Introduction to Polar Graphs - MathHands
Trigonometry Sec. 02 notes
MathHands.com M´ arquez
Introduction to Polar Graphs Main Idea Plot Points You should be very conformable plotting points in radians as well as degrees. Simply make a table of values with r’s and corresponding θ’s, then start plotting until you can make sense of the the graph. If connecting the points is uneasy, choose smaller angle intervals, i.e. instead of plotting and calculating the r every 30 degrees, plot every 5 degrees. Get to Know Your Calculator I am usually not a big calculator fan, but this may be a good time when we can use it appropriately to graph some of these functions. After all, there is limited beauty and creativity that occurs when plotting points. Get to know Famous Questions 1. Can you calculate what is the largest r value on a graph for a given equation? and for which angles does it occur? 2. For which angles θ does the graph go through the origin? 3. How is the graph affected if we restrict the values of θ? Get to know Famous Graphs Get to know Famous Graphs 1. Can you calculate what is the largest r value on a graph for a given equation? and for which angles does it occur? 2. For which angles θ does the graph go through the origin? 3. How is the graph affected if we restrict the values of θ?
r = sin θ
r = cos θ
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2007-2009 MathHands.com
r = cos(2θ)
r = cos(3θ)
r = 5 + 3 sin θ
r = sin(2θ)
r = sin(4θ)
r = 3 + 5 sin θ
r = sin(3θ)
r = cos(5θ)
math hands
pg. 1
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez
Introduction to Polar Graphs 1. Graph and Understand the graph of r = 5 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 2. Graph and Understand the graph of r = 5 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 3. Graph and Understand the graph of r = −3 cos θ
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2007-2009 MathHands.com
math hands
pg. 2
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 4. Graph and Understand the graph of r = 3 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 5. Graph and Understand the graph of r = −6 sin θ
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2007-2009 MathHands.com
math hands
pg. 3
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 6. Graph and Understand the graph of r = 5 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 90◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 7. Graph and Understand the graph of r = 5 sin θ. Limit the study of this graph to the values of θ ranging from −90◦ to 0◦ .
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2007-2009 MathHands.com
math hands
pg. 4
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 8. Graph and Understand the graph of r = −3 cos θ. Limit the study of this graph to the values of θ ranging from 180◦ to 270◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 9. Graph and Understand the graph of r = 3 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 180◦.
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2007-2009 MathHands.com
math hands
pg. 5
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 10. Graph and Understand the graph of r = −6 sin θ. Limit the study of this graph to the values of θ ranging from 90◦ to 180◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 11. Graph and Understand the graph of r = 5 + cos θ
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2007-2009 MathHands.com
math hands
pg. 6
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 12. Graph and Understand the graph of r = 4 − 2 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 13. Graph and Understand the graph of r = −3 + 2 cos θ
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2007-2009 MathHands.com
math hands
pg. 7
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 14. Graph and Understand the graph of r = 3 + 3 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 15. Graph and Understand the graph of r = 4 − 2 cos θ
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2007-2009 MathHands.com
math hands
pg. 8
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 16. Graph and Understand the graph of r = −2 + 3 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 17. Graph and Understand the graph of r = 3 + 2 sin θ
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2007-2009 MathHands.com
math hands
pg. 9
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 18. Graph and Understand the graph of r = 3 − 2 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 19. Graph and Understand the graph of r = −3 + 2 sin θ
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2007-2009 MathHands.com
math hands
pg. 10
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 20. Graph and Understand the graph of r = 3 + 3 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 21. Graph and Understand the graph of r = 2 − 3 sin θ
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2007-2009 MathHands.com
math hands
pg. 11
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 22. Graph and Understand the graph of r = −2 + 3 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 23. Graph and Understand the graph of r = 2 + 3 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 90◦ .
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2007-2009 MathHands.com
math hands
pg. 12
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 24. Graph and Understand the graph of r = 2 − 3 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 60◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 25. Graph and Understand the graph of r = −2 + 3 cos θ. Limit the study of this graph to the values of θ ranging from 90◦ to 180◦ .
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2007-2009 MathHands.com
math hands
pg. 13
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 26. Graph and Understand the graph of r = 5 + 5 sin θ. Limit the study of this graph to the values of θ ranging from 180◦ to 270◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 27. Graph and Understand the graph of r = 3 − 2 sin θ. Limit the study of this graph to the values of θ ranging from 135◦ to 225◦ .
