Trigonometric Functions and Graphs

Transcription

1 CHAPTER 5 Trigonometric Functions and Graphs You have seen different tpes of functions and how these functions can mathematicall model the real world. Man sinusoidal and periodic patterns occur within nature. Movement on the surface of Earth, such as earthquakes, and stresses within Earth can cause rocks to fold into a sinusoidal pattern. Geologists and structural engineers stud models of trigonometric functions to help them understand these formations. In this chapter, ou will stud trigonometric functions for which the function values repeat at regular intervals. Ke Terms periodic function period sinusoidal curve amplitude vertical displacement phase shift MHR Chapter 5

2 Career Link A geologist studies the composition, structure, and histor of Earth s surface to determine the processes affecting the development of Earth. Geologists appl their knowledge of phsics, chemistr, biolog, and mathematics to eplain these phenomena. Geological engineers appl geological knowledge to projects such as dam, tunnel, and building construction. Web Link To learn more about a career as a geologist, go to and follow the links. Chapter 5 MHR

3 5. Graphing Sine and Cosine Functions Focus on... sketching the graphs of = sin and = cos determining the characteristics of the graphs of = sin and = cos demonstrating an understanding of the effects of vertical and horizontal stretches on the graphs of sinusoidal functions solving a problem b analsing the graph of a trigonometric function Man natural phenomena are cclic, such as the tides of the ocean, the orbit of Earth around the Sun, and the growth and decline in animal populations. What other eamples of cclic natural phenomena can ou describe? You can model these tpes of natural behaviour with periodic functions such as sine and cosine functions. The Hopewell Rocks on the Ba of Fund coastline are sculpted b the cclic tides. Did You Know? The Ba of Fund, between New Brunswick and Nova Scotia, has the highest tides in the world. The highest recorded tidal range is 7 m at Burntcoat Head, Nova Scotia. Investigate the Sine and Cosine Functions Materials grid paper ruler. a) Cop and complete the table. Use our knowledge of special angles to determine eact values for each trigonometric ratio. Then, determine the approimate values, to two decimal places. One row has been completed for ou. Angle, θ = sin θ = cos θ π_ 6 π_ π_ π =.5.87 b) Etend the table to include multiples of the special angles in the other three quadrants. MHR Chapter 5

4 . a) Graph = sin θ on the interval θ [, π] b) Summarize the following characteristics of the function = sin θ. the maimum value and the minimum value the interval over which the pattern of the function repeats the zeros of the function in the interval θ [, π] the -intercept the domain and range. Graph = cos θ on the interval θ [, π] and create a summar similar to the one ou developed in step b). Reflect and Respond. a) Suppose that ou etended the graph of = sin θ to the right of π. Predict the shape of the graph. Use a calculator to investigate a few points to the right of π. At what value of θ will the net ccle end? b) Suppose that ou etended the graph of = sin θ to the left of. Predict the shape of the graph. Use a calculator to investigate a few points to the left of. At what value of θ will the net ccle end? 5. Repeat step for = cos θ. Did You Know? The sine function is based upon one of the trigonometric ratios originall calculated b the astronomer Hipparchus of Nicaea in the second centur B.C.E. He was tring to make sense of the movement of the stars and the moon in the night sk. Link the Ideas Sine and cosine functions are periodic functions. The values of these functions repeat over a specified period. A sine graph is a graph of the function = sin θ. You can also describe a sine graph as a sinusoidal curve. periodic function a function that repeats itself over regular intervals (ccles) of its domain = sin θ - 5π -π - π Period.5 -π - π _ π_ π π π -.5 5π θ period the length of the interval of the domain over which a graph repeats itself the horizontal length of one ccle on a periodic graph Period - One Ccle Trigonometric functions are sometimes called circular because the are based on the unit circle. sinusoidal curve the name given to a curve that fluctuates back and forth like a sine graph a curve that oscillates repeatedl up and down from a centre line 5. Graphing Sine and Cosine Functions MHR

5 The sine function, = sin θ, relates the measure of angle θ in standard position to the -coordinate of the point P where the terminal arm of the angle intersects the unit circle. π_ π_ π P = sin θ π, π π_ π π π θ 5π π 7π - The cosine function, = cos θ, relates the measure of angle θ in standard position to the -coordinate of the point P where the terminal arm of the angle intersects the unit circle. P π_ π_ (, cos ) P π_ π π π - 7π 7π (, cos 6 ) 6 The coordinates of point P repeat after point P travels completel around the unit circle. The unit circle has a circumference of π. Therefore, the smallest distance before the ccle of values for the functions = sin θ or = cos θ begins to repeat is π. This distance is the period of sin θ and cos θ. Eample Graph a Periodic Function Sketch the graph of = sin θ for θ 6 or θ π. Describe its characteristics. Solution To sketch the graph of the sine function for θ 6 or θ π, select values of θ and determine the corresponding values of sin θ. Plot the points and join them with a smooth curve. MHR Chapter 5

6 θ Degrees Radians π_ 6 π_ π_ π_ π_ π_ 5π_ 6 π 7π_ 6 5π_ π_ π_ 5π_ 7π_ π _ 6 π sin θ _ _ - _ _ = sin θ = sin θ θ Did You Know? The Indo-Asian mathematician Arabhata (76 55) made tables of half-chords that are now known as sine and cosine tables. - π_ π_ π_ π 5π π 7π π π 5π π π θ From the graph of the sine function, ou can make general observations about the characteristics of the sine curve: The curve is periodic. The curve is continuous. The domain is { θ θ R}. The range is { -, R}. The maimum value is +. The minimum value is -. The amplitude of the curve is. The period is 6 or π. The -intercept is. Which points would ou determine to be the ke points for sketching a graph of the sine function? In degrees, the θ-intercepts are, -5, -6, -8,, 8, 6,, or 8 n, where n I. The θ-intercepts, in radians, are, -π, -π, -π,, π, π,, or nπ, where n I. Look for a pattern in the values. amplitude (of a sinusoidal function) the maimum vertical distance the graph of a sinusoidal function varies above and below the horizontal central ais of the curve Your Turn Sketch the graph of = cos θ for θ 6. Describe its characteristics. 5. Graphing Sine and Cosine Functions MHR 5

7 Eample Determine the Amplitude of a Sine Function An function of the form = af() is related to = f() b a vertical stretch of a factor a about the -ais, including the sine and cosine functions. If a <, the function is also reflected in the -ais. a) On the same set of aes, graph = sin, =.5 sin, and = - sin for π. b) State the amplitude for each function. c) Compare each graph to the graph of = sin. Consider the period, amplitude, domain, and range. Solution a) Method : Graph Using Transformations Sketch the graph of = sin. For the graph of = sin, appl a vertical stretch b a factor of. For the graph of =.5 sin, appl a vertical stretch b a factor of.5. For the graph of = - sin, reflect in the -ais and appl a vertical stretch b a factor of. = sin =.5 sin = sin π_ π_ π π 5π - π 7π π - - = - sin Method : Use a Graphing Calculator Select radian mode. Use the following window settings: : [, π, π _ ] : [-.5,.5,.5] 6 MHR Chapter 5

8 b) Determine the amplitude of a sine function using the formula Amplitude = maimum value - minimum value. The amplitude of = sin is - (-), or. The amplitude of = sin is - (-), or..5 - (-.5) The amplitude of =.5 sin is, or.5. The amplitude of = - sin is - (-), or. How is the amplitude related to the range of the function? c) Function Period Amplitude Specified Domain Range = sin π { π, R} { -, R} = sin π { π, R} { -, R} =.5 sin π.5 { π, R} { -.5.5, R} = - sin π { π, R} { -, R} Changing the value of a affects the amplitude of a sinusoidal function. For the function = a sin, the amplitude is a. Your Turn a) On the same set of aes, graph = 6 cos and = - cos for π. b) State the amplitude for each graph. c) Compare our graphs to the graph of = cos. Consider the period, amplitude, domain, and range. d) What is the amplitude of the function =.5 cos? Period of = sin b or = cos b The graph of a function of the form = sin b or = cos b for b has a period different from π when b. To show this, remember that sin b or cos b will take on all possible values as b ranges from to π. Therefore, to determine the period of either of these functions, solve the compound inequalit as follows. π Begin with the interval of one ccle of = sin or = cos. b π Replace with b for the interval of one ccle of = sin b or = cos b. _ π Divide b b. b Solving this inequalit determines the length of a ccle for the sinusoidal curve, where the start of a ccle of = sin b is and the end is _ π b. Determine the period, or length of the ccle, b finding the distance from to _ π b. Thus, the period for = sin b or = cos b is _ π b, in radians, or _ 6, in degrees. Wh do ou use b to determine the period? b 5. Graphing Sine and Cosine Functions MHR 7

