Chapter 8: SINUSODIAL FUNCTIONS

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1 Chapter 8 Math 0 Chapter 8: SINUSODIAL FUNCTIONS Section 8.: Understanding Angles p. 8 How can we measure things? Eamples: Length - meters (m) or ards (d.) Temperature - degrees Celsius ( o C) or Fahrenheit (F) How can we measure angles? Up until now we can measure angles using degrees. There is an alternative unit of measurement for measuring angles, that is, RADIANS. RADIAN: One radian is the angle made b taking the radius and wrapping it along the edge (an arc) of the circle. r r INVESTIGATE: How man pieces of this length do ou think it would take to represent one complete circumference of the circle? To help, cut a piece of pipe cleaner/string to a length equal to radius CA and bend it around the circle starting at point A.. Approimatel, how man radius lengths are there in one complete circumference?. How man degrees in one complete circumference? radian ~ Degrees. Therefore, approimatel, how degrees are in one radian?

Chapter 8 Math 0 NOTE:. The size of the radius of a circle has NO effect on the size of radian.. The advantage of radians is that it is directl related to the radius of the circle.

2 Chapter 8 Math 0 NOTE:. The size of the radius of a circle has NO effect on the size of radian.. The advantage of radians is that it is directl related to the radius of the circle. This means that the units of the and ais is consistent and the graph of the sine curve will have its true shape, without vertical eaggeration. Let s consider a circle with radius unit (r = ). (This is called the unit circle!). What is the circumference of a circle?. What is the circumference of the unit circle?. How man degrees are there in a complete revolution of a circle?. Wh must the values from #. and #. be equal to each other? 5. State the proportion: 6. Divide both sides b π to determine the measure of radian: 7. Complete the following and add the equivalent radian measures on the unit circle below: a. = b. π = c. d. e. f. 6

3 Chapter 8 Math 0 Converting Degrees Radians. Degrees to Radians: o To convert from Degrees to Radians multipl b 80 o. Radians to Degrees: o To convert from Radians to Degrees multipl b 80 o Eamples:. Convert to radians: a. 0 o b. 0 o c. 50 o d. 660 o. Convert to degrees: a. b. 6 c. 9 d. 5. rad. For each pair of angle measures, determine which is greater: a. 50 o, rad b. 50 o, 7rad c., 8rad ASSIGN: p. 89, #,, 5, 7-0

4 Chapter 8 Math 0 Worksheet 8. Section : Selected Response. What is the best estimate for 6 in radians? A). B) 0.7 C).8 D).. What is 0 epressed in eact radians? A) π B) π C) π 8 D) π. What is the best estimate for 0. radians in degrees? A) 0.5 B) C) D) 6. What is 5π 8 epressed in degrees? A).5 B) 88 C) 900 D) 0 5. What is the best estimate for the central angle in degrees? A) 6 B) 7 C) 8 D) 9 6. What is the best estimate for the central angel in radians? A) 5π B) 6π 5 C) 5π D) 5π 6

5 Chapter 8 Math Imagine that it is now p.m. What time will it be when the minute hand has rotated through 00? A) :0 B) :50 C) :00 D) :0 8. Imagine that it is now p.m. What time will it be when the minute hand has rotated through 7π radians? A) :0 B) :5 C) :5 D) :5 Section : Constructed Response Answer all of the following showing all work. 9. Eddie is facing west. What direction will he be facing if he rotates 5 to his right? 0. For the following pair of angle measures, determine which is greater 75 or π? ANSWERS. A. D. D. A 5. B 6. D 7. B 8. B 9. SE 0. π

Chapter 8 Math 0 6 Section 8.: Eploring Graphs of Periodic Functions p.

Amplitude The distance from the midline to either the maimum or minimum value of a periodic

6 Chapter 8 Math 0 6 Section 8.: Eploring Graphs of Periodic Functions p. 9 Terms to Know: Periodic Function Midline / Sinusodial Ais A function whose graph repeats in regular intervals or ccles. The horizontal line halfwa between the maimum and minimum values of a periodic function. Amplitude The distance from the midline to either the maimum or minimum value of a periodic function; the amplitude is alwas epressed as a positive number. Period The length of the interval of the domain to complete one ccle. Sinusodial Function An periodic function whose graph has the same shape as that of = sin.