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2007-2009 MathHands.com
math hands
pg. 14
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 28. Graph and Understand the graph of r = −3 + 2 sin θ. Limit the study of this graph to the values of θ ranging from 360◦ to 540◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 29. Graph and Understand the graph of r = 5 cos(2θ)
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2007-2009 MathHands.com
math hands
pg. 15
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 30. Graph and Understand the graph of r = 6 cos(4θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 31. Graph and Understand the graph of r = 5 sin(2θ)
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2007-2009 MathHands.com
math hands
pg. 16
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 32. Graph and Understand the graph of r = 6 sin(4θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 33. Graph and Understand the graph of r = 5 cos(3θ)
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2007-2009 MathHands.com
math hands
pg. 17
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 34. Graph and Understand the graph of r = 6 cos(5θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 35. Graph and Understand the graph of r = 5 sin(3θ)
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2007-2009 MathHands.com
math hands
pg. 18
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 36. Graph and Understand the graph of r = 4 sin(5θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 37. Graph and Understand the graph of r = 5 sin(2θ). Limit the study of this graph to the values of θ ranging from 0◦ to 90◦ .
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2007-2009 MathHands.com
math hands
pg. 19
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 38. Graph and Understand the graph of r = 6 sin(4θ). Limit the study of this graph to the values of θ ranging from 90◦ to 180◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 39. Graph and Understand the graph of r = 5 cos(3θ). Limit the study of this graph to the values of θ ranging from 0◦ to 180◦ .
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2007-2009 MathHands.com
math hands
pg. 20
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 40. Graph and Understand the graph of r = 6 cos(5θ). Limit the study of this graph to the values of θ ranging from 180◦ to 270◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 41. Graph and Understand the graph of r2 = 25 sin2 θ √ Solution: Note, we can solve for r as r√= ± 25 sin2 θ... then we can plot each r separately, that is we graph √ r = 25 sin2 θ and then we graph r = − 25 sin2 θ
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2007-2009 MathHands.com
math hands
pg. 21
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
42. Graph and Understand the graph of r2 = 25 sin2 3θ √ 25 sin2 3θ... then we can plot each r separately, that is we graph Solution: Note, we can solve for r as r = ± √ √ 2 r = 25 sin 3θ and then we graph r = − 25 sin2 3θ 90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
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2007-2009 MathHands.com
math hands
pg. 22
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez
43. Understanding The graph of r = 2 − 3 cos θ has an inner loop. What values of θ trace precisely this inner loop? i.e. ?◦ ≤ θ ≤?◦ Solution: hint: solve for the values of θ when r = 0, that is when graph goes through the center
44. Understanding On the graph of r = 5 sin(3θ) (a) ’sin 3θ’ will take values between 1 and −1; never larger than one, never smaller than −1. Find the angles θ, at which sin 3θ take on its largest value 1, or is smallest value -1. (i.e solve sin 3θ = 1 and solve sin 3θ = −1 )
(b) Graph r = 5 sin(3θ) and mark the points obtained from part a). What can you say about these points? A) nothing B) they are the ’tip of the leaves’ C) they are not on the leaves. (c) Find the angles at which the graph goes through (0, 0) (d) Find the interval/s for theta that trace the second ’leaf’ Solution: hint: solve for the values of θ when r = 0, that is when graph goes through the center, such leaf starts and ends at center.. 45. Understanding On the graph of r = −5 + 2 cos θ
(a) ’cos θ’ will take values between 1 and −1; never larger than one, never smaller than −1. What is the largest or the smallest that the quantity ”−5 + 2 cos θ” can become?
(b) Find the angles at which ”−5 + 2 cos θ” takes on its largest value -3, or is smallest value -7. (c) Graph r = −5 + 2 cos θ and mark the points obtained from part b). What can you say about these points? A) nothing B) its the furthest point from the origin (d) Find the angles at which the graph goes through (0, 0) (e) Find the interval/s for theta that trace ONLY the outer leaf.
