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Overview of Second Semester Math Analysis

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1 Overview of Second Semester Math Analysis
Chapters 4, 5, and 6

2 Textbook: Precalculus Functions and Graphs:A Graphing Approach
Precalculus with Limits:A Graphing Approach By Ron Larson, Robert P. Hostetler, and Bruce H. Edwards Copyright 2001 by Houghton Mifflin Company.

3 Chapters Chapter 4 Chapter 5 Chapter 6 Online Notebook Organizer

4 Chapters Chapter 4 Chapter 5 Chapter 6 Online Notebook Organizer
Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry Trigonometric Functions of Any Angle Graphs of Sine and Cosine Functions Graphs of Other Trigonometric Functions Inverse Trigonometric Functions Chapter 4 Chapter 5 Chapter 6 Online Notebook Organizer

5 Chapters Chapter 4 Chapter 5 Chapter 6 Online Notebook Organizer
Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas Multiple-Angle and Product-Sum Formulas Chapter 5 Chapter 6 Online Notebook Organizer

6 Chapters Chapter 4 Chapter 5 Chapter 6 Online Notebook Organizer
Law of Sines Law of Cosines Areas of Right and Oblique Triangles Chapter 5 Chapter 6 Online Notebook Organizer

7 Chapters Chapter 4 Chapter 5 Chapter 6 Online Notebook Organizer
This online notebook organizer is a set of templates that will help you develop a section-by-section summary of the key concepts of the included chapters from the textbook. Click on the link below to use this tool: Chapter 5 Chapter 6 Online Notebook Organizer

8 Review Exercises a. 17.32 feet b. 34.64 feet c. 30 feet d. 28.28 feet
1. The height of a tree can be determined by measuring the length of the shadow of the tree and the angle of elevation of the sun from the tip of the shadow. Find the height of a tree that casts a 20 foot shadow when the angle of elevation is 600. a feet h=? 600 b feet c. 30 feet d feet

9 Review Exercises a. 17.32 feet b. 34.64 feet c. 30 feet d. 28.28 feet
1. The height of a tree can be determined by measuring the length of the shadow of the tree and the angle of elevation of the sun from the tip of the shadow. Find the height of a tree that casts a 20 foot shadow when the angle of elevation is 600. a feet h=? 600 b feet c. 30 feet d feet

10 Review Exercises a. 17.32 feet  b. 34.64 feet c. 30 feet
1. The height of a tree can be determined by measuring the length of the shadow of the tree and the angle of elevation of the sun from the tip of the shadow. Find the height of a tree that casts a 20 foot shadow when the angle of elevation is 600. a feet  h=? 600 b feet c. 30 feet d feet

11 Review Exercises a. 17.32 feet b. 34.64 feet  c. 30 feet
1. The height of a tree can be determined by measuring the length of the shadow of the tree and the angle of elevation of the sun from the tip of the shadow. Find the height of a tree that casts a 20 foot shadow when the angle of elevation is 600. a feet h=? 600 b feet  c. 30 feet d feet

12 Review Exercises a. 17.32 feet b. 34.64 feet c. 30 feet 
1. The height of a tree can be determined by measuring the length of the shadow of the tree and the angle of elevation of the sun from the tip of the shadow. Find the height of a tree that casts a 20 foot shadow when the angle of elevation is 600. a feet h=? 600 b feet c. 30 feet  d feet

13 Review Exercises a. 17.32 feet b. 34.64 feet c. 30 feet
1. The height of a tree can be determined by measuring the length of the shadow of the tree and the angle of elevation of the sun from the tip of the shadow. Find the height of a tree that casts a 20 foot shadow when the angle of elevation is 600. a feet h=? 600 b feet c. 30 feet d feet 

14 Review Exercises a. d = tan x / 10 b. d = 10 tan x c. d = 10 cot x
2. Standing on a dock 10 feet above the water line, you sight a ship heading directly toward you. Let x be the angle of depression to the ship and d the distance from the shore to the ship. Write d as a function of x. a. d = tan x / 10 b. d = 10 tan x x c. d = 10 cot x 10 feet d d. d = cot x / 10 Photo from

