You are not expected to transform y = tan(x) or solve problems that involve the tangent function.

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2 In this unit, we will develop the graphs for y = sin(x), y = cos(x), and later y = tan(x), and identify the characteristic features of each. Transformations of y = sin(x) and y = cos(x) are performed and are used to model and solve problems. You are not expected to transform y = tan(x) or solve problems that involve the tangent function.

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4 Going Around in Circles Mathematical Models for Rotating Wheels Ferris Wheels Car Wheels Here we look at the relationship between time and the height of a point on the outside of a rotating wheel.

5 Ferris Wheels Where do you start your ride when you get on a Ferris Wheel? You start at the bottom. As you begin to move, you start to rise above the ground so you are increasing your height. Your max height is reached when you reach the top and then your height gets less until you reach your minimum height at the bottom of the ride. Thus your height is constantly changing as time goes on.

6 Graph of Your Ferris Wheel Ride! What is the Max Height? What is the min height? How long does it take to make one complete revolution?

7 Components of the Graph NOTE: This graph is a SINUSOIDAL Graph

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10 Introduction to Sinusoidal Functions There are many examples in the real world that can be modelled by Sinusoidal Functions. Wheels rotating. Ocean waves. Important to surfers Tides Hours of Daylight Electrical Current Alternating Current

11 Periodic Functions Data that repeats itself is called periodic and is represented by a periodic function. Heart Beats The amount of time (distance between corresponding x-values) it takes for the data (y values) to repeat is called the period.

12 What is the period for this graph?

13 The main type of periodic function studied in this unit is sinusoidal. Sinusoidal functions have a wave shape repeating in horizontal direction. Similar to the Ferris wheel graph.

14 Describe the following graphs as Periodic and/or sinusoidal.

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16 Components of a Sinusoidal Graph Period Local Max Largest y- value Local Min Smallest y-value Period Horizontal distance required for graph to repeat itself Sinusoidal Axis Centre line of graph Amplitude Amplitude Distance from sinusoidal axis to max( or min) point SA

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20 Graph each function for the domain 0 θ 3π. a) y = sin θ b) y = cos θ

21 A) Complete the table of values. Round values to one decimal place. B) Plot the points and join them with a smooth curve.

22 b) y = cos θ

23 5 key points There's 5 key points we associate with the graphs, since these points will help us determine the characteristics of the graph (amplitude, domain, range, period, zeros, and sinusoidal axis). However we are not limited to these points.

24 Graph of y = sin x y-intercept: x-intercepts(zeros): max value: min value: Amplitude max min 2 Sinusoidal Axis max min 2 Period Domain Range min, max

25 Graph of y = cos x y-intercept: x-intercepts(zeros): max value: min value: Amplitude max min 2 Sinusoidal Axis max min 2 Period Domain Range min, max

26 Transformations of Sinusoidal Functions In Unit 2 we worked with transformations of graphs in the form y = af (b(x h)) + k. We will now apply that knowledge to sinusoidal graphs: y = a sin b (x c )+ d or y = a cos b (x c )+ d Note: Parameters h and k are now referred to as c and d. We will determine the end position of the five key points using transformations (mapping rule), and then extend the graph appropriately.

27 We need to know what characteristics of the graph change with each parameter. To achieve this, we will investigate how each parameter change affects the resulting graph one parameter at a time, and match these changes to the defining characteristics of sinusoidal graphs.

28 1. Determine how varying the value of a affects the graph of y = a sin x and y = a cos x. The value of a is a Vertical Stretch (and/or Reflection) Sketch the graphs of: A) y = 2sinx B) y= 0.5sinx y y x x

29 What characteristic(s) of the sinusoidal graph changed with a? Amplitude Max and min values Range For the function y = a sin x, (and y = a cos x) the amplitude is a.

