Chapter 4 Trigonometric Functions

Transcription

1 Chapter 4 Trigonometric Functions Overview: 4.1 Radian and Degree Measure 4.2 Trigonometric Functions: The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of Any Angle 4.5 Graphs of Sine and Cosine Functions 4.6 Graphs of Other Trig Functions 4.7 Inverse Trig Functions 4.8 Applications and Models

2 4.1 Radian and Degree Measure What You ll Learn: #74 - Describe angles. #75 - Use radian measure. #76 - Use degree measure and convert between degree and radian measure. #77 - Use angle to model and solve real-life problems.

3 Angles in Standard Position

4 Coterminal Angles Coterminal angles share the same initial and terminal ray. They may be opposite directions.

5 Referenced Angles You may reference an angle by the location of the terminal ray.

6 Radian Measure Two ways to measure angles: 1. Degrees : 360 o = 1 revolution 2. Radians : 2π radians = 1 revolution Radian the measure of a central angle of θ that intercepts an arc s equal in length to the radius r of the circle.

7 Common Angles These common angles in both radians and degrees should eventually be memorized!

8 Finding Coterminal Angles 1. θ = 3π 4 2. θ = 2π 3 3. θ = 13π 6

9 Finding Complementary and Supplementary Angles 1. θ = 3π 7 2. θ = 2π 5

10 Unit Circle with the commonly used degree and radian measurements.

11 How do degrees and radians relate? 1 degree = π revolution = 2π radians 180 o = π radians So radians or radian = π degrees

12 Converting DEG RAD Convert each angle in degrees to radians o o o o

13 Converting RAD DEG Convert each angle in radians to degrees π 6 radian 3π 2 radians 3. 3π 4 radians 4. 7π 3 radians

14 Arc Length For a circle of radius r, a central angle of θ radians subtends an arc whose length s is: s = r θ

15 Example Find arc length The minute hand of a clock is 6 inches long. How far does the tip of the minute hand move in 20 minutes?

16 Linear Speed Average Speed = distance time elapsed Or Linear Speed V = s t

17 Angular Motion Angular speed is how fast an object is traveling in a circular motion. This may be found by using the following formula: W = θ t

18 Linear and Circular If you combine the formulas, you derive a formula that determines an object s linear speed going in a circular motion. V = r w

19 Finding Linear Speed David is spinning a rock at the end of a 2 foot rope at the rate of 180 revolutions per minute (RPM). Find the linear speed of the rock when it is released.

20 Finding Linear Speed The second hand of a clock is centimeters long. Find the linear speed of the tip of this second hand.

21 Find Angular and Linear Speed A lawn roller with a 10 inch radius makes 1. 2 revolutions per second. a) Find the angular speed of the roller in radians per second. b) Find the speed of the tractor that is pulling the roller.

22 Finding angular and linear speed An object is traveling around a circle with a radius of 10 centimeters. If in 40 seconds a central angle of 2 3 radian is swept out, what is the angular speed of the object? What is its linear speed?

23 Homework Page 255 #5-8,13,17,19,31-33,39,40,45,46,49,50,53,59 #77,78,81,82,85,87,95-98

24 4.2 Trig Functions: The Unit Circle What You ll Learn: #78 - Identify a unit circle and describe its relationship to real numbers. #79 - Evaluate trig functions using the unit circle. #80 - Use domain and period to evaluate sine and cosine functions. #81 - Use a calculator to evaluate trig functions.

25 The Unit Circle Radius = 1 unit

26 Remember the 30 o 60 o 90 o triangle pattern from Geometry? If the hypotenuse is 1 unit long, then the (x, y) coordinates will be ( 3 2, 1 2 )

27 Remember the 45 o 45 o 90 o triangle pattern from Geometry? If the hypotenuse is 1 unit long, then the (x, y) coordinates will be ( 2 2, 2 2 )

28

29 The Six Trig Functions sin θ = opp. hyp. = y 1 or sin θ = y θ cos θ = adj. hyp. = x or cos θ = x 1 tan θ = opp. adj. = y x or tan θ = y x

30 Find the exact value cos(45 o ) = sin 120 o = cos 3π 2 = sin( 7π 6 ) = tan 135 o = tan( 2π 3 ) =

31 The Six Trig Functions θ csc θ = sec θ = cot θ = 1 sin θ = 1 y or csc θ = 1 y 1 cos θ = 1 x or sec θ = 1 x 1 tan θ = 1 y x or cot θ = x y