9 Eample Determine the Period of a Sine Function An function of the form = f(b) is related to = f() b a horizontal stretch b a factor of _ about the -ais, including b the sine and cosine functions. If b <, then the function is also reflected in the -ais. a) Sketch the graph of the function = sin for 6. State the period of the function and compare the graph to the graph of = sin. b) Sketch the graph of the function = sin _ for π. State the period of the function and compare the graph to the graph of = sin. Solution a) Sketch the graph of = sin. For the graph of = sin, appl a horizontal stretch b a factor of _. = sin = sin To find the period of a function, start from an point on the graph (for eample, the -intercept) and determine the length of the interval until one ccle is complete. From the graph of = sin, the period is 9. You can also determine this using the formula Period = _ 6. b Period = _ 6 b Period = _ 6 Substitute for b. Period = _ 6 Period = 9 Compared to the graph of = sin, the graph of = sin has the same amplitude, domain, and range, but a different period. 8 MHR Chapter 5

10 b) Sketch the graph of = sin. For the graph of = sin _, appl a horizontal stretch b a factor of. = sin _ = sin π_ π π π 5π π 7π π - From the graph, the period for = sin _ is π. Using the formula, Period = _ π b Period = π _ _ Period = π Period = π _ Substitute for b. Compared to the graph of = sin, the graph of = sin _ has the same amplitude, domain, and range, but a different period. Changing the value of b affects the period of a sinusoidal function. Your Turn a) Sketch the graph of the function = cos for 6. State the period of the function and compare the graph to the graph of = cos. b) Sketch the graph of the function = cos _ for 6π. State the period of the function and compare the graph to the graph of = cos. c) What is the period of the graph of = cos (-)? Eample Sketch the Graph of = a cos b a) Sketch the graph of = - cos for at least one ccle. b) Determine the amplitude the period the maimum and minimum values the -intercepts and the -intercept the domain and range 5. Graphing Sine and Cosine Functions MHR 9

11 Solution a) Method : Graph Using Transformations Compared to the graph of = cos, the graph of = - cos is stretched horizontall b a factor of _ about the -ais, stretched verticall b a factor of about the -ais, and reflected in the -ais. Begin with the graph of = cos. Appl a horizontal stretch of _ about the -ais. Wh is the horizontal stretch b a factor of _? = cos - π_ π = cos π π 5π π 7π π - - Then, appl a vertical stretch b a factor of. = cos = cos - π_ π π π 5π π 7π π - - Finall, reflect the graph of = cos in the -ais. = cos = - cos - π_ π π π 5π π 7π π - - MHR Chapter 5

12 Method : Graph Using Ke Points This method is based on the fact that one ccle of a cosine function = cos b, from to _ π, includes two -intercepts, two maimums, b and a minimum. These five points divide the period into quarters. Compare = - cos to = a cos b. Since a = -, the amplitude is -, or. Thus, the maimum value is and the minimum value is -. Since b =, the period is _ π, or π. One ccle will start at = and end at = π. Divide this ccle into four equal segments using the values, _ π, _ π, _ π, and π for. The ke points are (, -), ( _ π, ), _ ( π, ), _ ( π, ), and (π, -). How do ou know where the maimums or minimums will occur? Wh are there two minimums instead of two maimums? Connect the points in a smooth curve and sketch the graph through one ccle. The graph of = - cos repeats ever π units in either direction. = - cos - π_ π π π 5π π 7π π - - b) The amplitude of = - cos is. The period is π. The maimum value is. The minimum value is - The -intercept is -. The -intercepts are _ π, _ π, _ 5π, _ 7π or _ π + _ π n, n I. The domain of the function is { R}. The range of the function is { -, R}. 5. Graphing Sine and Cosine Functions MHR

13 Your Turn a) Graph = sin, showing at least two ccles. b) Determine the amplitude the period the maimum and minimum values the -intercepts and the -intercept the domain and range Ke Ideas To sketch the graphs of = sin θ and = cos θ for θ 6 or θ π, determine the coordinates of the ke points representing the θ-intercepts, maimum(s), and minimum(s). = sin θ = cos θ π_ π π - π θ π_ π π - π θ The maimum value is +. The minimum value is -. The amplitude is. The period is π. The -intercept is. The θ-intercepts for the ccle shown are, π, and π. The domain of = sin θ is {θ θ R}. The range of = sin θ is { -, R}. How are the characteristics different for = cos θ? Determine the amplitude and period of a sinusoidal function of the form = a sin b or = a cos b b inspecting graphs or directl from the sinusoidal function. You can determine the amplitude using the formula Amplitude = maimum value - minimum value. The amplitude is given b a. You can change the amplitude of a function b varing the value of a. How can ou determine the amplitude from the graph of the sine function? cosine function? The period is the horizontal length of one ccle on the graph of a function. It is given b _ π or _ 6. b b You can change the period of a function b varing the value of b. How can ou identif the period on the graph of a sine function? cosine function? MHR Chapter 5

14 Check Your Understanding Practise. a) State the five ke points for = sin that occur in one complete ccle from to π. b) Use the ke points to sketch the graph of = sin for -π π. Indicate the ke points on our graph. c) What are the -intercepts of the graph? d) What is the -intercept of the graph? e) What is the maimum value of the graph? the minimum value?. a) State the five ke points for = cos that occur in one complete ccle from to π. b) Use the ke points to sketch a graph of = cos for -π π. Indicate the ke points on our graph. c) What are the -intercepts of the graph? d) What is the -intercept of the graph? e) What is the maimum value of the graph? the minimum value?. Cop and complete the table of properties for = sin and = cos for all real numbers. Propert = sin = cos maimum minimum amplitude period domain range -intercept -intercepts. State the amplitude of each periodic function. Sketch the graph of each function. a) = sin θ b) = _ cos θ c) = - _ sin d) = -6 cos 5. State the period for each periodic function, in degrees and in radians. Sketch the graph of each function. a) = sin θ b) = cos _ θ c) = sin _ d) = cos 6 Appl 6. Match each function with its graph. a) = cos b) = cos c) = -sin d) = -cos A B C D π_ π π π - π_ π π - π π_ π π π - π_ π π π - 5. Graphing Sine and Cosine Functions MHR

15 7. Determine the amplitude of each function. Then, use the language of transformations to describe how each graph is related to the graph of = sin. a) = sin b) = -5 sin c) =.5 sin d) = - _ sin 8. Determine the period (in degrees) of each function. Then, use the language of transformations to describe how each graph is related to the graph of = cos. a) = cos b) = cos (-) c) = cos _ d) = cos _ 9. Without graphing, determine the amplitude and period of each function. State the period in degrees and in radians. a) = sin b) = - cos c) = _ 5 sin _ (- ) d) = cos _. a) Determine the period and the amplitude of each function in the graph. π_ π π π 5π π - - A B b) Write an equation in the form = a sin b or = a cos b for each function. 7π c) Eplain our choice of either sine or cosine for each function.. Sketch the graph of each function over the interval [-6, 6 ]. For each function, clearl label the maimum and minimum values, the -intercepts, the -intercept, the period, and the range. a) = cos b) = - sin π c) = _ sin d) = - _ cos. The points indicated on the graph shown represent the -intercepts and the maimum and minimum values. A B C D E F a) Determine the coordinates of points B, C, D, and E if = sin and A has coordinates (, ). b) Determine the coordinates of points C, D, E, and F if = cos and B has coordinates (, ). c) Determine the coordinates of points B, C, D, and E if = sin _ and A has coordinates (-π, ).. The second harmonic in sound is given b f() = sin, while the third harmonic is given b f () = sin. Sketch the curves and compare the graphs of the second and third harmonics for -π π. Did You Know? A harmonic is a wave whose frequenc is an integral multiple of the fundamental frequenc. The fundamental frequenc of a periodic wave is the inverse of the period length.. Sounds heard b the human ear are vibrations created b different air pressures. Musical sounds are regular or periodic vibrations. Pure tones will produce single sine waves on an oscilloscope. Determine the amplitude and period of each single sine wave shown. a) - π _ π_ π π π 5π - - π MHR Chapter 5