15 Review Exercises a. d = tan x / 10 b. d = 10 tan x c. d = 10 cot x
2. Standing on a dock 10 feet above the water line, you sight a ship heading directly toward you. Let x be the angle of depression to the ship and d the distance from the shore to the ship. Write d as a function of x. Photo from 10 feet x d a. d = tan x / 10 b. d = 10 tan x c. d = 10 cot x d. d = cot x / 10

16 Review Exercises a. d = tan x / 10  b. d = 10 tan x c. d = 10 cot x
2. Standing on a dock 10 feet above the water line, you sight a ship heading directly toward you. Let x be the angle of depression to the ship and d the distance from the shore to the ship. Write d as a function of x. Photo from 10 feet x d a. d = tan x / 10  b. d = 10 tan x c. d = 10 cot x d. d = cot x / 10

17 Review Exercises a. d = tan x / 10 b. d = 10 tan x  c. d = 10 cot x
2. Standing on a dock 10 feet above the water line, you sight a ship heading directly toward you. Let x be the angle of depression to the ship and d the distance from the shore to the ship. Write d as a function of x. Photo from 10 feet x d a. d = tan x / 10 b. d = 10 tan x  c. d = 10 cot x d. d = cot x / 10

18 Review Exercises a. d = tan x / 10 b. d = 10 tan x c. d = 10 cot x 
2. Standing on a dock 10 feet above the water line, you sight a ship heading directly toward you. Let x be the angle of depression to the ship and d the distance from the shore to the ship. Write d as a function of x. Photo from 10 feet x d a. d = tan x / 10 b. d = 10 tan x c. d = 10 cot x  d. d = cot x / 10

19 Review Exercises a. d = tan x / 10 b. d = 10 tan x c. d = 10 cot x
2. Standing on a dock 10 feet above the water line, you sight a ship heading directly toward you. Let x be the angle of depression to the ship and d the distance from the shore to the ship. Write d as a function of x. Photo from 10 feet x d a. d = tan x / 10 b. d = 10 tan x c. d = 10 cot x d. d = cot x / 10 

20 Review Exercises 3. Convert the angle measure 3 / 8 to degrees. a b c. 1350 d. 750 Photo from

21 Review Exercises 3. Convert the angle measure 3 / 8 to degrees. Photo from a b c. 1350 d. 750

22 Review Exercises a. 67.50  b. 1.180 c. 1350 d. 750
3. Convert the angle measure 3 / 8 to degrees. Photo from a  b c. 1350 d. 750

23 Review Exercises a. 67.50 b. 1.180  c. 1350 d. 750
3. Convert the angle measure 3 / 8 to degrees. Photo from a b  c. 1350 d. 750

24 Review Exercises a. 67.50 b. 1.180 c. 1350  d. 750
3. Convert the angle measure 3 / 8 to degrees. Photo from a b c  d. 750

25 Review Exercises a. 67.50 b. 1.180 c. 1350 d. 750 
3. Convert the angle measure 3 / 8 to degrees. Photo from a b c. 1350 d. 750 

26 Review Exercises a. ( 1, -1) b. ( 0, -1) c. ( -1, 0) d. ( -1, 1)
4. Find the point ( x, y ) on the unit circle that corresponds to the real number t = 17 / 6. a. ( 1, -1) b. ( 0, -1) c. ( -1, 0) d. ( -1, 1) Clip Art from

27 Review Exercises a. ( 1, -1) b. ( 0, -1) c. ( -1, 0) d. ( -1, 1)
4. Find the point ( x, y ) on the unit circle that corresponds to the real number t = 17 / 6. Clip Art from a. ( 1, -1) b. ( 0, -1) c. ( -1, 0) d. ( -1, 1)

28 Review Exercises a. ( 1, -1)  b. ( 0, -1) c. ( -1, 0) d. ( -1, 1)
4. Find the point ( x, y ) on the unit circle that corresponds to the real number t = 17 / 6. Clip Art from a. ( 1, -1)  b. ( 0, -1) c. ( -1, 0) d. ( -1, 1)

29 Review Exercises a. ( 1, -1) b. ( 0, -1)  c. ( -1, 0) d. ( -1, 1)
4. Find the point ( x, y ) on the unit circle that corresponds to the real number t = 17 / 6. Clip Art from a. ( 1, -1) b. ( 0, -1)  c. ( -1, 0) d. ( -1, 1)