30 C) y = -3cos x What is the amplitude? y x What is the Range? Note: There is also a vertical reflection about the x-axis

31 Example: Determine the amplitude of each function. A) y = -4sin x B) -2y= cos x C) 5y=40 sin x

32 2. Determine how varying the value of b affects the graph of y = sin bx and y = cos bx. The value of b is a Horizontal Stretch (and/or Reflection) Sketch the graphs of: A) y = sin(2x) B) y= sin (0.5x) y y x x

33 What characteristic(s) of the sinusoidal graph changed with b? Period For the function y = sin bx, (and y = cos bx) the new period is new period Radians 2 newperiod b old period b new period Degrees o 360 b

34 C) y = cos -4x What is the period? y x Note: There is also a Horizontal reflection about the y-axis. However, in this case there is no change since cosine is symmetric about the y-axis

35 Example: Determine the amplitude of each function. A) y = -4sin x B) -2y= cos 3x C) y=4 sin (0.1x)

36 Example: Sketch the graph of y = 3sin2x for at least two cycles. Determine the amplitude, period, max and min values, x intercepts, y intercepts, domain and range and equation of sinusoidal axis y y=sin x (x, y) y = 3sin2x x

37 Example: Sketch the graph of y = 3sin2x for at least two cycles. y Amplitude: Period: max and min values: x intercepts: x y intercepts: Domain Range Equation of sinusoidal axis.

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39 Transformations of Sinusoidal Functions

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41 1. Determine how varying the value of d affects the graph of y = sin x + d and y = cos x + d The value of d is a Vertical Translation Sketch the graphs of: A) y = sin x + 1 B) y= sin x - 2 y y x x

42 What characteristic(s) of the sinusoidal graph changed with d? Sinusoidal Axis Max and min values Range For the function y = sin x + d, (and y = cos x + d) the equation of the Sinusoidal Axis is y = d.

43 C) y = cos x y x What is the equation of the sinusoidal axis? What is the range?

44 Example: Determine the equation of the sinusoidal axis of each function. A) y = sin x +4 B) y - 2= cos x C) 5y= 5sin x-40

45 What is the equation of the function below? y x

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47 2. Determine how varying the value of c affects the graph of y = sin (x c) and y = cos (x c) The value of c is a Horizontal Translation Sketch the graphs of: 2 A) y = sin (x ) B) y= sin (x + ) 2 y y x x

48 What characteristic(s) of the sinusoidal graph changed with c? Starting point (middle of upward slope for sine) With sinusoidal functions the horizontal translation is referred to as the phase shift, and will determine how the 5 key points will be translated horizontally For the function y = sin(x c), (and y = cos (x c)) c is the phase shift.

49 2 C) y = cos (x - ) y x What does y = cos (x - 2 ) equal?

50 Example: Determine the phase shift of each function. A) y = sin (x + 4) B) y = cos (x - ) C) y= sin (4x-6 )

51 Example: Match each function with its corresponding function rule x y x y x y x y

52 Page 250 #1 a,b, 2a,b, 5

53 y = a sin b (x c )+ d OR y = a cos b (x c )+ d Mapping Rule:

54 Parent Graph y sin x x y 0 o 0 90 o o o o 0 Period = Amplitude = Sinusoidal Axis: Max y value = Min y value = Range: Domain:

55 OR in Radians Period = Amplitude = Sinusoidal Axis: Max y value = Min y value = Range: Domain:

56 Examples: Graph the following ( y 2) sin( x 45 ) Mapping rule: a: b: c: d: (x,y) Period: Amplitude: Sinusoidal Axis:

57 2. y 2cos( x 30 ) 1 Mapping rule: a: b: c: d: (x,y) Period: Amplitude: Sinusoidal Axis: Starting Point:

58 3 3. 2( y 1) cos( x ) 4 Mapping rule: a: b: c: d: (x,y) y x Period: Amplitude: Sinusoidal Axis: Starting Point: max y value: min y value: Range:

59 ( y 3) sin ( x 2 ) 3 2 Mapping rule: a: (x,y) b: c: d: Period: Amplitude: Sinusoidal Axis: Starting Point: Jump: max y value: min y value: y x

60 y = a sin b (x c )+ d Summary Eq of Sinusoidal Axis: Amplitude: Local Max: Local Min: Period: Start: Jump: Range: min, max c y y d d a d a d c (sin) a a c (cos) d a o or b b o or b b period 4 period 4 d a

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62 1. Reading a Sinusoidal Equation Reflection? Vertical Stretch: Vertical Translation: Horizontal Stretch: Horizontal Translation: Amplitude: Sinusoidal Axis Period: Starting Point: Max Value: Min Value: Range: Sketch Graph 1 ( 6) sin2( 135 ) 5 y x y y x