16 b) - π _ π_ π π π 5π - π c) Draw a line radiating from the centre of the circle to each mark. d) Draw a vertical line to complete a right triangle for each of the angles that ou measured. - Did You Know? Pure tone audiometr is a hearing test used to measure the hearing threshold levels of a patient. This test determines if there is hearing loss. Pure tone audiometr relies on a patient s response to pure tone stimuli. 5. Sstolic and diastolic pressures mark the upper and lower limits in the changes in blood pressure that produce a pulse. The length of time between the peaks relates to the period of the pulse. Pressure (in millimetres of mercur) 6 8 Sstolic Pressure Diastolic Pressure Blood Pressure Variation Time (in seconds) a) Determine the period and amplitude of the graph. b) Determine the pulse rate (number of beats per minute) for this person. 6. MINI LAB Follow these steps Materials to draw a sine curve. paper Step Draw a large circle. a) Mark the centre of compass the circle. ruler b) Use a protractor and mark ever 5 from to 8 along the circumference of the circle. protractor grid paper Step Recall that the sine ratio is the length of the opposite side divided b the length of the hpotenuse. The hpotenuse of each triangle is the radius of the circle. Measure the length of the opposite side for each triangle and complete a table similar to the one shown. Angle, Opposite Hpotenuse 5 5 sin = opposite hpotenuse Step Draw a coordinate grid on a sheet of grid paper. a) Label the -ais from to 6 in increments of 5. b) Label the -ais from - to +. c) Create a scatter plot of points from our table. Join the dots with a smooth curve. Step Use one of the following methods to complete one ccle of the sine graph: complete the diagram from 8 to 6 etend the table b measuring the lengths of the sides of the triangle use the smmetr of the sine curve to complete the ccle 5. Graphing Sine and Cosine Functions MHR 5

17 7. Sketch one ccle of a sinusoidal curve with the given amplitude and period and passing through the given point. a) amplitude, period 8, point (, ) b) amplitude.5, period 5, point (, ) 8. The graphs of = sin θ and = cos θ show the coordinates of one point. Determine the coordinates of four other points on the graph with the same -coordinate as the point shown. Eplain how ou determined the θ-coordinates. a) π, ( ) - π -π - π _ π_ π π π - θ. Consider the function = sin. a) Use the graph of = sin to sketch a prediction for the shape of the graph of = sin. b) Use graphing technolog or grid paper and a table of values to check our prediction. Resolve an differences. c) How do ou think the graph of = sin + will differ from the graph of = sin? d) Graph = sin + and compare it to our prediction.. Is the function f() = 5 cos + sin sinusoidal? If it is sinusoidal, state the period of the function. Did You Know? b) - π ( ) π_, 6 -π - π _ π_ π π π - θ In 8, French mathematician Joseph Fourier discovered that an wave could be modelled as a combination of different tpes of sine waves. This model applies even to unusual waves such as square waves and highl irregular waves such as human speech. The discipline of reducing a comple wave to a combination of sine waves is called Fourier analsis and is fundamental to man of the sciences. 9. Graph = sin θ and = cos θ on the same set of aes for -π θ π. a) How are the two graphs similar? b) How are the different? c) What transformation could ou appl to make them the same graph? Etend. If = f() has a period of 6, determine the period of = f ( _ ).. Determine the period, in radians, of each function using two different methods. a) = - sin b) = - _ cos _ π 6. If sin θ =., determine the value of sin θ + sin (θ + π) + sin (θ + π). Create Connections C MINI LAB Eplore the relationship between the unit circle and the sine and cosine graphs with a graphing calculator. Step In the first list, enter the angle values from to π b increments of _ π. In the second and third lists, calculate the cosine and sine of the angles in the first list, respectivel. 6 MHR Chapter 5

18 Step Graph the second and third lists for the unit circle. Step Graph the first and third lists for the sine curve. Step Graph the first and second lists for the cosine curve. Step 5 a) Use the trace feature on the graphing calculator and trace around the unit circle. What do ou notice about the points that ou trace? What do the represent? b) Move the cursor to trace the sine or cosine curve. How do the points on the graph of the sine or cosine curve relate to the points on the unit circle? Eplain. C The value of (cos θ) + (sin θ) appears to be constant no matter the value of θ. What is the value of the constant? Wh is the value constant? (Hint: Use the unit circle and the Pthagorean theorem in our eplanation.) C The graph of = f() is sinusoidal with a period of passing through the point (, ). Decide whether each of the following can be determined from this information, and justif our answer. a) f() b) f() c) f(8) C Identif the regions that each of the following characteristics fall into. Sine = sin Sine and Cosine Cosine = cos a) domain { R} b) range { -, R} c) period is π d) amplitude is e) -intercepts are n(8 ), n I f) -intercepts are 9 + n(8 ), n I g) -intercept is h) -intercept is i) passes through point (, ) j) passes through point (, ) k) a maimum value occurs at (6, ) l) a maimum value occurs at (9, ) m) n) C5 a) Sketch the graph of = cos for -π π. How does the graph compare to the graph of = cos? b) Sketch the graph of = sin for -π π. How does the graph compare to the graph of = sin? 5. Graphing Sine and Cosine Functions MHR 7

19 5. Transformations of Sinusoidal Functions Focus on... graphing and transforming sinusoidal functions identifing the domain, range, phase shift, period, amplitude, and vertical displacement of sinusoidal functions developing equations of sinusoidal functions, epressed in radian and degree measure, from graphs and descriptions solving problems graphicall that can be modelled using sinusoidal functions recognizing that more than one equation can be used to represent the graph of a sinusoidal function Electric power and the light waves it generates are sinusoidal waveforms. The pistons and connecting rods of a steam train drive the wheels with a motion that is sinusoidal. The motion of a bod attached to a suspended spring, the motion of the plucked string of a musical instrument, and the pendulum of a clock produce oscillator motion that ou can model with sinusoidal functions. To use the functions = sin and = cos in applied situations, such as these and the ones in the images shown, ou need to be able to transform the functions. Ocean waves created b the winds ma be modelled b sinusoidal curves. Investigate Transformations of Sinusoidal Functions Materials grid paper graphing technolog A: Graph = sin θ + d or = cos θ + d. On the same set of aes, sketch the graphs of the following functions for θ 6. = sin θ = sin θ + = sin θ -. Using the language of transformations, compare the graphs of = sin θ + and = sin θ to the graph of = sin θ.. Predict what the graphs of = sin θ + and = sin θ - will look like. Justif our predictions. 8 MHR Chapter 5

20 Reflect and Respond. a) What effect does the parameter d in the function = sin θ + d have on the graph of = sin θ when d >? b) What effect does the parameter d in the function = sin θ + d have on the graph of = sin θ when d <? 5. a) Predict the effect varing the parameter d in the function = cos θ + d has on the graph of = cos θ. b) Use a graph to verif our prediction. B: Graph = cos (θ - c) or = sin (θ - c) Using Technolog 6. On the same set of aes, sketch the graphs of the following functions for -π θ π. = cos θ = cos ( θ + _ π ) = cos (θ - π) 7. Using the language of transformations, compare the graphs of = cos ( θ + _ π ) and = cos (θ - π) to the graph of = cos θ. 8. Predict what the graphs of = cos ( θ - _ π ) and = cos (θ + _ π ) will look like. Justif our predictions. Reflect and Respond 9. a) What effect does the parameter c in the function = cos (θ - c) have on the graph of = cos θ when c >? b) What effect does the parameter c in the function = cos (θ - c) have on the graph of = cos θ when c <?. a) Predict the effect varing the parameter c in the function = sin (θ - c) has on the graph of = sin θ. b) Use a graph to verif our prediction. Link the Ideas You can translate graphs of functions up or down or left or right and stretch them verticall and/or horizontall. The rules that ou have applied to the transformations of functions also appl to transformations of sinusoidal curves. 5. Transformations of Sinusoidal Functions MHR 9

21 Eample Graph = sin ( - c) + d a) Sketch the graph of the function = sin ( - ) +. b) What are the domain and range of the function? c) Use the language of transformations to compare our graph to the graph of = sin. Solution a) b) Domain: { R} Range: {, R} vertical displacement the vertical translation of the graph of a periodic function phase shift the horizontal translation of the graph of a periodic function c) The graph has been translated units up. This is the vertical displacement. The graph has also been translated to the right. This is called the phase shift. Your Turn a) Sketch the graph of the function = cos ( + 5 ). b) What are the domain and range of the function? c) Use the language of transformations to compare our graph to the graph of = cos. Eample Graph = a cos (θ c) + d a) Sketch the graph of the function = cos (θ + π) over two ccles. b) Use the language of transformations to compare our graph to the graph of = cos θ. Indicate which parameter is related to each transformation. MHR Chapter 5