30 Review Exercises a. ( 1, -1) b. ( 0, -1) c. ( -1, 0)  d. ( -1, 1)
4. Find the point ( x, y ) on the unit circle that corresponds to the real number t = 17 / 6. Clip Art from a. ( 1, -1) b. ( 0, -1) c. ( -1, 0)  d. ( -1, 1)

31 Review Exercises a. ( 1, -1) b. ( 0, -1) c. ( -1, 0) d. ( -1, 1) 
4. Find the point ( x, y ) on the unit circle that corresponds to the real number t = 17 / 6. Clip Art from a. ( 1, -1) b. ( 0, -1) c. ( -1, 0) d. ( -1, 1) 

32 Review Exercises a. period: 6, amplitude: /2
5. Determine the period and amplitude of the graph shown below. 3 2 1 -1 -2 -3 a. period: 6, amplitude: /2 π 8 b. period: /4, amplitude: 3 c. period: /2 , amplitude: 3 d. period: 3, amplitude: /2

33 Review Exercises a. period: 6, amplitude: /2
5. Determine the period and amplitude of the graph shown below. π 8 3 2 1 -1 -2 -3 a. period: 6, amplitude: /2 b. period: /4, amplitude: 3 c. period: /2 , amplitude: 3 d. period: 3, amplitude: /2

34 Review Exercises a. period: 6, amplitude: /2 
5. Determine the period and amplitude of the graph shown below. π 8 3 2 1 -1 -2 -3 a. period: 6, amplitude: /2  b. period: /4, amplitude: 3 c. period: /2 , amplitude: 3 d. period: 3, amplitude: /2

35 Review Exercises a. period: 6, amplitude: /2
5. Determine the period and amplitude of the graph shown below. π 8 3 2 1 -1 -2 -3 a. period: 6, amplitude: /2 b. period: /4, amplitude: 3  c. period: /2 , amplitude: 3 d. period: 3, amplitude: /2

36 Review Exercises a. period: 6, amplitude: /2
5. Determine the period and amplitude of the graph shown below. π 8 3 2 1 -1 -2 -3 a. period: 6, amplitude: /2 b. period: /4, amplitude: 3 c. period: /2 , amplitude: 3  d. period: 3, amplitude: /2

37 Review Exercises a. period: 6, amplitude: /2
5. Determine the period and amplitude of the graph shown below. π 8 3 2 1 -1 -2 -3 a. period: 6, amplitude: /2 b. period: /4, amplitude: 3 c. period: /2 , amplitude: 3 d. period: 3, amplitude: /2

38 Review Exercises a. 93.4 feet b. 100.9 feet c. 103.8 feet
6. A surveyor wishes to find the distance across the river. The bearings from 2 points 70 feet apart on the same bank of the river to a tree on the opposite bank are N and N 340 W. Find the width of the river. N a feet b feet ? c feet 560 d feet Photo from

39 Review Exercises a. 93.4 feet b. 100.9 feet c. 103.8 feet
6. A surveyor wishes to find the distance across the river. The bearings from 2 points 70 feet apart on the same bank of the river to a tree on the opposite bank are N and N 340 W. Find the width of the river. N Photo from ? 560 a feet b feet c feet d feet

40 Review Exercises a. 93.4 feet  b. 100.9 feet c. 103.8 feet
6. A surveyor wishes to find the distance across the river. The bearings from 2 points 70 feet apart on the same bank of the river to a tree on the opposite bank are N and N 340 W. Find the width of the river. N Photo from ? 560 a feet  N b feet c feet d feet

41 Review Exercises a. 93.4 feet b. 100.9 feet  c. 103.8 feet
6. A surveyor wishes to find the distance across the river. The bearings from 2 points 70 feet apart on the same bank of the river to a tree on the opposite bank are N and N 340 W. Find the width of the river. N Photo from ? 560 a feet N b feet  c feet d feet

42 Review Exercises a. 93.4 feet b. 100.9 feet c. 103.8 feet 
6. A surveyor wishes to find the distance across the river. The bearings from 2 points 70 feet apart on the same bank of the river to a tree on the opposite bank are N and N 340 W. Find the width of the river. N Photo from ? 560 a feet N b feet c feet  d feet

43 Review Exercises a. 93.4 feet b. 100.9 feet c. 103.8 feet
6. A surveyor wishes to find the distance across the river. The bearings from 2 points 70 feet apart on the same bank of the river to a tree on the opposite bank are N and N 340 W. Find the width of the river. N Photo from ? 560 a feet N b feet c feet d feet