32 Find the exact value sec(45 o ) = csc 120 o = sec 3π 2 = csc( 7π 6 ) = cot 135 o = cot( 2π 3 ) =

33 Quadrantal Angles Evaluate the six trig functions for the quadrantal angles: θ = 0 o or 0π θ = 90 o or π 2 θ = 270 o or 3π 2 θ = 360 o or 2π θ = 180 o or π

34 Quadrantal Angles θ = 0 o or 0π θ = 90 o or π 2 θ = 180 o or π θ = 270 o or 3π 2 θ = 360 o or 2π sin θ cos θ tan θ csc θ sec θ cot θ 0 1 = = undef. 1 1 = = undef. 1 0 = undef. 1 1 = = undef. 0 1 = = undef. 1 1 = = undef. 1 1 = = undef. 0 1 = = undef. 0 1 = = = undef. 1 1 = = undef.

35 Evaluating the 6 Trig Functions Evaluate the six trig functions at each real number: 1. t = π 6 2. t = 5π 4 3. t = 7π 4 4. t = π 3

36 Homework Page 264 #1,5-12,(13-27 ODD) Memorize Unit Circle

37 Periodic Functions A function f is periodic if there exists a positive real number c such that f t + c = f(t) for all t in the domain of f. This number c is called the period. Sine and Cosine have a period of 2π. Example: cos π 2 = cos π 2 + 2π = cos(5π 2 ) Tangent has a period of π. Example: tan π 4 = tan π 4 + π = tan(5π 4 )

38 Using the Period to Evaluate Evaluate sin( 13π ) using its period as an aid. 6

39 Even-Odd Properties ODD FUNCTIONS sin θ = sinθ csc θ = cscθ cot θ = cotθ tan θ = tanθ EVEN FUNCTIONS cos θ = cosθ sec θ = secθ

40 Example- Using Even/Odd Properties Find the exact value of: 1. sin 45 0 = 2. cos π = 3. cot 3π 2 = 4. tan 37π 4 =

41 Signs of Trig Functions

42 Fundamental Trig Identities Fundamental Identities sinθ = y cosθ = x tanθ = y x cscθ = 1 y secθ = 1 x cotθ = x y Reciprocal Identities cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ Quotient Identities tanθ = sinθ cosθ cotθ = cosθ sinθ

43 Approximating Trig Values Evaluate each of the following using a calculator. 1. sin π 6 2. tan 54 o 3. sec(1. 3) 4. cot(4)

44 Homework Page 264 #29-38,43-46

45 4.3 Right Triangle Trigonometry What You ll Learn: #82 - Evaluate trig functions of acute angles. #83 - Use the fundamental trig identities. #84 - Use a calculator to evaluate trig functions. #85 - Use trig functions to model and solve real-life problems.

46 Identities and Conditional Equations The following are examples of identities: sin 2 x + cos 2 x = 1 csc x = 1 sin x The following are examples of conditional equations: 2x + 4 = 0 sin x = 1 sin x = cos x

47 Identity Two functions f and g are said to be identically equal if f x = g(x) For every value of x for which both functions are defined. Such an equation is referred to as an identity. An equation that is not an identity is called a conditional equation.

48 Already Established Trig Identities Quotient Identities tan θ = sin θ cos θ cot θ = cos θ sin θ Reciprocal Identities csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ Pythagorean Identities sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ

49 Right Triangle in Standard Position r y x

50 Establishing New Identities Establish the identity: csc θ tan θ = sec θ

51 Establishing New Identities Establish the identity: sin 2 θ + cos 2 ( θ) = 1

52 Establishing New Identities Establish the identity: 1+tan θ 1+cot θ = tan θ

53 SOH CAH TOA

54 Find the exact values of all 6 trig functions of θ from a right triangle 12 θ

55 Complimentary Angles : Cofunctions

56 Examples : Cofunctions sin 30 o = cos 90 o 30 o tan 40 o = cot 90 o 40 o = cos 60 o = cot 50 o sec 80 o = csc(90 o 80 o ) = csc 10 o