22 Solution a) - π π π π θ - - b) Since a is, the graph has been reflected about the θ-ais and then stretched verticall b a factor of two. The d-value is, so the graph is translated unit down. The sinusoidal ais is defined as =. Finall, the c-value is -π. Therefore, the graph is translated π units to the left. Your Turn a) Sketch the graph of the function = sin ( θ π _ ) + over two ccles. b) Compare our graph to the graph of = sin θ. Did You Know? In this chapter, the parameters for horizontal and vertical translations are represented b c and d, respectivel. Eample Graph = a sin b( - c) + d Sketch the graph of the function = sin ( - π _ ) + over two ccles. What are the vertical displacement, amplitude, period, phase shift, domain, and range for the function? Solution First, rewrite the function in the standard form = a sin b( - c) + d. = sin ( - π _ ) + Method : Graph Using Transformations Step : Sketch the graph of = sin for one ccle. Appl the horizontal and vertical stretches to obtain the graph of = sin. Compared to the graph of = sin, the graph of = sin is a horizontal stretch b a factor of _ and a vertical stretch b a factor of. For the function = sin, b =. Period = _ π b = _ π = π So, the period is π. 5. Transformations of Sinusoidal Functions MHR

23 For the function = sin, a =. So, the amplitude is. = sin - π π π = sin Step : Appl the horizontal translation to obtain the graph of = sin ( - π _ ). The phase shift is determined b the value of parameter c for a function in the standard form = a sin b( - c) + d. Compared to the graph of = sin, the graph of = sin ( - _ π ) is translated horizontall _ π units to the right. The phase shift is _ π units to the right. = sin = sin ( - ) π_ - π π π Step : Appl the vertical translation to obtain the graph of = sin ( - π _ ) +. The vertical displacement is determined b the value of parameter d for a function in the standard form = a sin b( - c) + d. Compared to the graph of = sin ( - _ π ), the graph of = sin ( - _ π ) + is translated up units. The vertical displacement is units up. 6 = sin ( - π_ ) + Would it matter if the order = sin ( - π_ ) of the transformations were changed? Tr a different order for the transformations. - π π π π MHR Chapter 5

24 Compared to the graph of = sin, the graph of = sin ( - π _ ) + is horizontall stretched b a factor of _ verticall stretched b a factor of horizontall translated _ π units to the right verticall translated units up The vertical displacement is units up. The amplitude is. The phase shift is _ π units to the right. The domain is { R}. The range is { - 5, R}. Method : Graph Using Ke Points You can identif five ke points to graph one ccle of the sine function. The first, third, and fifth points indicate the start, the middle, and the end of the ccle. The second and fourth points indicate the maimum and minimum points. Comparing = sin ( - _ π ) + to = a sin b( - c) + d gives a =, b =, c = _ π, and d =. The amplitude is a, or. The period is _ π, or π. b The vertical displacement is d, or. Therefore, the equation of the sinusoidal ais or mid-line is =. You can use the amplitude and vertical displacement to determine the maimum and minimum values. The maimum value is d + a = + = 5 The minimum value is d - a = - = - Determine the values of for the start and end of one ccle from the function = a sin b( - c) + d b solving the compound inequalit b( - c) π. ( - π _ ) π - _ π π π_ _ π How does this inequalit relate to the period of the function? 5. Transformations of Sinusoidal Functions MHR

25 Divide the interval _ π _ π into four equal segments. B doing this, ou can locate five ke values of along the sinusoidal ais. π_, _ 7π, _ 5π 6, _ π, _ π 6 Use the above information to sketch = sin ( - π_ ) + one ccle of the graph, and then a second ccle. Note the five ke points and how ou can use them to sketch one π π π ccle of the graph of the function. For the graph of the function = sin ( - π _ ) +, the vertical displacement is units up the amplitude is the phase shift is _ π units to the right the domain is { R} the range is { - 5, R} Your Turn Sketch the graph of the function = cos ( + π) - over two ccles. What are the vertical displacement, amplitude, period, phase shift, domain, and range for the function? Eample Determine an Equation From a Graph The graph shows the function = f(). a) Write the equation of the function in the form = a sin b( - c) + d, a >. b) Write the equation of the - π _ π_ π π π function in the form = a cos b( - c) + d, a >. c) Use technolog to verif our solutions. 5π π Solution a) Determine the values of the parameters a, b, c, and d. Locate the sinusoidal ais or mid-line. Its position determines the value of d. Thus, d =. - π _ a = π_ π π π 5π d = π MHR Chapter 5

26 Use the sinusoidal ais from the graph or use the formula to determine the amplitude. How can ou use the maimum and minimum values of the graph to find the value of d? Amplitude = maimum value - minimum value a = - a = The amplitude is. Determine the period and the value of b. Method : Count the Number of Ccles in π Determine the number of ccles in a distance of π. In this function, there are three ccles. Therefore, the value of b is and the period is π _. - π _ - Period π_ π First Ccle π π 5π π Third Ccle Second Ccle Method : Determine the Period First Locate the start and end of one ccle of the sine curve. Recall that one ccle of = sin starts at (, ). How is that point transformed? How could this information help ou determine the start for one ccle of this sine curve? The start of the first ccle of the sine curve that is closest to the -ais is at = _ π and the end is at = 5π _ 6. The period is _ 5π 6 - _ π 6, or _ π. Solve the equation for b. Period = _ π b π_ = _ π b b = Choose b to be positive. 6 Determine the phase shift, c. Locate the start of the first ccle of the sine curve to the right of the -ais. Thus, c = π _ 6. Substitute the values of the parameters a =, b =, c = π _ 6, and d = into the equation = a sin b( - c) + d. The equation of the function in the form = a sin b( - c) + d is = sin ( - π _ 6 ) Transformations of Sinusoidal Functions MHR 5

27 b) To write an equation in the form = a cos b( - c) + d, determine the values of the parameters a, b, c, and d using steps similar to what ou did for the sine function in part a). a = b = c = _ π d = _ Wh is c = π? Are there other possible values for c? Period - π _ π_ π - π π 5π π The equation of the function in the form = a cos b( - c) + d is = cos ( - π _ ) +. How do the two equations compare? Could other equations define the function = f()? c) Enter the functions on a graphing calculator. Compare the graphs to the original and to each other. The graphs confirm that the equations for the function are correct. Your Turn The graph shows the function = f(). π_ π - π a) Write the equation of the function in the form = a sin b( - c) + d, a >. b) Write the equation of the function in the form = a cos b( - c) + d, a >. c) Use technolog to verif our solutions. 6 MHR Chapter 5

28 Eample 5 Interpret Graphs of Sinusoidal Functions Prince Rupert, British Columbia, has the deepest natural harbour in North America. The depth, d, in metres, of the berths for the ships can be approimated b the equation d(t) = 8 cos _ π t +, where t is the 6 time, in hours, after the first high tide. a) Graph the function for two ccles. b) What is the period of the tide? c) An ocean liner requires a minimum of m of water to dock safel. From the graph, determine the number of hours per ccle the ocean liner can safel dock. d) If the minimum depth of the berth occurs at 6 h, determine the depth of the water. At what other times is the water level at a minimum? Eplain our solution. Solution a) d Depth of Berths for Prince Rupert Harbour Depth (m) Time (h) t Wh should ou set the calculator to radian mode when graphing sinusoidal functions that represent real-world situations? b) Use b = _ π to determine the period. 6 Period = _ π b Period = _ π _ π 6 Period = The period for the tides is h. What does the period of h represent? 5. Transformations of Sinusoidal Functions MHR 7

29 c) To determine the number of hours an ocean liner can dock safel, draw the line = to represent the minimum depth of the berth. Determine the points of intersection of the graphs of = and d(t) = 8 cos π _ 6 t +. More precise answers can be obtained using technolog. The points of intersection for the first ccle are approimatel (.76, ) and (9.6, ). The depth is greater than m from h to approimatel.76 h and from approimatel 9. h to h. The total time when the depth is greater than m is , or 5.5 h, or about 5 h min per ccle. d) To determine the berth depth at 6 h, substitute the value of t = 6 into the equation. d(t) = 8 cos _ π 6 t + d(6) = 8 cos _ π (6) + 6 d(6) = 8 cos π + d(6) = 8(-) + d(6) = You can use the graph to verif the solution. The berth depth at 6 h is m. Add h (the period) to 6 h to determine the net time the berth depth is m. Therefore, the berth depth of m occurs again at 8 h. Your Turn The depth, d, in metres, of the water in the harbour at New Westminster, British Columbia, is approimated b the equation d(t) =.6 cos _ π t +.7, where t is the time, in hours, after the first high tide. a) Graph the function for two ccles starting at t =. b) What is the period of the tide? c) If a boat requires a minimum of.5 m of water to launch safel, for how man hours per ccle can the boat safel launch? d) What is the depth of the water at 7 h? At what other times is the water level at this depth? Eplain our solution. 8 MHR Chapter 5