63 Example y 2cos ( x ) y Reflection? Vertical Stretch: Vertical Translation: Horizontal Stretch: Horizontal Translation: Amplitude: Sinusoidal Axis Period: Starting Point: Max Value: Min Value: Sketch Graph x

64 Example 3. Reflection? 2 sin( 4 12) 2 3 y x Vertical Stretch: Vertical Translation: Horizontal Stretch: Horizontal Translation: Amplitude: Sinusoidal Axis Period: Starting Point: Max Value: Min Value: y x Range: Sketch Graph

65 4. Write an equation for each transformation of y = sin x given the mapping rules below: A) ( x, y ) ( x 2, y 3) B) ( x, y ) ( x 4,3y 1) 1 C ) ( x, y ) x, 4y 3 2 D) ( x, y ) 5x 120, y 11 7 Page 250 #4

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68 1. Find the equation of the graph below as a transformation of y = sinx. Amplitude Sinusoidal Axis: Period: Formulae max min VS = a 2 max min VT = d 2 2 newperiod b 2 b newperiod We need to find the values of a, b, c, and d for the equation : y = a sin b (x c )+ d o 360 or b newperiod Starting Point: phase shift = HT c NOTE: There are an infinite number of correct choices for the phase shift, the convention is to use the smallest positive value. Equation: Bonus: What is the equation if we used y = cos x?

69 2. Find the equation of the graph below Amplitude Sinusoidal Axis: Period: Starting Point: Equation:

70 3. Find the equation of the graph that models the height of the piece of tape as the can rolls across the floor. y cm x Tape Distance Rolled(cm) Equation: BONUS: What is the diameter of the can? SUPER-DUPER Bonus: What is the height of the tape when the can rolls 42 cm?

71 Multiple Choice Questions 1. If the amplitude of a sinusoidal function is 3, and the maximum point is 4, what is the equation of the sinusoidal axis? A. y = 7 B. y = 1 C. y = -1 D. y = If the equation of the sinusoidal axis is y = 1 and the maximum point of a sinusoidal function is 4, what is the minimum point on the graph? A. - 4 B. - 3 C. - 2 D. 7

72 Multiple Choice Questions 3. If the starting point of a sine wave is -45 and the next cycle begins at 315, what is the period of the function in degrees? 4. If a maximum point of a sinusoidal function is 1 and a minimum point is - 6, what is the equation of the sinusoidal axis? A. 180 B. 270 C. 315 D. 360 A. y = -5 B. y = -5/2 C. y = 7 D. y = 7/2

73 For the following multiple choice questions, use the equation: 2 ( y - 5 ) = sin 3 ( x o ) 5. What is the amplitude of the above equation? 7. What is the maximum point on the graph? A. ½ B. 1 C. 2 D What is the period of the above function, in degrees? A. 120 B. 180 C. 360 D A. 4.5 B. 5 C. 5.5 D.7 8. What is the range of the function? A. {y/ -5 < y < 5} B. {y/ 3 < y < 7} C. {y/ -7 < y < 7} D.{y/ 4.5 < y < 5.5}

74 Multiple Choice Questions 9. If the horizontal stretch factor of a sinusoidal function is 3, what is the period of its graph, in degrees? A. 60 B. 120 C. 540 D What is the horizontal stretch factor of a sinusoidal function whose period is 90 degrees? A. ¼ B. ½ C. 2 D. 4

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77 1. A pebble is stuck in the grooves of a car tire. As the car moves at a speed of 60 inches/sec, the height of the pebble changes. If the tire has a 15-inch radius, graph the path traveled by the pebble. Determine the equation of the graph.

78 2. Jackie, Nicole and Megan are playing skip rope. As the rope rotates it is observed that its maximum height is 3m and its first maximum occurs 1s after starting. If the first minimum occurs 2s after the maximum at a height of 0.2m. Determine the equation that expresses the skip ropes height above the ground in terms of time

79 3. A mother puts her child on a Merry-Go-Round. To watch her child, she stands at a point that is initially 3 metres away from the child, which is the closest distance between the mother and child. At 6 seconds, the child is 15 metres from his mother, which is the farthest distance between the mother and child. Assuming the distance between the mother and child varies sinusoidally with time, determine the relation that models this situation. distance (m) y x time (seconds)