7 Chapter 8 Math 0 7 Section 8. Eploration Sine Graph. Complete the table of values below for the function sin (DEGREE MODE) Sketch the graph of sin on the graph below: Complete the tables below b using the graph. If ou wanted to quickl graph the sine curve, which five points would allow ou to easil graph the entire curve? sin Five Ke Points Period Sinusoidal Ais (midline) Amplitude Domain Range Local Maimums Local Minimums -intercepts -intercepts

8 Chapter 8 Math 0 8 Section 8. Eploration Cosine Graph. Complete the table of values below for the function cos (DEGREE MODE) Sketch the graph of cos on the graph below: Complete the tables below b using the graph. If ou wanted to quickl graph the cosine curve, which five points would allow ou to easil graph the entire curve? cos Five Ke Points Period Sinusoidal Ais (midline) Amplitude Domain Range Local Maimums Local Minimums -intercepts -intercepts

9 Chapter 8 Math 0 9 OBSERVATIONS: For SINE, one complete wave can be seen from an X-INTERCEPT at (0, 0) to the X-INTERCEPT at (60 o, 0). For COSINE, one complete wave can be seen from the MAXIMUM point (0, ) to the net MAXIMUM point at (60 o, ). The graph of the left. cos is related to the graph of sin b a shift of 90 o to Label the following as periodic, sinusoidal or both. a. c. b. d. CONCLUSION: All sinusoidal functions are periodic but not all periodic functions are sinusoidal. ASSIGN: p. 9, #, 5-8

10 Chapter 8 Math 0 0 Worksheet 8.. Complete the following table for = sin θ. Period Sinusoidal Ais (midline) Amplitude Domain Range In Degrees or Radians (if possible) Local Maimum Local Minimum -intercept(s) -intercept(s) Draw the graph of = sin θ using the five ke points Draw the graph of = cos θ using the five ke points

11 Chapter 8 Math 0 Section 8.: The Graphs of Sinusoidal Functions p. 97 Equation of the midline: is the average of the maimum and minimum values: = maimum value + minimum value Amplitude: is the positive vertical distance between the midline and either a maimum or minimum value. It is also half of the vertical distance between a maimum value and a minimum value. maimum value minimum value Amplitude = Eamples:. For the sinusoidal function shown, determine: (Eample, p. 99) a. Range: b. Equation of Midline/Sinusoidal Ais: c. Amplitude: d. Period:

12 Chapter 8 Math 0. For the sinusoidal function shown, determine: (Eample, p. 99) a. Range: b. Equation of Midline/Sinusoidal Ais: c. Amplitude: d. Period:. While riding a Ferris wheel, Mason s height above the ground in terms of time can be represented b the following graph. a. How far is the Ferris wheel off the ground? b. What is the range of the function? What does it represent? c. What is the height of the Ferris wheel? d. What is the equation of the midline? What does it represent? e. How long does it take for the Ferris wheel to make one complete revolution? What characteristic does this correspond to?

Chapter 8 Math 0. Aleis and Colin own a car and a pickup truck. The noticed that the odometers of the two vehicles gave different values for the same distance.

13 Chapter 8 Math 0. Aleis and Colin own a car and a pickup truck. The noticed that the odometers of the two vehicles gave different values for the same distance. As part of their investigation into the cause, the put a chalk mark on the outer edge of a tire on each vehicle. The following graphs show the height of the tires as the rotated while the vehicles were driven at the same slow, constant speed. What can ou determine about the characteristics of the tires from these graphs? (Eample, p. 50) ASSIGN: p. 507, #, 5, 7 0, - 5

14 Chapter 8 Math 0 Worksheet 8.. What is the midline of the following graph? A) = B) = C) = D) = 5. What is the amplitude of the following graph? A) B) C) D) 5. What is the period of the following graph? A) 0 B) 0 C) 00 D) 60. What is the range of the following graph? A) { 5, εr} B) {, εr} C) { 0, εr} D) { εr} 5. A sinusoidal graph has an amplitude of 0 and a maimum at the point (8, 5). What is the midline of the graph? A) = 0 B) = -5 C) = D) = 8

15 Chapter 8 Math A sinusoidal graph has a maimum at the point (, -8) and the net minimum is at the point (7, -0). What is the period of the graph? A) B) C) D) 6 7. Sketch a possible graph of a sinusoidal function with the following set of characteristics. Eplain our decision. Domain: { 0 080, ε R} Maimum Value: 5 Minimum Value: -5 Period: 70 -intercept: The graph of a sinusoidal function is shown. Describe the graph b determining its range, the equation of the midline, its amplitude and its period. Show an work.