30 Ke Ideas You can determine the amplitude, period, phase shift, and vertical displacement of sinusoidal functions when the equation of the function is given in the form = a sin b( - c) + d or = a cos b( - c) + d. For: = a sin b( - c) + d = a cos b( - c) + d How does changing each parameter affect the graph of a function? a π b d c - π _ π_ π_ - π π 5π π 7π Vertical stretch b a factor of a changes the amplitude to a reflected in the -ais if a < Horizontal stretch b a factor of _ b changes the period to _ 6 (in degrees) or _ π (in radians) b b reflected in the -ais if b < Horizontal phase shift represented b c to right if c > to left if c < Vertical displacement represented b d up if d > down if d < maimum value + minimum value d = You can determine the equation of a sinusoidal function given its properties or its graph. 5. Transformations of Sinusoidal Functions MHR 9

31 Check Your Understanding Practise. Determine the phase shift and the vertical displacement with respect to = sin for each function. Sketch a graph of each function. a) = sin ( - 5 ) + b) = sin ( + π) c) = sin ( + _ π ) + 5 d) = sin ( + 5 ) - e) = - sin (6 + ) - f) = sin _ ( - _ π ) -. Determine the phase shift and the vertical displacement with respect to = cos for each function. Sketch a graph of each function. a) = cos ( - ) + b) = cos ( - π _ ) c) = cos ( + _ 5π 6 ) + 6 d) = cos ( + 5 ) + e) = cos ( - π) + f) = cos ( - π _ 6 ) + 7. a) Determine the range of each function. i) = cos ( - π _ ) + 5 ii) = - sin ( + π) - iii) =.5 sin + iv) = _ cos ( + 5 ) + _ b) Describe how to determine the range when given a function of the form = a cos b( - c) + d or = a sin b( - c) + d.. Match each function with its description in the table. a) = - cos ( + ) - b) = sin ( - ) - c) = sin ( - ) - d) = sin ( - 9) - e) = sin ( + π) - Amplitude Period Phase Shift Vertical Displacement A _ π right down B π right down C π right down D π left down E π _ π_ left down 5. Match each function with its graph. a) = sin ( - _ π ) b) = sin ( + _ π ) c) = sin - d) = sin + A B C - π _ π_ π π - π _ π_ π π - - π _ π_ π π - π π π D - π_ π_ π π - π 5 MHR Chapter 5

32 Appl 6. Write the equation of the sine function in the form = a sin b( - c) + d given its characteristics. a) amplitude, period π, phase shift _ π to the right, vertical displacement 6 units down b) amplitude.5, period π, phase shift π_ to the left, vertical displacement 6 unit up c) amplitude _, period 7, no phase shift, vertical displacement 5 units down 7. The graph of = cos is transformed as described. Determine the values of the parameters a, b, c, and d for the transformed function. Write the equation for the transformed function in the form = a cos b( - c) + d. a) vertical stretch b a factor of about the -ais, horizontal stretch b a factor of about the -ais, translated units to the left and units up b) vertical stretch b a factor of _ about the -ais, horizontal stretch b a factor of _ about the -ais, translated units to the right and 5 units down c) vertical stretch b a factor of _ about the -ais, horizontal stretch b a factor of about the -ais, reflected in the -ais, translated _ π units to the right and unit down 8. When white light shines through a prism, the white light is broken into the colours of the visible light spectrum. Each colour corresponds to a different wavelength of the electromagnetic spectrum. Arrange the colours, in order from greatest to smallest period. Blue Red Green Indigo Violet Orange Yellow 9. The piston engine is the most commonl used engine in the world. The height of the piston over time can be modelled b a sine curve. Given the equation for a sine curve, = a sin b( - c) + d, which parameter(s) would be affected as the piston moves faster? Height (cm) π_ π_ π π - Time (s) 5π π 5. Transformations of Sinusoidal Functions MHR 5

33 . Victor and Stewart determined the phase shift for the function f() = sin ( - 6) +. Victor said that the phase shift was 6 units to the right, while Stewart claimed it was units to the right. a) Which student was correct? Eplain our reasoning. b) Graph the function to verif our answer from part a).. A famil of sinusoidal graphs with equations of the form = a sin b( - c) + d is created b changing onl the vertical displacement of the function. If the range of the original function is { -, R}, determine the range of the function with each given value of d. a) d = b) d = - c) d = - d) d = 8. Sketch the graph of the curve that results after appling each transformation to the graph of the function f() = sin. a) f ( - _ π ) b) f ( + _ π ) c) f() + d) f() -. The range of a trigonometric function in the form = a sin b( - c) + d is { - 5, R}. State the values of a and d.. For each graph of a sinusoidal function, state i) the amplitude ii) the period iii) the phase shift iv) the vertical displacement v) the domain and range vi) the maimum value of and the values of for which it occurs over the interval π vii) the minimum value of and the values of for which it occurs over the interval π a) a sine function - π -π - π _ π_ π π π b) a cosine function - π c) a sine function -π - π _ π_ π π π π _ π_ π π π 5π π - 5 MHR Chapter 5

34 5. Determine an equation in the form = a sin b( - c) + d for each graph. a) b) c) -π -π - π _ - - π π π_ 6. For each graph, write an equation in the form = a cos b( - c) + d. a) b) -π -π - π _ π_ π π - - π _ π_ π π - 7. a) Graph the function f() = cos ( - π _ ). b) Consider the graph. Write an equation of the function in the form = a sin b( - c) + d. c) What conclusions can ou make about the relationship between the two equations of the function? 8. Given the graph of the function f() = sin, what transformation is required so that the function g() = cos describes the graph of the image function? 9. For each start and end of one ccle of a cosine function in the form = cos b( - c), i) state the phase shift, period, and -intercepts ii) state the coordinates of the minimum and maimum values a) 9 b) _ π _ 5π. The Wave is a spectacular sandstone formation on the slopes of the Coote Buttes of the Paria Canon in Northern Arizona. The Wave is made from 9 million-ear-old sand dunes that have turned to red rock. Assume that a ccle of the Wave ma be approimated using a cosine curve. The maimum height above sea level is 5 ft and the minimum height is 5 ft. The beginning of the ccle is at the.75 mile mark of the canon and the end of this ccle is at the.75 mile mark. Write an equation that approimates the pattern of the Wave. c) π π π π 5π 5. Transformations of Sinusoidal Functions MHR 5

35 . Compare the graphs of the functions = sin _ π ( - ) - and = cos _ π ( - _ 7 ) -. Are the graphs equivalent? Support our answer graphicall.. Noise-cancelling headphones are designed to give ou maimum listening pleasure b cancelling ambient noise and activel creating their own sound waves. These waves mimic the incoming noise in ever wa, ecept that the are out of snc with the intruding noise b 8. sound waves created b headphones noise created b outside source combining the two sound waves results in silence Suppose that the amplitude and period for the sine waves created b the outside noise are and _ π, respectivel. Determine the equation of the sound waves the headphones produce to effectivel cancel the ambient noise.. The overhang of the roof of a house is designed to shade the windows for cooling in the summer and allow the Sun s ras to enter the house for heating in the winter. The Sun s angle of elevation, A, in degrees, at noon in Estevan, Saskatchewan, can be modelled b the formula A = -.5 sin _ 6 ( + ) +, 65 where is the number of das elapsed beginning with Januar. a) Use technolog to sketch the graph showing the changes in the Sun s angle of elevation throughout the ear. b) Determine the Sun s angle of elevation at noon on Februar. c) On what date is the angle of elevation the greatest in Estevan?. After eercising for 5 min, a person has a respirator ccle for which the rate of air flow, r, in litres per second, in the lungs is approimated b r =.75 sin _ π t, where t is the time, in seconds. a) Determine the time for one full respirator ccle. b) Determine the number of ccles per minute. c) Sketch the graph of the rate of air flow function. d) Determine the rate of air flow at a time of s. Interpret this answer in the contet of the respirator ccle. e) Determine the rate of air flow at a time of 7.5 s. Interpret this answer in the contet of the respirator ccle. Etend 5. The frequenc of a wave is the number of ccles that occur in s. Adding two sinusoidal functions with similar, but unequal, frequencies results in a function that pulsates, or ehibits beats. Piano tuners often use this phenomenon to help them tune a piano. a) Graph the function = cos + cos.9. b) Determine the amplitude and the period of the resulting wave. 6. a) Cop each equation. Fill in the missing values to make the equation true. i) sin ( - ) = cos ( - ) ii) sin ( - _ π ) = cos ( - ) iii) - cos ( - _ π ) = sin ( + ) iv) cos (- + 6π) = sin ( + ) b) Choose one of the equations in part a) and eplain how ou got our answer. 5 MHR Chapter 5