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2007-2009 MathHands.com
math hands
pg. 23
MathHands.com M´ arquez
Introduction to Polar Graphs Main Idea Plot Points You should be very conformable plotting points in radians as well as degrees. Simply make a table of values with r’s and corresponding θ’s, then start plotting until you can make sense of the the graph. If connecting the points is uneasy, choose smaller angle intervals, i.e. instead of plotting and calculating the r every 30 degrees, plot every 5 degrees. Get to Know Your Calculator I am usually not a big calculator fan, but this may be a good time when we can use it appropriately to graph some of these functions. After all, there is limited beauty and creativity that occurs when plotting points. Get to know Famous Questions 1. Can you calculate what is the largest r value on a graph for a given equation? and for which angles does it occur? 2. For which angles θ does the graph go through the origin? 3. How is the graph affected if we restrict the values of θ? Get to know Famous Graphs Get to know Famous Graphs 1. Can you calculate what is the largest r value on a graph for a given equation? and for which angles does it occur? 2. For which angles θ does the graph go through the origin? 3. How is the graph affected if we restrict the values of θ?
r = sin θ
r = cos θ
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2007-2009 MathHands.com
r = cos(2θ)
r = cos(3θ)
r = 5 + 3 sin θ
r = sin(2θ)
r = sin(4θ)
r = 3 + 5 sin θ
r = sin(3θ)
r = cos(5θ)
math hands
pg. 1
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez
Introduction to Polar Graphs 1. Graph and Understand the graph of r = 5 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 2. Graph and Understand the graph of r = 5 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 3. Graph and Understand the graph of r = −3 cos θ
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2007-2009 MathHands.com
math hands
pg. 2
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 4. Graph and Understand the graph of r = 3 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 5. Graph and Understand the graph of r = −6 sin θ
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2007-2009 MathHands.com
math hands
pg. 3
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 6. Graph and Understand the graph of r = 5 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 90◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 7. Graph and Understand the graph of r = 5 sin θ. Limit the study of this graph to the values of θ ranging from −90◦ to 0◦ .
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2007-2009 MathHands.com
math hands
pg. 4
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 8. Graph and Understand the graph of r = −3 cos θ. Limit the study of this graph to the values of θ ranging from 180◦ to 270◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 9. Graph and Understand the graph of r = 3 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 180◦.
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2007-2009 MathHands.com
math hands
pg. 5
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 10. Graph and Understand the graph of r = −6 sin θ. Limit the study of this graph to the values of θ ranging from 90◦ to 180◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 11. Graph and Understand the graph of r = 5 + cos θ
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2007-2009 MathHands.com
math hands
pg. 6
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 12. Graph and Understand the graph of r = 4 − 2 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 13. Graph and Understand the graph of r = −3 + 2 cos θ
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2007-2009 MathHands.com
math hands
pg. 7
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 14. Graph and Understand the graph of r = 3 + 3 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 15. Graph and Understand the graph of r = 4 − 2 cos θ
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2007-2009 MathHands.com
math hands
pg. 8
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 16. Graph and Understand the graph of r = −2 + 3 cos θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 17. Graph and Understand the graph of r = 3 + 2 sin θ
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2007-2009 MathHands.com
math hands
pg. 9
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 18. Graph and Understand the graph of r = 3 − 2 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 19. Graph and Understand the graph of r = −3 + 2 sin θ
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2007-2009 MathHands.com
math hands
pg. 10
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 20. Graph and Understand the graph of r = 3 + 3 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 21. Graph and Understand the graph of r = 2 − 3 sin θ
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2007-2009 MathHands.com
math hands
pg. 11
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 22. Graph and Understand the graph of r = −2 + 3 sin θ
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 23. Graph and Understand the graph of r = 2 + 3 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 90◦ .
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2007-2009 MathHands.com
math hands
pg. 12
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 24. Graph and Understand the graph of r = 2 − 3 cos θ. Limit the study of this graph to the values of θ ranging from 0◦ to 60◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 25. Graph and Understand the graph of r = −2 + 3 cos θ. Limit the study of this graph to the values of θ ranging from 90◦ to 180◦ .
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2007-2009 MathHands.com
math hands
pg. 13
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 26. Graph and Understand the graph of r = 5 + 5 sin θ. Limit the study of this graph to the values of θ ranging from 180◦ to 270◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 27. Graph and Understand the graph of r = 3 − 2 sin θ. Limit the study of this graph to the values of θ ranging from 135◦ to 225◦ .