16 Chapter 8 Math Kira is sitting in an inner tube in the wave pool. The depth of the water below her, in terms of time, during a series of waves can be represented b the graph shown. A) How long does it take for one complete wave to pass? B) What is the approimate depth of the water below Kira after s? C) How high is each wave? D) What is the depth of the water below Kira when no waves are being generated? E) What is the depth of the water below Kira at 9s? Answers: A A B A 5B 6D 8.range { 5, R} midline = Amplitude period 9A) seconds B).m C) 0.8m D) m E) m

17 Chapter 8 Math 0 7 Section 8.: The Equations of Sinusoidal Functions p. 56 INVESTIGATION: Using technolog, we will eplore how the parameters a, b, c and d affect the graph of sinusoidal functions written in the form: asin b( c) d and acos b( c) d (A) The Effect of a in = asin on the graph = sin where a > 0. Below are the sketches of the graphs of: sin, sin, sin, 0.5sin sin sin sin 0.5sin Compare the amplitudes in each graph with its equation.. Describe the affect the value of a has on the graph of sin.. Will the value of a affect the cosine graph in the same wa that it affects the sine graph?

18 Chapter 8 Math 0 8 (B) The Effect of d in = sin + d on the graph = sin 5. Compare the sketches of the graph of: sin and sin to the graph of sin sin sin How does each graph change when compared to sin? 7. How is the value of d related to the equation of the midline? 8. Is the shape of the graph or the location of the graph affected b the parameter d? 9. Is the period affected b changing the value of d? 0. Will the value of d affect the cosine graph in the same wa that it affects the sine graph?

19 Chapter 8 Math 0 9 (C) The Effect of b in = sinb on the graph = sin. Compare the sketch of the graphs of: sin sin and sin 0.5 to the graph of sin sin What is the period of sin? What is the b value?. What is the period of sin? What is the b value?. What is the period of sin 0.5? What is the b value? 5. What is affected b the value of b? 6. Write an equation that relates the b value to the period of the function. Period or Period 7. Will the value of b affect the cosine graph in the same wa that it affects the sine graph?

20 Chapter 8 Math 0 0 (D) The Effect of c in = sin( c) on the graph = sin o o 8. Sketch a graph of: sin( 60 ) and sin( 0 ) and compare it to the graph of sin o o 0 sin( 60 ) sin( 0 ) sin( ( 0 )) maimum maimum minimum - minimum How is the c value affecting the graph? This horizontal shift is also called the phase shift of the graph. In order to determine the phase shift of the graph, ou need to compare a KEY POINT on the sine graph to determine if it has shifted left or right. One ke point on the SINE graph is the point (0, 0) which intersects the midline at = 0 going from a minimum point to a maimum. 0. If the c value is positive, the graph shifts to the.. If the c value is negative, the graph shifts to the.. Will the value of c affect the cosine graph in the same wa that it affects the sine graph?

21 Chapter 8 Math 0 (E) The Effect of c in = cos( c) on the graph = cos o o. Compare the graphs of: cos( 90 ) and cos( 60 ) to sin o o 0 cos( 90 ) cos( 60 ) cos( ( 60 )) maimum maimum One KEY point on the COSINE graph is the point (0, ) which is a maimum point.. If the c value is positive, the graph shifts to the. 5. If the c value is negative, the graph shifts to the. CONCLUSION: For the sinusoidal functions written in the form: asin b( c) d and acos b( c) d. a = amplitude =. b affects the period: ma min 60 o period or b period b. c = horizontal shift (need to compare a Ke point). d = vertical translation, = d = midline/sinusoidal ais ma min = equation of the 5. maimum value = d + a; minimum value = d a 6. Range: {min value ma value}

22 Chapter 8 Math 0 NOTE: Beware of the brackets! cos( ) versus cos cos( ) results in a horizontal translation cos results in a vertical translation Eamples:. For the function, cos, state: (Eample, p. 58) a. amplitude: b. equation of the midline: c. range: d. period: e. phase shift: o. For the function, sin ( 5 ), state: (Eample, p.59) a. amplitude: b. equation of the midline: c. range: d. period: e. phase shift: o f. How would the graph of cos( 5 ) be the same? How would it be different?