36 7. Determine the equation of the sine function with a) amplitude, maimum ( - π _, 5 ), and nearest maimum to the right at _ ( π, 5 ) b) amplitude, minimum ( π _, - ), and nearest maimum to the right at _ ( π, ) c) minimum (-π, ) and nearest maimum to the right at (, 7) d) minimum (9, -6) and nearest maimum to the right at (5, ) 8. The angle, P, in radians, between a pendulum and the vertical ma be modelled b the equation P = a cos bt, where a represents the maimum angle that the pendulum swings from the vertical; b is the horizontal stretch factor; and t is time, in seconds. The period of a pendulum ma be approimated b the formula Period = π L_ g, where L is the pendulum length and g is the acceleration due to gravit (9.8 m/s ). a) Sketch the graph that models the position of the pendulum in the diagram from t 5. C Sketch the graphs of = -sin and = sin (-). a) Compare the two graphs. How are the alike? different? b) Eplain wh this happens. c) How would ou epect the graphs of = -cos and = cos (-) to compare? d) Check our hpothesis from part c). If it is incorrect, write a correct statement about the cosine function. Did You Know? An even function satisfies the propert f(-) = f() for all in the domain of f(). An odd function satisfies the propert f(-) = -f() for all in the domain of f(). C Triangle ABC is inscribed between the graphs of f() = 5 sin and g() = 5 cos. Determine the area of ABC. C f() = 5 sin A B a cm 8 cm b) Determine the position of the pendulum after 6 s. Epress our answer to the nearest tenth of a centimetre. Create Connections C Consider a sinusoidal function of the form = a sin b( - c) + d. Describe the effect that each of the parameters a, b, c, and d has on the graph of the function. Compare this to what ou learned in Chapter Function Transformations. g() = 5 cos C The equation of a sine function can be epressed in the form = a sin b( - c) + d. Determine the values of the parameters a, b, c, and/or d, where a > and b >, for each of the following to be true. a) The period is greater than π. b) The amplitude is greater than unit. c) The graph passes through the origin. d) The graph has no -intercepts. e) The graph has a -intercept of a. f) The length of one ccle is. 5. Transformations of Sinusoidal Functions MHR 55

37 5. The Tangent Function Focus on... sketching the graph of = tan determining the amplitude, domain, range, and period of = tan determining the asmptotes and -intercepts for the graph of = tan solving a problem b analsing the graph of the tangent function You can derive the tangent of an angle from the coordinates of a point on a line tangent to the unit circle at point (, ). These values have been tabulated and programmed into scientific calculators and computers. This allows ou to appl trigonometr to surveing, engineering, and navigation problems. Did You Know? Tangent comes from the Latin word tangere, to touch. Tangent was first mentioned in 58 b T. Fincke, who introduced the word tangens in Latin. E. Gunter (6) used the notation tan, and J.H. Lambert (77) discovered the fractional representation of this function. Investigate the Tangent Function Materials grid paper ruler protractor compass graphing technolog A: Graph the Tangent Function A tangent line to a curve is a line that touches a curve, or a graph of a function, at a single point.. On a piece of grid paper, draw and label the -ais and -ais. Draw a circle of radius so that its centre is at the origin. Draw a tangent to the circle at the point where the -ais intersects the circle on the right side.. To sketch the graph of the tangent function over the interval θ 6, ou can draw angles in standard position on the unit circle and etend the terminal arm to the right so that it intersects the tangent line, as shown in the diagram. The -coordinate of the point of intersection represents the value of the tangent function. Plot points represented b the coordinates (angle measure, -coordinate of point of intersection). 56 MHR Chapter 5

38 θ unit θ - a) Begin with an angle of. Where does the etension of the terminal arm intersect the tangent line? b) Draw the terminal arm for an angle of 5. Where does the etension of the terminal arm intersect the tangent line? c) If the angle is 9, where does the etension of the terminal arm intersect the tangent line? d) Use a protractor to measure various angles for the terminal arm. Determine the -coordinate of the point where the terminal arm intersects the tangent line. Plot the ordered pair (angle measure, -coordinate on tangent line) on a graph like the one shown above on the right. What can ou conclude about the value of tan 9? How do ou show this on a graph? Angle Measure coordinate on Tangent Line. Use graphing technolog to verif the shape of our graph. Reflect and Respond. When θ = 9 and θ = 7, the tangent function is undefined. How does this relate to the graph of the tangent function? 5. What is the period of the tangent function? 6. What is the amplitude of the tangent function? What does this mean? 7. Eplain how a point P(, ) on the unit circle relates to the sine, cosine, and tangent ratios. B: Connect the Tangent Function to the Slope of the Terminal Arm 8. The diagram shows an angle θ in standard position whose terminal arm intersects the tangent AB at point B. Epress the ratio of tan θ in terms of the sides of AOB. B θ A(, ) unit 5. The Tangent Function MHR 57

39 9. Using our knowledge of special triangles, state the eact value of tan 6. If θ = 6 in the diagram, what is the length of line segment AB?. Using the measurement of the length of line segment AB from step 9, determine the slope of line segment OB.. How does the slope of line segment OB relate to the tangent of an angle in standard position? Reflect and Respond. How could ou use the concept of slope to determine the tangent ratio when θ =? when θ = 9?. Using a calculator, determine the values of tan θ as θ approaches 9. What is tan 9?. Eplain the relationship between the terminal arm of an angle θ and the tangent of the line passing through the point (, ) when θ = 9. (Hint: Can the terminal arm intersect the tangent line?) Link the Ideas The value of the tangent of an angle θ is the slope of the line passing through the origin and the point on the unit circle (cos θ, sin θ). You can think of it as the slope of the terminal arm of angle θ in standard position. tan θ = sin θ _ cos θ When sin θ =, what is tan θ? Eplain. When cos θ =, what is tan θ? Eplain. The tangent ratio is the length of the line segment tangent to the unit circle at the point A(, ) from the -ais to the terminal arm of angle θ at point Q. From the diagram, the distance AQ is equal to the -coordinate of point Q. Therefore, point Q has coordinates (, tan θ). P(cos θ, sin θ) θ Q(, tan θ) A(, ) How could ou show that the coordinates of Q are (, tan θ)? 58 MHR Chapter 5

40 Eample Graph the Tangent Function Graph the function = tan θ for -π θ π. Describe its characteristics. Solution The function = tan θ is known as the tangent function. Using the unit circle, ou can plot values of against the corresponding values of θ. Between asmptotes, the graph of = tan θ passes through a point with -coordinate -, a θ-intercept, and a point with -coordinate. Period 8 6 π -π π - - π_ π_ π θ You can observe the properties of the tangent function from the graph. The curve is not continuous. It breaks at θ = -_ π, θ = -_ π, θ = _ π, and θ = _ π, where the function is undefined. tan θ = when θ = -π, θ = -π, θ =, θ = π, and θ = π. tan θ = when θ = -_ 7π, θ = - _ π, θ = _ π, and θ = _ 5π. tan θ = - when θ = -_ 5π, θ = - _ π, θ = _ π, and θ = _ 7π. The graph of = tan θ has no amplitude because it has no maimum or minimum values. The range of = tan θ is { R}. 5. The Tangent Function MHR 59

41 As point P moves around the unit circle in either a clockwise or a counterclockwise direction, the tangent curve repeats for ever interval of π. The period for = tan θ is π. The tangent is undefined whenever cos θ =. This occurs when θ = _ π + nπ, n I. At these points, the value of the tangent approaches infinit and is undefined. When graphing the tangent, use dashed lines to show where the value of the tangent is undefined. These vertical lines are called asmptotes. The domain of = tan θ is { θ θ _ π + nπ, θ R, n I }. For tangent graphs, the distance between an two consecutive vertical asmptotes represents one complete period. Wh is tan θ undefined for cos θ =? Your Turn Graph the function = tan θ, θ 6. Describe how the characteristics are different from those in Eample. Eample Model a Problem Using the Tangent Function A small plane is fling at a constant altitude of 6 m directl toward an observer. Assume that the ground is flat in the region close to the observer. a) Determine the relation between the horizontal distance, in metres, from the observer to the plane and the angle, in degrees, formed from the vertical to the plane. b) Sketch the graph of the function. c) Where are the asmptotes located in this graph? What do the represent? d) Eplain what happens when the angle is equal to. Wh is this assumption made? Solution a) Draw a diagram to model the situation. Let d represent the horizontal distance from the observer to the plane. Let θ represent the angle formed b the vertical and the line of sight to the plane. plane d 6 m θ observer tan θ = d_ 6 d = 6 tan θ 6 MHR Chapter 5