44 Review Exercises a. 4500.3 feet b. 1005.2 feet c. 500.75 feet
7.A sea-to-air guided missile shot from a submarine breaks the water surface at an angle of elevation of traveling at 520 feet per second. If the missile continues at a constant angle and at the same speed, how far above sea level will it be after 20 seconds? a feet b feet c feet d feet Photo from

45 Review Exercises a. 4500.3 feet b. 1005.2 feet c. 500.75 feet
7.A sea-to-air guided missile shot from a submarine breaks the water surface at an angle of elevation of traveling at 520 feet per second. If the missile continues at a constant angle and at the same speed, how far above sea level will it be after 20 seconds? Photo from a feet b feet c feet d feet

46 Review Exercises a. 4500.3 feet  b. 1005.2 feet c. 500.75 feet
7.A sea-to-air guided missile shot from a submarine breaks the water surface at an angle of elevation of traveling at 520 feet per second. If the missile continues at a constant angle and at the same speed, how far above sea level will it be after 20 seconds? Photo from a feet  N b feet c feet d feet

47 Review Exercises a. 4500.3 feet b. 1005.2 feet  c. 500.75 feet
7.A sea-to-air guided missile shot from a submarine breaks the water surface at an angle of elevation of traveling at 520 feet per second. If the missile continues at a constant angle and at the same speed, how far above sea level will it be after 20 seconds? Photo from a feet N b feet  c feet d feet

48 Review Exercises a. 4500.3 feet b. 1005.2 feet c. 500.75 feet 
7.A sea-to-air guided missile shot from a submarine breaks the water surface at an angle of elevation of traveling at 520 feet per second. If the missile continues at a constant angle and at the same speed, how far above sea level will it be after 20 seconds? Photo from a feet N b feet c feet  d feet

49 Review Exercises a. 4500.3 feet b. 1005.2 feet c. 500.75 feet
7.A sea-to-air guided missile shot from a submarine breaks the water surface at an angle of elevation of traveling at 520 feet per second. If the missile continues at a constant angle and at the same speed, how far above sea level will it be after 20 seconds? Photo from a feet N b feet c feet d feet 

50 Review Exercises sin x=1/csc x cos x = 1/sec x sin x/cos x = tan x
8. Given cot x is undefined, and cos x > 0, find csc x. sin x=1/csc x cos x = 1/sec x a. 0 sin x/cos x = tan x b. 1 c. -1 tan x = 1/cot x d. undefined

51 Review Exercises sin x=1/csc x cos x = 1/sec x sin x/cos x = tan x
8. Given cot x is undefined, and cos x > 0, find csc x. sin x=1/csc x cos x = 1/sec x tan x = 1/cot x sin x/cos x = tan x a. 0 b. 1 c. -1 d. undefined

52 Review Exercises sin x=1/csc x cos x = 1/sec x sin x/cos x = tan x
8. Given cot x is undefined, and cos x > 0, find csc x. sin x=1/csc x cos x = 1/sec x tan x = 1/cot x sin x/cos x = tan x a. 0  N b. 1 c. -1 d. undefined

53 Review Exercises sin x=1/csc x cos x = 1/sec x sin x/cos x = tan x
8. Given cot x is undefined, and cos x > 0, find csc x. sin x=1/csc x cos x = 1/sec x tan x = 1/cot x sin x/cos x = tan x a. 0 N b. 1  c. -1 d. undefined

54 Review Exercises sin x=1/csc x cos x = 1/sec x sin x/cos x = tan x
8. Given cot x is undefined, and cos x > 0, find csc x. sin x=1/csc x cos x = 1/sec x tan x = 1/cot x sin x/cos x = tan x a. 0 N b. 1 c. -1  d. undefined

55 Review Exercises sin x=1/csc x cos x = 1/sec x sin x/cos x = tan x
8. Given cot x is undefined, and cos x > 0, find csc x. sin x=1/csc x cos x = 1/sec x tan x = 1/cot x sin x/cos x = tan x a. 0 N b. 1 c. -1 d. undefined 