23 Chapter 8 Math 0. Match graph and graph below with the corresponding equation. (Eample, p. 5) o o i. cos( 90 ) iii. 5sin( 60 ) o o ii. sin( 60 ) iv. cos( 60 )

24 Chapter 8 Math 0 o. Ashle created the following graph for the equation sin( 90 ). Identif her error(s) and construct the correct graph

25 Chapter 8 Math The temperature of an air-conditioned home on a hot da can be modeled using the function, f() =.5 cos + 0,where is the time in minutes after the air conditioner turns on and t() is the temperature in degrees Celsius. a. What are the maimum and minimum temperatures in the home? b. What is the temperature 0 minutes after the air conditioner has been turned on? c. What is the period of the function? Interpret this value in this contet?

26 Chapter 8 Math 0 6 Section 8.5: Modeling Data with Sinusoidal Functions p. 5 When a sinusoidal regression occurs the calculator generates an equation = asin(b + c) = d in radian measure. Eample: The average precipitation was recorded and a sinusoidal regression was created for the data which resulted in the following SinReg =a sin(b+c)+d a=.99 b=.50 c=.58 d =.9 a. Using the data in the screenshot above, determine the equation of the sinusoidal regression function that models the change in precipitation. Use the form = asin(b + c) + d. b. Use our equation from Part a to determine what the amount of precipitation when = 7

27 Chapter 8 Math 0 7 Unit 8: Sinusoidal Functions Worksheet 8. Section : Selected Response. Which function below has the greatest amplitude? A) = sin ( + 90 ) + 5 B) = sin ( - 90 ) C) = sin ( + 90 ) D) = sin 0.5( - 90 ). Which function has the greatest period? A) = sin ( + 90 ) + 5 B) = sin ( - 90 ) C) = sin ( + 90 ) D) = sin 0.5( - 90 ). Which function has the greatest maimum value? A) = sin ( + 90 ) + 5 B) = sin ( - 90 ) C) = sin ( + 90 ) D) = sin 0.5( - 90 ). What is the amplitude for the following function: = sin ( + 90 ) A) B) C) D) 5 5. What is the period for the following function: = cos + A) 80 B) 60 C) 70 D) What is the midline of the following function: = 0.5 sin( -) A) = - B) = 0.5 C) = 0 D) = 7. What is the range of the following function: = sin ( + 90 ) A) {, εr} B) {, εr} C) {, εr} D) { εr}

Chapter 8 Math 0 8 8. What is the domain of the following function: = 0.5 sin( ) Section : A) {, εr} B) { 0.5 0.5, εr} C) {, εr} D) { ε R} Constructed Response.

28 Chapter 8 Math What is the domain of the following function: = 0.5 sin( ) Section : A) {, εr} B) { , εr} C) {, εr} D) { ε R} Constructed Response. What is the amplitude of the following function: = cos( π) 5. What is the midline of the following function: = 5 sin.5( + 60 ) 5. What is the range of the following function: = 0 cos ( - 80 ) +. Match each graph with the corresponding equation. Eplain our answers: i) = cos ( - 0 ) + ii) = cos ( 60 ) + iii) = cos( + 60 ) + iv) = cos( 0 ) v) = cos( - 60 ) vi) = cos( + 60 ) 5. Describe the graph of the following function b stating the amplitude, equation of the midline, range and period. = sin() The following graph represents the rise and fall of sea level in part of the Ba of Fund, where t is the time, in hours, and h(t) represents the height relative to the mean sea level. A) What is the range of the tide levels? 6 B) What does the equation of the midline represent in the graph? C) What is the period of the graph? - - D) The equation of the sinusoidal function is represented b: h(t) = 6.5 sin π t. 6 Calculate the period from the equation and compare it to our answer in c. - 6