42 b) The graph represents the horizontal distance between the plane and the observer. As the plane flies toward the observer, that distance decreases. As the plane moves from directl overhead to the observer s left, the distance values become negative. The domain of the function is {θ -9 < θ < 9, θ R}. d 6 8 d = 6 tan θ θ 8 6 c) The asmptotes are located at θ = 9 and θ = -9. The represent when the plane is on the ground to the right or left of the observer, which is impossible, because the plane is fling in a straight line at a constant altitude of 6 m. d) When the angle is equal to, the plane is directl over the head of the observer. The horizontal distance is m. Your Turn A small plane is fling at a constant altitude of 5 m directl toward an observer. Assume the ground is flat in the region close to the observer. a) Sketch the graph of the function that represents the relation between the horizontal distance, in metres, from the observer to the plane and the angle, in degrees, formed b the vertical and the line of sight to the plane. b) Use the characteristics of the tangent function to describe what happens to the graph as the plane flies from the right of the observer to the left of the observer. 5. The Tangent Function MHR 6

43 Ke Ideas You can use asmptotes and three points to sketch one ccle of a tangent function. To graph = tan, draw one asmptote; draw the points where = -, =, and = ; and then draw another asmptote. The tangent function = tan has the following characteristics: The period is π. The graph has no maimum or minimum values. The range is { R}. Vertical asmptotes occur at The domain is = π _ + nπ, n I. { π _ + nπ, R, n I }. The -intercepts occur at = nπ, n I. The -intercept is. How can ou determine the location of the asmptotes for the function = tan? π_ π_ = tan π π π 5π Check Your Understanding Practise. For each diagram, determine tan θ and the value of θ, in degrees. Epress our answer to the nearest tenth, when necessar. a) c) θ Q(, ) θ d) Q(, -.7) Q(, ) b) θ θ Q(, -.7) 6 MHR Chapter 5

44 . Use the graph of the function = tan θ to determine each value π - π -π - π _ π_ π π π θ - a) tan π _ b) tan _ π c) tan _ (- 7π ) d) tan e) tan π f) tan 5π _ Does = tan have an amplitude? Eplain.. Use graphing technolog to graph = tan using the following window settings: : [-6, 6, ] and : [-,, ]. Trace along the graph to locate the value of tan when = 6. Predict the other values of that will produce the same value for tan within the given domain. Verif our predictions. Appl 5. In the diagram, PON and QOA are similar triangles. Use the diagram to justif the statement tan θ = _ sin θ cos θ. P θ N Q(, tan θ) A(, ) 6. Point P(, ) is plotted where the terminal arm of angle θ intersects the unit circle. a) Use P(, ) to determine the slope of the terminal arm. b) Eplain how our result from part a) is related to tan θ. c) Write our results for the slope from part a) in terms of sine and cosine. d) From our answer in part c), eplain how ou could determine tan θ when the coordinates of point P are known. 7. Consider the unit circle shown. B(,) P(, ) = (cos θ, sin θ) A (-, ) θ A(, ) M B (, -) a) From POM, write the ratio for tan θ. b) Use cos θ and sin θ to write the ratio for tan θ. c) Eplain how our answers from parts a) and b) are related. 5. The Tangent Function MHR 6

45 8. The graph of = tan θ appears to be vertical as θ approaches 9. a) Cop and complete the table. Use a calculator to record the tangent values as θ approaches 9. θ tan θ b) What happens to the value of tan θ as θ approaches 9? c) Predict what will happen as θ approaches 9 from the other direction. θ tan θ 9. A securit camera securit scans a long d camera straight fence that 5 m encloses a section of a militar base. midpoint The camera is fence of fence mounted on a post that is located 5 m from the midpoint of the fence. The camera makes one complete rotation in 6 s. a) Determine the tangent function that represents the distance, d, in metres, along the fence from its midpoint as a function of time, t, in seconds, if the camera is aimed at the midpoint of the fence at t =. b) Graph the function in the interval -5 t 5. c) What is the distance from the midpoint of the fence at t = s, to the nearest tenth of a metre? d) Describe what happens when t = 5 s.. A rotating light on top of a lighthouse sends out ras of light in opposite directions. As the beacon rotates, the ra at angle θ makes a spot of light that moves along the shore. The lighthouse is located 5 m from the shoreline and makes one complete rotation ever min. lighthouse beacon θ light ra 5 m d shore a) Determine the equation that epresses the distance, d, in metres, as a function of time, t, in minutes. b) Graph the function in part a). c) Eplain the significance of the asmptote in the graph at θ = 9. Did You Know? The Fisgard Lighthouse was the first lighthouse built on Canada s west coast. It was built in 86 before Vancouver Island became part of Canada and is located at the entrance to Esquimalt harbour. 6 MHR Chapter 5

46 . A plane fling at an altitude of km over level ground will pass directl over a radar station. Let d be the ground distance from the antenna to a point directl under the plane. Let represent the angle formed from the vertical at the radar station to the plane. Write d as a function of and graph the function over the interval π _.. Andrea uses a pole of known height, a piece of string, a measuring tape, and a calculator for an assignment. She places the pole in a vertical position in the school field and runs the string from the top of the pole to the tip of the shadow formed b the pole. Ever 5 min, Andrea measures the length of the shadow and then calculates the slope of the string and the measure of the angle. She records the data and graphs the slope as a function of the angle. θ string shadow pole a) What tpe of graph would ou epect Andrea to graph to represent her data? b) When the Sun is directl overhead and no shadow results, state the slope of the string. How does Andrea s graph represent this situation? Etend. a) Graph the line = _, where >. Mark an angle θ that represents the angle formed b the line and the positive -ais. Plot a point with integral coordinates on the line = _. b) Use these coordinates to determine tan θ. c) Compare the equation of the line with our results in part b). Make a conjecture based on our findings.. Have ou ever wondered how a calculator or computer program evaluates the sine, cosine, or tangent of a given angle? The calculator or computer program approimates these values using a power series. The terms of a power series contain ascending positive integral powers of a variable. The more terms in the series, the more accurate the approimation. With a calculator in radian mode, verif the following for small values of, for eample, =.5. a) tan = + _ + _ _ 77 5 b) sin = - _ 6 + _ 5 - _ 7 5 c) cos = - _ + _ - _ 6 7 Create Connections C How does the domain of = tan differ from that of = sin and = cos? Eplain wh. C a) On the same set of aes, graph the functions f() = cos and g() = tan. Describe how the two functions are related. b) On the same set of aes, graph the functions f() = sin and g() = tan. Describe how the two functions are related. C Eplain how the equation tan ( + π) = tan relates to circular functions. 5. The Tangent Function MHR 65

47 5. Equations and Graphs of Trigonometric Functions Focus on... using the graphs of trigonometric functions to solve equations analsing a trigonometric function to solve a problem determining a trigonometric function that models a problem using a model of a trigonometric function for a real-world situation One of the most useful characteristics of trigonometric functions is their periodicit. For eample, the times of sunsets, sunrises, and comet appearances; seasonal temperature changes; the movement of waves in the ocean; and even the qualit of a musical sound can be described using trigonometric functions. Mathematicians and scientists use the periodic nature of trigonometric functions to develop mathematical models to predict man natural phenomena. Investigate Trigonometric Equations Materials marker ruler compass stop watch centimetre grid paper Work with a partner.. On a sheet of centimetre grid paper, draw a circle of radius 8 cm. Draw a line tangent to the bottom of the circle. Start Distance 66 MHR Chapter 5

48 . Place a marker at the three o clock position on the circle. Move the marker around the circle in a counterclockwise direction, measuring the time it takes to make one complete trip around the circle.. Move the marker around the circle a second time stopping at time intervals of s. Measure the vertical distance from the marker to the tangent line. Complete a table of times and distances. Aim to complete one revolution in s. You ma have to practice this several times to maintain a consistent speed. Time (s) Distance (cm) 8. Create a scatterplot of distance versus time. Draw a smooth curve connecting the points. 5. Write a function for the resulting curve. 6. a) From our initial starting position, move the marker around the circle in a counterclockwise direction for s. Measure the vertical distance of the marker from the tangent line. Label this point on our graph. b) Continue to move the marker around the circle to a point that is the same distance as the distance ou recorded in part a). Label this point on our graph. c) How do these two points relate to our function in step 5? d) How do the measured and calculated distances compare? 7. Repeat step 6 for other positions on the circle. Did You Know? A scatter plot is the result of plotting data that can be represented as ordered pairs on a graph. Reflect and Respond 8. What is the connection between the circular pattern followed b our marker and the graph of distance versus time? 9. Describe how the circle, the graph, and the function are related. Link the Ideas You can represent phenomena with periodic behaviour or wave characteristics b trigonometric functions or model them approimatel with sinusoidal functions. You can identif a trend or pattern, determine an appropriate mathematical model to describe the process, and use it to make predictions (interpolate or etrapolate). You can use graphs of trigonometric functions to solve trigonometric equations that model periodic phenomena, such as the swing of a pendulum, the motion of a piston in an engine, the motion of a Ferris wheel, variations in blood pressure, the hours of dalight throughout a ear, and vibrations that create sounds. 5. Equations and Graphs of Trigonometric Functions MHR 67