56 Review Exercises a. /2, 3/2 b. 0,  c. /4, 3/4, 5/4, 7/4 d. /2
9. Find all solutions in the interval [0, 2 ) : csc2 x - (cos4 x + cos2 x sin2 x + sin4 x) = 0. a. /2, 3/2 b. 0,  c. /4, 3/4, 5/4, 7/4 Photo from d. /2

57 Review Exercises a. /2, 3/2 b. 0,  c. /4, 3/4, 5/4, 7/4 d. /2
9. Find all solutions in the interval [0, 2 ) : csc2 x - (cos4 x + cos2 x sin2 x + sin4 x) = 0. Photo from a. /2, 3/2 b. 0,  c. /4, 3/4, 5/4, 7/4 d. /2

58 Review Exercises a. /2, 3/2  b. 0,  c. /4, 3/4, 5/4, 7/4
9. Find all solutions in the interval [0, 2 ) : csc2 x - (cos4 x + cos2 x sin2 x + sin4 x) = 0. Photo from a. /2, 3/2  N b. 0,  c. /4, 3/4, 5/4, 7/4 d. /2

59 Review Exercises a. /2, 3/2 b. 0,   c. /4, 3/4, 5/4, 7/4
9. Find all solutions in the interval [0, 2 ) : csc2 x - (cos4 x + cos2 x sin2 x + sin4 x) = 0. Photo from a. /2, 3/2 N b. 0,   c. /4, 3/4, 5/4, 7/4 d. /2

60 Review Exercises a. /2, 3/2 b. 0,  c. /4, 3/4, 5/4, 7/4 
9. Find all solutions in the interval [0, 2 ) : csc2 x - (cos4 x + cos2 x sin2 x + sin4 x) = 0. Photo from a. /2, 3/2 N b. 0,  c. /4, 3/4, 5/4, 7/4  d. /2

61 Review Exercises a. /2, 3/2 b. 0,  c. /4, 3/4, 5/4, 7/4
9. Find all solutions in the interval [0, 2 ) : csc2 x - (cos4 x + cos2 x sin2 x + sin4 x) = 0. Photo from a. /2, 3/2 N b. 0,  c. /4, 3/4, 5/4, 7/4 d. /2 

62 Review Exercises a. 115.9 feet b. 200.3 feet c. 175.6 feet d. 100 feet
10. Find the height of a giant helium balloon used in a Thanksgiving Day parade given that two guy wires are attached as shown in the figure below. a feet b feet c feet 300 490 d. 100 feet 100 feet Photo from

63 Review Exercises a. 115.9 feet b. 200.3 feet c. 175.6 feet d. 100 feet
10. Find the height of a giant helium balloon used in a Thanksgiving Day parade given that two guy wires are attached as shown in the figure below. 300 490 100 feet a feet b feet c feet d. 100 feet Photo from

64 Review Exercises a. 115.9 feet  b. 200.3 feet c. 175.6 feet
10. Find the height of a giant helium balloon used in a Thanksgiving Day parade given that two guy wires are attached as shown in the figure below. 300 490 100 feet a feet  N b feet c feet d. 100 feet Photo from

65 Review Exercises a. 115.9 feet b. 200.3 feet  c. 175.6 feet
10. Find the height of a giant helium balloon used in a Thanksgiving Day parade given that two guy wires are attached as shown in the figure below. 300 490 100 feet a feet N b feet  c feet d. 100 feet Photo from

66 Review Exercises a. 115.9 feet b. 200.3 feet c. 175.6 feet 
10. Find the height of a giant helium balloon used in a Thanksgiving Day parade given that two guy wires are attached as shown in the figure below. 300 490 100 feet a feet N b feet c feet  d. 100 feet Photo from

67 Review Exercises a. 115.9 feet b. 200.3 feet c. 175.6 feet
10. Find the height of a giant helium balloon used in a Thanksgiving Day parade given that two guy wires are attached as shown in the figure below. 300 490 100 feet a feet N b feet c feet d. 100 feet  Photo from

68 Edit 241 Final Project EmergingTechnologies for Teachers
References: Precalculus Functions and Graphs: A Graphing Approach by Ron Larson, Robert P. Hostetler, and Bruce H. Edwards Test Item File by Anne Larson Quinn Student Success Organizer (online) from Houghton Mifflin Company Sources of photos and clip arts cited on the slides. Rebecca T. Macasaet


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