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

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Presentation on theme: "Overview of Second Semester Math Analysis Chapters 4, 5, and 6."— Presentation transcript:

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

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

4 Chapters  Chapter 5 Chapter 5  Chapter 6 Chapter 6  Chapter 4 Chapter 4  Online Notebook Organizer Online Notebook Organizer

5 Chapters  Chapter 5 Chapter 5  Chapter 6 Chapter 6  Chapter 4 Chapter 4 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  Online Notebook Organizer Online Notebook Organizer

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

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

8 Chapters  Chapter 5 Chapter 5  Chapter 6 Chapter 6  Chapter 4 Chapter 4 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: /larson/precalculus_functions_aga/3e /students/sso.html  Online Notebook Organizer

9 Review Exercises 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 h=? a feet b feet c. 30 feet d feet

10 Review Exercises 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 a feet b feet c. 30 feet d feet 60 0 h=?

11 Review Exercises 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 a feet  b feet c. 30 feet 60 0 h=? d feet

12 Review Exercises 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 a feet b feet  c. 30 feet d feet 60 0 h=?

13 Review Exercises 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 a feet b feet c. 30 feet  d feet 60 0 h=?

14 Review Exercises 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 a feet b feet c. 30 feet d feet  60 0 h=?

15 Review Exercises 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 c. d = 10 cot x d. d = cot x / 10 Photo from 10 feet x d

16 Review Exercises 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 c. d = 10 cot x d. d = cot x / 10 Photo from 10 feet x d

17 Review Exercises 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 c. d = 10 cot x d. d = cot x / 10 Photo from 10 feet x d

18 Review Exercises 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  c. d = 10 cot x d. d = cot x / 10 Photo from 10 feet x d

19 Review Exercises 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 c. d = 10 cot x  d. d = cot x / 10 Photo from 10 feet x d

20 Review Exercises 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 c. d = 10 cot x d. d = cot x / 10  Photo from 10 feet x d

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

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

23 Review Exercises 3. Convert the angle measure 3  / 8 to degrees. a  b c d Photo from

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

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

26 Review Exercises 3. Convert the angle measure 3  / 8 to degrees. a b c d  Photo from

27 Review Exercises 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

28 Review Exercises 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

29 Review Exercises 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

30 Review Exercises 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

31 Review Exercises 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

32 Review Exercises 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

33 Review Exercises 5. Determine the period and amplitude of the graph shown below. a. period: 6, amplitude:  /2 b. period:  /4, amplitude: 3 c. period:  /2, amplitude: 3 d. period: 3, amplitude:  /2 π8π8 3π 8 7π 8 5π

34 Review Exercises 5. Determine the period and amplitude of the graph shown below. π8π8 3π 8 7π 8 5π a. period: 6, amplitude:  /2 b. period:  /4, amplitude: 3 c. period:  /2, amplitude: 3 d. period: 3, amplitude:  /2

35 Review Exercises 5. Determine the period and amplitude of the graph shown below. π8π8 3π 8 7π 8 5π a. period: 6, amplitude:  /2  b. period:  /4, amplitude: 3 c. period:  /2, amplitude: 3 d. period: 3, amplitude:  /2

36 Review Exercises 5. Determine the period and amplitude of the graph shown below. π8π8 3π 8 7π 8 5π a. period: 6, amplitude:  /2 b. period:  /4, amplitude: 3  c. period:  /2, amplitude: 3 d. period: 3, amplitude:  /2

37 Review Exercises 5. Determine the period and amplitude of the graph shown below. π8π8 3π 8 7π 8 5π a. period: 6, amplitude:  /2 b. period:  /4, amplitude: 3 c. period:  /2, amplitude: 3  d. period: 3, amplitude:  /2

38 Review Exercises 5. Determine the period and amplitude of the graph shown below. π8π8 3π 8 7π 8 5π a. period: 6, amplitude:  /2 b. period:  /4, amplitude: 3 c. period:  /2, amplitude: 3 d. period: 3, amplitude:  /2 

39 Review Exercises 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 34 0 W. Find the width of the river. a feet b feet c feet d feet Photo from N ? 56 0

40 Review Exercises 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 34 0 W. Find the width of the river. a feet b feet c feet d feet N Photo from N ? 56 0

41 Review Exercises 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 34 0 W. Find the width of the river. a feet  b feet c feet d feet N N Photo from N ? 56 0

42 Review Exercises 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 34 0 W. Find the width of the river. a feet b feet  c feet d feet N N Photo from N ? 56 0

43 Review Exercises 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 34 0 W. Find the width of the river. a feet b feet c feet  d feet N N Photo from N ? 56 0

44 Review Exercises 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 34 0 W. Find the width of the river. a feet b feet c feet d feet  N N Photo from N ? 56 0

45 Review Exercises 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

46 Review Exercises 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

47 Review Exercises 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 N Photo from

48 Review Exercises 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 N Photo from

49 Review Exercises 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 N Photo from

50 Review Exercises 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  N Photo from

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

52 Review Exercises 8. Given cot x is undefined, and cos x > 0, find csc x. a. 0 b. 1 c. -1 d. undefined

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

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

55 Review Exercises 8. Given cot x is undefined, and cos x > 0, find csc x. a. 0 b. 1 c. -1  d. undefined N

56 Review Exercises 8. Given cot x is undefined, and cos x > 0, find csc x. a. 0 b. 1 c. -1 d. undefined  N

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

58 Review Exercises 9. Find all solutions in the interval [0, 2  ) : csc 2 x - (cos 4 x + cos 2 x sin 2 x + sin 4 x) = 0. a.  /2, 3  /2 b. 0,  c.  /4, 3  /4, 5  /4, 7  /4 d.  /2 Photo from

59 Review Exercises 9. Find all solutions in the interval [0, 2  ) : csc 2 x - (cos 4 x + cos 2 x sin 2 x + sin 4 x) = 0. a.  /2, 3  /2  b. 0,  c.  /4, 3  /4, 5  /4, 7  /4 d.  /2 N Photo from

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

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

62 Review Exercises 9. Find all solutions in the interval [0, 2  ) : csc 2 x - (cos 4 x + cos 2 x sin 2 x + sin 4 x) = 0. a.  /2, 3  /2 b. 0,  c.  /4, 3  /4, 5  /4, 7  /4 d.  /2  N Photo from

63 Review Exercises a feet b feet c feet d. 100 feet feet Photo from 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.

64 Review Exercises a feet b feet c feet d. 100 feet feet Photo from 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.

65 Review Exercises 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 d. 100 feet N feet Photo from

66 Review Exercises 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 d. 100 feet N feet Photo from

67 Review Exercises 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  d. 100 feet N feet Photo from

68 Review Exercises 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 d. 100 feet  N feet Photo from

69 Edit 241 Final Project EmergingTechnologies for Teachers Rebecca T. Macasaet References: 1.Precalculus Functions and Graphs: A Graphing Approach by Ron Larson, Robert P. Hostetler, and Bruce H. Edwards 2.Test Item File by Anne Larson Quinn 3.Student Success Organizer (online) from Houghton Mifflin Company Sources of photos and clip arts cited on the slides.


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