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2007-2009 MathHands.com
math hands
pg. 14
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 28. Graph and Understand the graph of r = −3 + 2 sin θ. Limit the study of this graph to the values of θ ranging from 360◦ to 540◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 29. Graph and Understand the graph of r = 5 cos(2θ)
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2007-2009 MathHands.com
math hands
pg. 15
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 30. Graph and Understand the graph of r = 6 cos(4θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 31. Graph and Understand the graph of r = 5 sin(2θ)
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2007-2009 MathHands.com
math hands
pg. 16
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 32. Graph and Understand the graph of r = 6 sin(4θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 33. Graph and Understand the graph of r = 5 cos(3θ)
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2007-2009 MathHands.com
math hands
pg. 17
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 34. Graph and Understand the graph of r = 6 cos(5θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 35. Graph and Understand the graph of r = 5 sin(3θ)
c
2007-2009 MathHands.com
math hands
pg. 18
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 36. Graph and Understand the graph of r = 4 sin(5θ)
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 37. Graph and Understand the graph of r = 5 sin(2θ). Limit the study of this graph to the values of θ ranging from 0◦ to 90◦ .
c
2007-2009 MathHands.com
math hands
pg. 19
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 38. Graph and Understand the graph of r = 6 sin(4θ). Limit the study of this graph to the values of θ ranging from 90◦ to 180◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 39. Graph and Understand the graph of r = 5 cos(3θ). Limit the study of this graph to the values of θ ranging from 0◦ to 180◦ .
c
2007-2009 MathHands.com
math hands
pg. 20
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez 90◦ 60◦
120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 40. Graph and Understand the graph of r = 6 cos(5θ). Limit the study of this graph to the values of θ ranging from 180◦ to 270◦ .
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
Solution: 41. Graph and Understand the graph of r2 = 25 sin2 θ √ Solution: Note, we can solve for r as r√= ± 25 sin2 θ... then we can plot each r separately, that is we graph √ r = 25 sin2 θ and then we graph r = − 25 sin2 θ
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2007-2009 MathHands.com
math hands
pg. 21
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez
90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
42. Graph and Understand the graph of r2 = 25 sin2 3θ √ 25 sin2 3θ... then we can plot each r separately, that is we graph Solution: Note, we can solve for r as r = ± √ √ 2 r = 25 sin 3θ and then we graph r = − 25 sin2 3θ 90◦ 60◦ 120◦
30◦ 150◦
0◦ 180◦
330◦ 210◦
300◦ 240◦ 270◦
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2007-2009 MathHands.com
math hands
pg. 22
Trigonometry Sec. 02 exercises
MathHands.com M´ arquez
43. Understanding The graph of r = 2 − 3 cos θ has an inner loop. What values of θ trace precisely this inner loop? i.e. ?◦ ≤ θ ≤?◦ Solution: hint: solve for the values of θ when r = 0, that is when graph goes through the center
44. Understanding On the graph of r = 5 sin(3θ) (a) ’sin 3θ’ will take values between 1 and −1; never larger than one, never smaller than −1. Find the angles θ, at which sin 3θ take on its largest value 1, or is smallest value -1. (i.e solve sin 3θ = 1 and solve sin 3θ = −1 )
(b) Graph r = 5 sin(3θ) and mark the points obtained from part a). What can you say about these points? A) nothing B) they are the ’tip of the leaves’ C) they are not on the leaves. (c) Find the angles at which the graph goes through (0, 0) (d) Find the interval/s for theta that trace the second ’leaf’ Solution: hint: solve for the values of θ when r = 0, that is when graph goes through the center, such leaf starts and ends at center.. 45. Understanding On the graph of r = −5 + 2 cos θ
(a) ’cos θ’ will take values between 1 and −1; never larger than one, never smaller than −1. What is the largest or the smallest that the quantity ”−5 + 2 cos θ” can become?
(b) Find the angles at which ”−5 + 2 cos θ” takes on its largest value -3, or is smallest value -7. (c) Graph r = −5 + 2 cos θ and mark the points obtained from part b). What can you say about these points? A) nothing B) its the furthest point from the origin (d) Find the angles at which the graph goes through (0, 0) (e) Find the interval/s for theta that trace ONLY the outer leaf.
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2007-2009 MathHands.com
math hands
pg. 23