29 Chapter 8 Math The temperature of an air-conditioned home on a hot da can be modelled using the function t() =.5sin ( π ) + 5, where is the time in hours after the air 6 conditioner turns on and t() is the temperature in degrees Celsius. A) What are the maimum and minimum temperatures in the home? B) What is the temperature 0 hours after the air conditioner has been turned on? C) What is the period of the function? How would ou interpret this value in this contet? Answers: SECTION :B D A B 5D 6C 7C 8D SECTION :.. = -5. { 8, R}. i) A v) B 5. Amplitude midline 5 0 =.5 range {..6, R} period 80 o 6.A) { , R} B) = 0 when the water is calm. C) hours D) period same 7. A) ma 7.5 min.5 B).8 C)

30 Chapter 8 Math 0 0 Chapter 8 REVIEW Section : Multiple Choice ) What is radians in degrees? 0 (A) 0 (B) 5 (C) 08 (D) 600 ) What is (A) 6 5 radians in degrees? (B) 80 (C) 00 (D) 600 ) What is. radians in degrees? (A) (B) 8 (C) 75 (D) ) What is 0 in radians? (A) (B) (C) (D) 5) What is 0 in radians? (A) 0. (B). (C).0 (D) 7.6 6) What is the domain of the function sin? (A) 0, R (B) 0, R (C) R (D), R 7) What is the domain of the function cos? (A), R (B), R (B) 0, R (D) R 8) What is the range of cos? (A) R (B), R (C), R (D) 0

31 Chapter 8 Math 0 9) What is the range of sin? (A) R (B), R (C), R (D) 0 0) Which is an int of cos? (A) 0 (B) 90 (C) 80 (D) 60 ) Which is an int of sin? (A) 0 (B) 90 (C) 70 (D) 00 ) How can the graph of cos be translated so that we get the graph of sin? (A) 90 to the left (B) 90 to the right (C) 5 to the left (D) 5 to the right ) Which graph is periodic? (A) (B) (C) (D)

32 Chapter 8 Math 0 ) Which graph is sinusoidal? (A) 5 (B) (C) 5 (D) ) What is the equation of the midline for the graph shown? (A) (B) (C) (D) ) What is the equation of the midline for the graph shown? (A).5 (B) (C) 0.75 (D)

33 Chapter 8 Math 0 7) What is the amplitude for the graph shown? (A) - (B) - (C) (D) ) What is the amplitude for the graph shown? (A) 0.5 (B) 0.75 (C) 0.5 (D) ) What is the period for the graph shown? (A) (B) (C) (D) ) What is the period for the graph shown? (A) 5 (B) 90 (C) (D)

34 Chapter 8 Math 0 While riding a Ferris wheel, Lil s height above the ground in terms of time can be represented b the following graph: Height above ground (in m) Use the graph to answer #-#5 ) At what height does Lil board the Ferris Wheel? (A) m (B) m (C) 5 m (D) 9 m ) What is the maimum height above the ground? (A) m (B) m (C) 5 m (D) 9 m ) What is the height of the ale of the Ferris Wheel? (A) m (B) m (C) 5 m (D) 9 m ) What is the radius of the Ferris Wheel? Time (seconds) (A) m (B) m (C) 5 m (D) 9 m 5) What length of time does it take for the Ferris Wheel to complete rotation? (A) seconds (B) seconds (C) 6 seconds (D) 8 seconds 6) What is the amplitude of sin 0? (A) (B) (C) (D) 0 7) What is the amplitude of 0.5 cos. (A) 0.5 (B). (C) (D)

35 Chapter 8 Math 0 5 8) What is the equation of the midline of sin. 5? (A) (B). (C) (D) 5 9) What is the equation of the midline of 0.75 cos 0.5 5? (A) (B) 0.5 (C) 0.75 (D) 5 0) What is the period of sin? (A) (B) (C) (D) 8 ) What is the period of sin0.5? (A) 90 (B) 80 (C) 60 (D) 70 ) What is the period of sin? (A) (B) (C) ) What is the phase shift of sin. 5? (D) 6 (A). radians to the right (B). radians to the left (C) 5 radians to the right (D) 5 radians to the left ) What is the phase shift of 0.75 cos 0.5 5? (A) 5 to the right (B) 5 to the left (C) to the right (D) to the left 5) What is the maimum value of sin 0? (A) (B) 5 (C) 6 (D) 7 6) What is the minimum value of 0.75 cos 0.5 5? (A) -.75 (B) -.5 (C) - (D) -.5

