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Quick Review Solutions. Why 360 º ? Navigation In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel.

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Presentation on theme: "Quick Review Solutions. Why 360 º ? Navigation In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel."— Presentation transcript:

1 Quick Review Solutions

2 Why 360 º ?

3 Navigation In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel measured clockwise from due north.

4 Radian A central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius.

5 Example Working with Radian Measure How many radians are in 60 degrees?

6 Example Working with Radian Measure How many radians are in 60 degrees?

7 Degree-Radian Conversion

8 Arc Length Formula (Radian Measure)

9 Arc Length Formula (Degree Measure)

10 Example Perimeter of a Pizza Slice

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12 Angular and Linear Motion Angular speed is measured in units like revolutions per minute. Linear speed is measured in units like miles per hour.

13 Quick Review Solutions 1. Solve for x. x Solve for x. 6 3 x

14 Standard Position An acute angle θ in standard position, with one ray along the positive x-axis and the other extending into the first quadrant.

15 Trigonometric Functions

16 Example Evaluating Trigonometric Functions of 45 º Find the values of all six trigonometric functions for an angle of 45 º.

17 Example Evaluating Trigonometric Functions of 45 º Find the values of all six trigonometric functions for an angle of 45 º.

18 Example Evaluating Trigonometric Functions of 60 º Find the values of all six trigonometric functions for an angle of 60º.

19 Common Calculator Errors When Evaluating Trig Functions Using the calculator in the wrong angle mode (degree/radians) Using the inverse trig keys to evaluate cot, sec, and csc Using function shorthand that the calculator does not recognize Not closing parentheses

20 Example Solving a Right Triangle

21 Example Evaluating Trigonometric Functions of 60 º Find the values of all six trigonometric functions for an angle of 60º.

22 Initial Side, Terminal Side

23 Positive Angle, Negative Angle

24 Coterminal Angles Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.

25 Example Finding Coterminal Angles

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27 Example Evaluating Trig Functions Determined by a Point in QI

28 Trigonometric Functions of any Angle

29 Evaluating Trig Functions of a Nonquadrantal Angle θ 1. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. 2. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ. 3. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator. 4. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. 5. Use the coordinates of point P and the definitions to determine the six trig functions.

30 Example Evaluating More Trig Functions

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32 Example Using one Trig Ration to Find the Others

33 Unit Circle The unit circle is a circle of radius 1 centered at the origin.

34 Trigonometric Functions of Real Numbers

35 Periodic Function

36 The 16-Point Unit Circle

37 Quick Review Solutions

38 What youll learn about The Basic Waves Revisited Sinusoids and Transformations Modeling Periodic Behavior with Sinusoids … and why Sine and cosine gain added significance when used to model waves and periodic behavior.

39 Sinusoid

40 Amplitude of a Sinusoid

41 Period of a Sinusoid

42 Example Horizontal Stretch or Shrink and Period

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44 Frequency of a Sinusoid

45 Example Combining a Phase Shift with a Period Change

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47 Graphs of Sinusoids

48 Constructing a Sinusoidal Model using Time

49 Quick Review

50 Quick Review Solutions

51 What youll learn about The Tangent Function The Cotangent Function The Secant Function The Cosecant Function … and why This will give us functions for the remaining trigonometric ratios.

52 Asymptotes of the Tangent Function

53 Zeros of the Tangent Function

54 Asymptotes of the Cotangent Function

55 Zeros of the Cotangent Function

56 The Secant Function

57 The Cosecant Function

58 Basic Trigonometry Functions

59 Quick Review Solutions

60 What youll learn about Combining Trigonometric and Algebraic Functions Sums and Differences of Sinusoids Damped Oscillation … and why Function composition extends our ability to model periodic phenomena like heartbeats and sound waves.

61 Example Combining the Cosine Function with x 2

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65 Sums That Are Sinusoids Functions

66 Example Identifying a Sinusoid

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70 Damped Oscillation

71 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.7 Inverse Trigonometric Functions

72 Quick Review

73 Quick Review Solutions

74 What youll learn about Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric Functions Applications of Inverse Trigonometric Functions … and why Inverse trig functions can be used to solve trigonometric equations.

75 Inverse Sine Function

76 Inverse Sine Function (Arcsine Function)

77 Example Evaluate sin -1 x Without a Calculator

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81 Inverse Cosine (Arccosine Function)

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83 Inverse Tangent Function (Arctangent Function)

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85 End Behavior of the Tangent Function

86 Composing Trigonometric and Inverse Trigonometric Functions

87 Example Composing Trig Functions with Arcsine

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89 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.8 Solving Problems with Trigonometry

90 Quick Review 1. Solve for a. a 3 23 º

91 Quick Review

92 Quick Review Solutions 1. Solve for a. a 3 23 º

93 Quick Review Solutions

94 What youll learn about More Right Triangle Problems Simple Harmonic Motion … and why These problems illustrate some of the better- known applications of trigonometry.

95 Angle of Elevation, Angle of Depression An angle of elevation is the angle through which the eye moves up from horizontal to look at something above. An angle of depression is the angle through which the eye moves down from horizontal to look at something below.

96 Example Using Angle of Elevation The angle of elevation from the buoy to the top of the Barnegat Bay lighthouse 130 feet above the surface of the water is 5 º. Find the distance x from the base of the lighthouse to the buoy. 130 x 5º5º

97 Example Using Angle of Elevation The angle of elevation from the buoy to the top of the Barnegat Bay lighthouse 130 feet above the surface of the water is 5 º. Find the distance x from the base of the lighthouse to the buoy. 130 x 5º5º

98 Simple Harmonic Motion

99 Example Calculating Harmonic Motion A mass oscillating up and down on the bottom of a spring (assuming perfect elasticity and no friction or air resistance) can be modeled as harmonic motion. If the weight is displaced a maximum of 4 cm, find the modeling equation if it takes 3 seconds to complete one cycle.

100 Example Calculating Harmonic Motion A mass oscillating up and down on the bottom of a spring (assuming perfect elasticity and no friction or air resistance) can be modeled as harmonic motion. If the weight is displaced a maximum of 4 cm, find the modeling equation if it takes 3 seconds to complete one cycle.

101 Chapter Test 5 C 12 α A B

102 Chapter Test

103

104 Chapter Test Solutions 5 C 12 α A B

105 Chapter Test Solutions

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