57 Using Cofunctions Find the exact value of each expression: 1. sin 62 o cos 28 o 2. tan 35 o cot 55 o

58 Solving a Right Triangle Find the missing values, if A A = 40 o, b = 2. 2 b 40 o c Find: a= c = B = C a B

59 Solving a Right Triangle If a = 3 and b = 2, find c, A, and B. A B C

60 Find the height of the building. Applications

61 Angles of Elevation / Depression

62 Angles of Elevation / Depression 20 o

63 Applications How tall is the light post?

64 Applications θ Find the angle the shooter has to raise his head in order to see the rim.

65 Applications How tall is the statue on top of the building? 400 feet 45 o o

66 Applications Find the distance between the two people. 50 m X

67 Homework Page 274 #1,9,27,29,33-37,47,48,53-56,59,62,63

68 4.4 Trig Functions of Any Angle What You ll Learn: #86 - Evaluate trig functions of any angle. #87 - Use reference angles to evaluate trig functions. #88 - Evaluate trig functions of real numbers.

69 Right Triangle in Standard Position What if the circle had a radius other than 1 unit? You ll have this: sin θ = y r cos θ = x r etc. r x y

70 A Non-Unit Circle (Radius of not 1)

71 Example Let ( 3, 4) be a point on the terminal side of θ. Find the sine, cosine, and tangent of θ.

72 Example Given tan θ = 5 4 secant of θ. and cos θ > 0, find sine and

73 Example Using Trig Identities Let θ be an angle in Quadrant II such that sin θ = 1 3. Find cos θ and tanθ by using trigonometric identities.

74 Example Using Triangles Let θ be an angle in Quadrant II such that sin θ = 1 3. Find cos θ and tan θ by using triangles.

75 Homework Page 284 #1,7,13-18,29-36,(53-69 ODD), 89,99

76 Graph

77 4.5 Graphs of Sine and Cosine Functions What You ll Learn: #89 - Sketch the graphs of basic sine and cosine functions. #90 - Use amplitude and period to help sketch the graphs of sine and cosine functions. #91 - Sketch translations of graphs of sine and cosine functions. #92 - Use sine and cosine functions to model real-life data.

78 First off Typically we display the trig functions like this: y = sinθ y = cotθ Now we will display them as (x,y) equations for the purpose of graphing them on the (x,y) grid. y = sin x y = cot x

79 The Graph of y = sin x x y = sin x (x, y) 0 π 6 π 2 5π 6 π 7π 6 3π 2 11π 6 2π

80 One Period of y = sin x

81 The Graph of y = cos x x y = cos x (x, y) 0 π 3 π 2 2π 3 π 4π 3 3π 2 5π 3 2π

82 One Period of y = cos x

83 Transformations f x = sin x + C Vertical Translation The added (or subtracted) constant will translate the graph up (or down) that many units. For Example: Graph each of the following. y = cos x 1 y = sin x + 2

84 y = cos x 1 y = sin x + 2 Graph

85 Transformations y = sin(x + C) Horizontal Translation The added (or subtracted) constant INSIDE the trig function will translate the function right (or left). For Example: Graph each of the following. y = cos(x + π) y = sin(x π 2 )

86 y = cos(x + π) y = sin(x π 2 ) Graph

87 Transformations y = C sin(x) Amplitude The constant multiple in front will change how high and how low the wave will displace from the middle. It may also negate (or flip) the entire graph. For Example: Graph each of the following. y = 2cos(x) y = 3sin(x)

88 y = 2cos(x) y = 3sin(x) Graph

89 Transformations y = a sin(bx) Period The constant multiple in front of the variable and inside the trig function will affect the period. Period = 2π b For Example: Graph each of the following. y = cos(2x) y = sin( x 2 )

90 y = cos(2x) y = sin( x 2 ) Graph

91 Examples Find the period and amplitude of each trig function: 1. y = 5 cos(3x) 2. y = 3 sin( 1 x) 3 3. y = 3 cos(πx) 2

92 Examples Write the sine function that follows the given conditions: 1) Amplitude = 3 and Period = π 2 2) Amplitude = 5, period = 6 and flipped. 3) Period = 3π and vertical shift of down 3 units.