36 Chapter 8 Math 0 6 7) What is the maimum value of cos 0.5 7? (A) (B) (C) 8 (D) 8) What is the minimum value of.5 sin5 6 (A) -5.5 (B) -.5 (C) (D) 6.5 9) What is the range of sin? (A) 5, R (B), R (C), R (D) 0) What is the range of.5 cos..5? R (A), R (B) 6, R (C) , R (D) R

37 Chapter 8 Math 0 7 Section : Constructed Response- Answer all questions in the space provided. Be sure to show all workings to receive full credit! ) The table below shows the height of a biccle pedal over time: Time (s) Height (in (a) Graph the function on the grid below: (b) What is the height of the ale of the biccle pedal? (c) What is the length of time taken to complete one rotation? (d) Suppose another cclist pedals the same bike at a rate of revolution per second. How would the graph change? (e) Write the equation of the function in the form a cos b d

38 Chapter 8 Math 0 8 H e i g h t ( f t ) ) The height of the water in a harbour since PM is shown in the graph below: (a) (b) What is the maimum height of the water? What is the minimum height of the water? Time (h) (c) What is the length of time between low tide and high tide? (d) John wants to go collect mussels (the tide must be relativel low for this). He thinks that 6:00 PM would be a good time to go. Do ou agree or disagree with John? Justif. ) The height of water in a wave pool oscillated between a maimum of ft and a minimum of 5 ft. The wave generator pumps 6 waves per minute. (a) What is the equation of the midline? (b) What is the amplitude? (c) What is the period? (d) Write the equation of the function in the form a sinb d

Chapter 8 Math 0 9 models the average monthl 6 temperature for Phoeni, Arizona, in degrees Fahrenheit. In this equation, t denotes the number of months, with ) The equation 70.5 9.

39 Chapter 8 Math 0 9 models the average monthl 6 temperature for Phoeni, Arizona, in degrees Fahrenheit. In this equation, t denotes the number of months, with ) The equation sin t average monthl temperature for Jul? t representing Januar. What is the 5) A horse on a carousel goes up and down as the carousel goes round and round. The height of the horse in inches, h, as a function of time (s) is given b h 9 sin t 6 7 (a) (b) Determine the height of the horse after 8 seconds. The children have been complaining that the carousel is too slow. How would the equation change the conductor makes the carousel go around faster? 6) Bob collects data on the average monthl temperatures of his hometown. He uses his graphing calculator to perform a Sinusoidal Regression. The screenshot of his calculator is shown below (where represents the temperature in degrees Fahrenheit and representing Januar: represents the month with t (a) Write the equation of the curve of best fit. (b) Use our equation to determine the temperature in June.

40 Chapter 8 Math 0 0 7) Match the graph with its equation: Graph Graph Graph Graph Equation : sin Equation : sin Equation : sin0.5 Equation : = sin +

41 Chapter 8 Math 0 Answers: ) B ) C ) B ) D 5) C 6) C 7) D 8) B 9) B 0) B ) A ) B ) B ) C 5) C 6) B 7) C 8) C 9) D 0) D ) A ) D ) C ) B 5) D 6) B 7) A 8) D 9) A 0) B ) D ) B ) A ) B 5) B 6) A 7) D 8) A 9) B 0) B ) (a) (b) Height of ale = midline = in (c) (d) Period = time for one revolution = s Period would be s. Waves would be closer together. One complete wave would be shown from 0- second. (e) 7cos ) (a) Maimum height is ft (b) Minimum height is ft (c) 6 hours (d) No, because the height of water is over ft. ) (a) 9 (b) ft (c) 0 s (d) sin ) 90 F 5) (a) 5 in (b) The b value would decrease 6) (a).sin(0.5.) 5.66 (b) 7 F 7) Graph - Equation, Graph - Equation Graph Equation, Graph - Equation

Math 3201 Unit 8: SINUSODIAL FUNCTIONS NAME: Up until now we can measure angles using degrees.

Math 3201 Unit 8: SINUSODIAL FUNCTIONS NAME: Up until now we can measure angles using degrees. Math 0 Unit 8: SINUSODIAL FUNCTIONS NAME: Sectin 8.: Understanding Angles p. 8 Hw can we measure things? Eamples: Length - meters (m) r ards (d.) Temperature - degrees Celsius ( C) r Fahrenheit (F) Hw