80 4. Determine the equation that models the hours of daylight for one complete year in Corner Brook, NL We need some data. When is the shortest day? Dec 21 What is the length of the shortest day? 8 hours 14 min 8.23 h What day of the year is this? 355 When is the longest day? Jun 21 What is the length of the longest day? 16 hours 12 min h What day of the year is this? 172

81 What is the average daylight hours? hours What is the equation? What is the hours of daylight for today? Page #20,26A, 27A) C)

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83 What is the slope of the terminal arm for angle on the unit circle as it rotates from 0 to 90? Lets start with at 0 o r = 1 y rise y 0 slope m 0 run x 1 What happens as the angle increases? What is the slope at 10 o? x slope y x sin cos sin10 cos10 o o What is a quicker way to calculate the slope? m sin cos tan tan10

84 r = 1 x y Find the slope of the terminal arm for the following values of. 30 o 45 o 60 o 75 o 89 o 89.9 o Slope What happens to the slope as the angle approaches 90? Slope approaches infinity. What is the slope of the line at 90? Undefined ( ) Which other angles would provide the same slopes as those above? 30 o 45 o 60 o 75 o 89 o 89.9 o Other Angles

85 r = 1 y What are the slopes for rotations from 0 to 90? 0 o -30 o -45 o -75 o -89 o -90 o Slope Compare these to slopes when the terminal arm is rotated past o 91 o 105 o 135 o 150 o 180 o Slope x Is the tangent function periodic? Yes. What is its period? 180 o or π radians

86 Recall: For the function: y = tan x tanx sinx cosx Where is tan x undefined? Places where cos x = 0 x k, k Z 2 Plot these as Vertical Asymptotes Note: An Asymptote is a line that a graph approaches, but never reaches. There are: Vertical Asymptotes where a function s y values approach infinity Horizontal Asymptotes where a function goes to a certain y-value as the x values approach infinity (or negative infinity) Slant Asymptotes where a function approaches a line as the x values approach infinity (or negative infinity)

87 Where would tan x cross the x- axis? (x- intercepts) Places where sin x = 0 x k, k Z Plot these as Zeros

88 Graphing the Tan Function What table of values do we use to graph the tan function? y x y What is the period? x What are the intervals of increasing or decreasing for the tan graph? 4 2 Note: A function is increasing if it goes up as you travel from left to right across the graph. It is decreasing if the function goes down. Domain: Range: x k, k Z 2,

89 Does the graph of y = tan(x) have an amplitude? amplitude is a characteristic of sine and cosine graphs and it depends on a maximum and a minimum height. Since the function y = tan(x) has no maximum or minimum, it cannot have an amplitude.

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92 In the previous unit, we solved first and second degree trigonometric equations algebraically. 2 2 cos cos 1cos Such as: In this unit, we continue to solve equations algebraically. We also use the graphs of trigonometric functions to solve equations. For assessment purposes, you will analyze given graphs to determine solutions. We will also solve trigonometric equations for which the argument may include a horizontal stretch or a horizontal translation. Such as: sin

93 Using graphs to solve trigonometric equations. Example 1: Using the graph shown, determine the general solution for -3sin(2(x-π/4))+3=3 y y=-3sin(2(x-π/4))+3 x

94 Example 2: Using the graph shown, determine the general solution for 2sin x 1 0 2

95 Example 3. Solve y 4cos 2 x for x ε[0,2π] x y=4cos(2(x-π/6))+1

96 Using algebra to solve trigonometric equations involving horizontal stretches and translations. Lets solve the last 3 problems using algebra 1. Determine the general solution for -3sin(2(x-π/4))+3=3 Solution: First lets simplify what we are solving. Let 2(x-π/4)) = m 2 x k Solve -3sin(m)+3=3 4 3sin m 0 x k sinm m sin (0) k 2k 2k x m 0, (2k 1) m k, where k x, where k 4

97 Remember to check solutions by evaluating each side of the original equation at all (or a few) of the solution values.

98 2. Determine the general solution for 2sin x 1 0 2

99 4cos 2 x 1 3 for x 0,2 6

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101 4. Solve: x cos 4 90, for x 0,

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103 FUN One! 5. Solve: 3sin 4 x 2 2,for x,2 Express answers as exact values, or rounded to 2 decimals.

104 Text Page 275 # 1-3, 5, 13a) 14