar coordinates for the point where

-2 2.

17. A

2,3

__________, __________, __________

18. B

6

11,4

__________, __________, __________

19. C

3

2,2

__________, __________, __________

Warm Up

Write the equation of the following:

1) _____________________ 2) _____________________

3) ______________________ 4) ______________________

POLAR GRAPHS: CARDIOIDS AND LIMAONS

Remember: 2periodb

and

4

periodincrements

I. r = a + acos Make a table and graph: r = 2 + 2cos This graph is called a __________________.

Sketch the graph. 1. r = 2 2cos 2. r = 2 + 2sin 3. r = 2 2sin

(0, -1)

(1, 0) (-1, 0)

(0, 1)

II. r = a + bcos, a > b Make a table and graph: r = 3 + 2cos This graph is called a ________________________. Sketch the graph: 4. r = 4 + cos 5. r = 4 + 2cos

III. r = a + bcos, a < b Make a table and graph: r = 2 + 4cos This graph is called a _________________________. Sketch the graph: 1. r = 3 + 4cos 2. r = 1 + 4cos

CARDIOIDS AND LIMAONS

For all equations: a>0, b> 0

Cardioid r = a acos r = a asin

Limaon (No Loop) a>b

r = a bcos r = a bsin

Limaon (With Loop) a b

Name ____________________________ Date__________ Period_____

WORKSHEET: CARDIOIDS AND LIMACONS

Graph each equation without using a calculator and name the type of graph.

1. r = 3 + 3sin _________________ 2. r = 2 + 3cos __________________

3. r = 5 2sin _________________ 4. r = 3 3cos _________________ 5. r = 2 + 5sin _________________

6. r = 2 sin _________________ 7. r = 1 + 4sin __________________ 8. r = 3sin _________________ 9. r = 4 3cos _________________ 10. r = 2 + 5sin _________________ 11. rcos = -5 _________________

12. r = -6 _________________ 13. = 4

__________________

14. r = 1 cos _________________ 15. r = 6 + cos _________________

For each point (r, ), give rectangular coordinates (x, y).

16.

4

5,6 __________ 17. ,10 __________

For each point (x, y), give two sets of polar coordinates (r, ), where

0 2.

18. (-2, 32 ) ________________ 19. (0, -13.5) ________________

Warm Up


1) _____________________ 2) _____________________

3) ______________________ 4) ______________________

POLAR GRAPHS: ROSE CURVES AND LEMNISCATES

I. r = acos(n) 1. Make a table and graph: r = 4cos(2) This graph is called a __________________ with _____ petals.

Sketch the graph. 1. r = 3cos(2) 2. r = 2cos(2) 3. r = 4sin(5) 4. r = 5cos(3)

The value of a determines the __________________________. II. r

2 = a

2cos(2)

1) r2 = 9cos(2) 2) r

2 = 16sin(2)

This graph is called a __________________.

Graph:

3) r2 = 4sin(2) 4) r

2 = 25cos(2)

ROSE CURVES AND LEMNISCATES

Rose Curves r = acos(n) r = asin(n) a> 0, n>0

n Even 2n Petals

n Odd n Petals

Lemniscates r2 = a

2cos(2)

r = acos(2) r = asin(2)

r = acos(3)

r = asin(3)

r2 = a

2cos(2)

r2 = a

2sin(2)

Name ____________________________ Date__________ Period_____

WORKSHEET: ROSE CURVES AND LEMNISCATES

Graph each equation without using a calculator and name the type of graph. Check your graph on your calculator.

1. r = 3cos3 _________________ 2. r = 3sin2 _________________

3. r = 4cos 4 _________________ 4. r2 = 9sin2 _________________ 5. rcos = 4 _________________

6. r2 = 4cos2 _________________ 7. r = 4 cos _________________ 8. r = 4sin3 _________________ 9. r = 2 _________________

10. r = 2 + 3cos _________________ 11. 4

3 _________________

12. 6

7 _________________ 13. rsin = -3 __________________

14. r = 1 + 3sin _________________ 15. r = 2 2cos _________________


16.

2

3,5.4 __________ 17.

3

4,2 __________


0 2.

18. (3, 3 ) ________________ 19. (0, 10) ________________


20. A = ____________, ____________ ____________, ____________ 21. B = ____________, ____________ ____________, ____________

A

B

Warm Up


1) _____________________ 2) _____________________

3) ______________________ 4) ______________________

POLAR GRAPHS: PUTTING IT ALL TOGETHER

Lines: = a rcos = a rsin = a

Circles: r = a acosr asinr Cardioids and Limaons: bcosar bsinar Rose Curves: Lemniscates:

nacosr nasinr 2cosar 22

2sinar 22


1. ____________________ 2. ____________________

3. ____________________ 4. ____________________

Name ____________________________ Date__________ Period_____

WORKSHEET: PUTTING IT ALL TOGETHER


1. ________________________ 2. _________________________

3. ________________________ 4. _________________________

5. ________________________ 6. _________________________

7. ________________________ 8. _________________________

9. ________________________ 10. _________________________

11. ________________________ 12. _________________________

13. a) Graph r = 3sin3.

b) Find the zeros of r.

Name _____________________________ Date_________ Pd ________

REVIEW 10.7-10.8

POLAR EQUATIONS AND GRAPHS

Plot each point.

15. A

4

7,2 2. B

3

2,3

3. C

6

11,5 4. D

2

3,4

5. E

3

4,1 6. F ,4


7. P = ____________, ____________ ____________, ____________ 8. Q = ____________, ____________ ____________, ____________ Name each point two ways using

polar coordinates where 2 4.

9. Q = ____________, ____________

P

Q


10.

6

7,10 __________ 11.

2

3,5.9 __________


0 2.

12. (5, -5) ______________________ 13. (-16, 0) ______________________

14. 2,32 ___________________ 15. 6,2 ____________________ Identify and graph each polar equation.

16. r = 1 3sin _________________ 17. r = 5sin2 _________________

18. r = 1 4cos _________________ 19. rcos = 2 ___________________ 20. r = cos ____________________ 21. r2 = 25sin2 _________________

22. r = -4sin _________________ 23. r = 2 + 2sin __________________

24. 3

4 _________________ 25. r = 4cos3 _________________

26. r = -2 _________________ 27. r = 3 + 2cos _______________ Write the equation for each graph.

28. ________________________ 29. _________________________

30. ________________________ 31. _________________________

32. ________________________ 33. _________________________

Analyzing Polar Equations 34. a) Graph r = 2 + 4sin.

b) Find the zeros of r.

c) When is r = 4?

In addition to the review, make sure to study all homework and notes.

Documents

Polar Coordinates Packet 2012