93 Mathematical Modeling Example 8 on Page 293 Throughout the day, the depth of the water at the end of a dock varies with the tides. The table shows the depths (in feet) at various times during the morning. a) Use a trig function to model this data. b) A boat needs at least 10 feet of water to moor at the dock. During what times in the evening can it safely dock? Time Depth, y 12:00am 3.1 2: : : : : Noon 0.9

94 Homework Page 294 #1-5, #63-66 #74

95 Bell Work 11/5 Find the exact value of each of the following trig functions WITHOUT LOOKING AT YOUR UNIT CIRCLE. 1. cos π 4 = 2. sin 5π 6 = 3. cot π = 4. sec 5π 4 =

96 Multiple Transformations 1. y = 3 cos x y = sin(x π ) 2 3. y = 2 sin x + π 1 4. y = 3cos 4x + 2

97 4.6 Graphs of Other Trig Functions What You ll Learn: #93 - Sketch the graphs of tangent functions. #94 - Sketch the graphs of cotangent functions. #95 - Sketch the graphs of secant and cosecant functions. #96 - Sketch the graphs of damped trig functions.

98 The other four trig functions Tangent (y = tan x) Cotangent (y = cot x) Cosecant (y = csc x) Secant (y = sec c)

99 The Graph of y = tan x

100 Transformations of y = tan x Graph each of the following: y = tan(x π) y = tan(x + π 4 )

101 The Graph of y = cot x

102 Transformations of y = cot x Graph the following function: y = cot(2x)

103 Graph of y = csc x

104 Graph of y = sec x

105 Transformations of y = sec x & y = csc x Graph each of the following trig functions. 1. y = sec(x + π 2 ) 2. y = csc x 1

106 Transformations of y = sec x & y = csc x Graph each of the following trig functions. 1. y = 2sec(x π 2 ) 2. y = csc 4x + 2

107 Homework Page 305 Page 305 #1-6,11,13,17,29

108 4.7 Inverse Trig Functions What You ll Learn: #97 - Evaluate inverse sine functions. #98 - Evaluate other inverse trig functions. #99 - Evaluate compositions of trig functions.

109 One-To-One Functions To have an inverse, a function MUST be one-to-one. The vertical line test determines if an equation is a function. One-to-one functions must pass the horizontal line test.

110 The Inverse Sine Function y = sin 1 x

111 Closer Up : y = sin 1 x

112 Inverse Sine Function y = sin θ Switch the y and the θ to find inverse. θ = sin 1 y

113 The Inverse Sine Function y = sin 1 x Range π 2 y π 2 Domain 1 x 1

114 Another look at the graph Inverse Properties sin 1 sin x) = x sin(sin 1 x) = x Just like this: x = x

115 Example Find the exact values of: 1. sin 1 (1) 2. sin sin 1 ( 3 2 )

116 The Inverse Cosine Function y = cos 1 x

117 A closer look

118 The Inverse Cosine Function y = cos 1 x Range 0 y π Domain 1 x 1

119 Example Find the exact values of: 1. cos 1 (1) 2. cos cos 1 ( 3 2 )

120 Example W/ Calculator Find the approximate value of: sin 1 0.2

121 Inverse Tangent Function

122 Inverse Tangent Function y = tan 1 x Range π 2 < y < π 2 Domain < x <

123 Example Find the exact value of: 1. tan 1 ( 3) 2. tan 1 ( 1)

124 Composite Trig Functions Find the exact value of: cos 1 [cos π 12 ] = π 12 cos[cos 1 ( 0.4)] = 0.4

125 csc 1 θ, sec 1 θ, and cot 1 θ The Restrictions y = csc 1 x y = sec 1 x y = cot 1 x Range: π 2 y π 2 Range: 0 y π Range: 0 < y < π

126 Best Way to Think of These sec 1 (2) cos Flip both the trig function and its argument.

127 Examples Find the exact value of each: 1. csc 1 (2) 2. sec 1 ( ) 3. cot 1 (1)

128 Approximating Use a calculator to approximate each expression in radians. Round to two decimal places. sec 1 (3) cot 1 ( 1 2 )

129 Using a triangle Find the exact value of the trig expression: sin(arc tan 3 4 )

130 Homework Page 316 #1-7, (13-17 odd), 27,31-34, 39,76

131 4.8 Applications and Models What You ll Learn: #100 - Solve real-life problems involving right triangles. #101 - Solve real-life problems involving directional bearings. #102 - Solve real-life problems involving harmonic motion. In-Class Assignment Page 326 #1,8,21,22,36,37

132 Chapter 4 Review Assignment Page 333 #7,8,15,16,27,28,31,33,35,37,39,40,45,47,49,53,54,58,61,70,77,88,99,103,109,121,137,