49 Eample Solve a Trigonometric Equation in Degrees Determine the solutions for the trigonometric equation cos - = for the interval 6. Solution Method : Solve Graphicall Graph the related function f() = cos -. Use the graphing window [, 6, ] b [-,, ]. The solutions to the equation cos - = for the interval 6 are the -intercepts of the graph of the related function. The solutions for the interval 6 are = 5, 5, 5, and 5. Method : Solve Algebraicall cos - = cos = cos = _ cos = ± Wh is the ± smbol used? For cos = or _, the angles in the interval 6 that satisf the equation are 5 and 5. For cos = -, the angles in the interval 6 that satisf the equation are 5 and 5. The solutions for the interval 6 are = 5, 5, 5, and 5. Your Turn Determine the solutions for the trigonometric equation sin - = for the interval MHR Chapter 5

50 Eample Solve a Trigonometric Equation in Radians Determine the general solutions for the trigonometric equation 6 = 6 cos _ π +. Epress our answers to the nearest hundredth. 6 Solution Method : Determine the Zeros of the Function Rearrange the equation 6 = 6 cos _ π + so that one side is equal to. 6 6 cos _ π 6 - = Graph the related function = 6 cos _ π -. Use the window [-,, ] 6 b [-,, ]. Wh should ou set the calculator to radian mode? The solutions to the equation 6 cos _ π - = are the -intercepts. 6 The -intercepts are approimatel =.5 and = The period of the function is radians. So, the -intercepts repeat in multiples of radians from each of the original intercepts. The general solutions to the equation 6 = 6 cos _ π + are n radians and n radians, where n is an integer. Method : Determine the Points of Intersection Graph the functions = 6 cos _ π + and = 6 using a window 6 [-,, ] b [-,, ]. 5. Equations and Graphs of Trigonometric Functions MHR 69

51 The solution to the equation 6 = 6 cos _ π + is given b the points 6 of intersection of the curve = 6 cos _ π + and the line = 6. In 6 the interval, the points of intersection occur at.5 and The period of the function is radians. The points of intersection repeat in multiples of radians from each of the original intercepts. The general solutions to the equation 6 = 6 cos _ π + are n radians and n radians, where n is an integer. Method : Solve Algebraicall Did You Know? No matter in which quadrant θ falls, -θ has the same reference angle and both θ and -θ are located on the same side of the -ais. Since cos θ is positive on the right side of the -ais and negative on the left side of the -ais, cos θ = cos (-θ). II I III θ -θ IV 6 = 6 cos _ π 6 + = 6 cos _ π 6 _ 6 = cos _ π 6 _ = cos _ π 6 cos - ( _ ) = π _ 6.9 = π _ 6 =.59 Since the cosine function is positive in quadrants I and IV, a second possible value of can be determined. In quadrant IV, the angle is π - _ π 6. _ = cos ( π - _ π 6 ) cos - ( _ ) = π - _ π 6 π_ 6 = π - ( cos- _ ) = - _ π 6 ( cos- _ ) = 9.69 Two solutions to the equation 6 = 6 cos _ π + are.5 and The period of the function is radians, then the solutions repeat in multiples of radians from each original solution. The general solutions to the equation 6 = 6 cos _ π + are n radians and n radians, where n is an integer. Your Turn Determine the general solutions for the trigonometric equation = 6 sin _ π MHR Chapter 5

52 Eample Model Electric Power The electricit coming from power plants into our house is alternating current (AC). This means that the direction of current flowing in a circuit is constantl switching back and forth. In Canada, the current makes 6 complete ccles each second. The voltage can be modelled as a function of time using the sine function V = 7 sin πt. a) What is the period of the current in Canada? b) Graph the voltage function over two ccles. Eplain what the scales on the aes represent. c) Suppose ou want to switch on a heat lamp for an outdoor patio. If the heat lamp requires V to start up, determine the time required for the voltage to first reach V. Solution a) Since there are 6 complete ccles in each second, each ccle takes _ 6 s. So, the period is _ 6. b) To graph the voltage function over two ccles on a graphing calculator, use the following window settings: : [-.,.5,.] : [-,, 5] The -ais represents the number of volts. Each tick mark on the -ais represents 5 V. The -ais represents the time passed. Each tick mark on the -ais represents. s. c) Graph the line = and determine the first point of intersection with the voltage function. It will take approimatel. s for the voltage to first reach V. Your Turn In some Caribbean countries, the current makes 5 complete ccles each second and the voltage is modelled b V = 7 sin πt. a) Graph the voltage function over two ccles. Eplain what the scales on the aes represent. b) What is the period of the current in these countries? c) How man times does the voltage reach V in the first second? Did You Know? Tidal power is a form of hdroelectric power that converts the energ of tides into electricit. Estimates of Canada s tidal energ potential off the Canadian Pacific coast are equivalent to approimatel half of the countr s current electricit demands. Did You Know? The number of ccles per second of a periodic phenomenon is called the frequenc. The hertz (Hz) is the SI unit of frequenc. In Canada, the frequenc standard for AC is 6 Hz. Voltages are epressed as root mean square (RMS) voltage. RMS is the square root of the mean of the squares of the values. The RMS voltage is given peak voltage b. What is the RMS voltage for Canada? 5. Equations and Graphs of Trigonometric Functions MHR 7

53 Eample Model Hours of Dalight Iqaluit is the territorial capital and the largest communit of Nunavut. Iqaluit is located at latitude 6 N. The table shows the number of hours of dalight on the st da of each month as the da of the ear on which it occurs for the capital (based on a 65-da ear). Wh is the st da of each month chosen for the data in the table? Jan Feb Hours of Dalight b Da of the Year for Iqaluit, Nunavut Mar Apr Ma June Jul Aug Sept Oct Nov Dec a) Draw a scatter plot for the number of hours of dalight, h, in Iqaluit on the da of the ear, t. b) Which sinusoidal function will best fit the data without requiring a phase shift: h(t) = sin t, h(t) = -sin t, h(t) = cos t, or h(t) = -cos t? Eplain. c) Write the sinusoidal function that models the number of hours of dalight. d) Graph the function from part c). e) Estimate the number of hours of dalight on each date. i) March 5 (da 7) ii) Jul (da 9) iii) December 5 (da 9) Solution a) Graph the data as a scatter plot. b) Note that the data starts at a minimum value, climb to a maimum value, and then decrease to the minimum value. The function h(t) = -cos t ehibits this same behaviour. c) The maimum value is.8, and the minimum value is.. Use these values to find the amplitude and the equation of the sinusoidal ais. Amplitude = maimum value - minimum value a =.8 -. a = MHR Chapter 5

54 The sinusoidal ais lies halfwa between the maimum and minimum values. Its position will determine the value of d. maimum value + minimum value d =.8 +. d = d =.585 Determine the value of b. You know that the period is 65 das. Period = _ π b 65 = _ π b b = _ π 65 Choose b to be positive. Determine the phase shift, the value of c. For h(t) = -cos t the minimum value occurs at t =. For the dalight hours curve, the actual minimum occurs at da 55, which represents a -da shift to the left. Therefore, c = -. The number of hours of dalight, h, on the da of the ear, t, is given b the function h(t) = -8.5 cos ( π _ 65 (t + ) ) d) Graph the function in the same window as our scatter plot. Wh is the period 65 das? e) Use the value feature of the calculator or substitute the values into the equation of the function. i) The number of hours of dalight on March 5 (da 7) is approimatel.56 h. ii) The number of hours of dalight on Jul (da 9) is approimatel. h. iii) The number of hours of dalight on December 5 (da 9) is approimatel.65 h. 5. Equations and Graphs of Trigonometric Functions MHR 7

55 Your Turn Windsor, Ontario, is located at latitude N. The table shows the number of hours of dalight on the st da of each month as the da of the ear on which it occurs for this cit. Hours of Dalight b Da of the Year for Windsor, Ontario a) Draw a scatter plot for the number of hours of dalight, h, in Windsor, Ontario on the da of the ear, t. b) Write the sinusoidal function that models the number of hours of dalight. c) Graph the function from part b). d) Estimate the number of hours of dalight on each date. i) March ii) Jul iii) December e) Compare the graphs for Iqaluit and Windsor. What conclusions can ou draw about the number of hours of dalight for the two locations? Ke Ideas You can use sinusoidal functions to model periodic phenomena that do not involve angles as the independent variable. You can adjust the amplitude, phase shift, period, and vertical displacement of the basic trigonometric functions to fit the characteristics of the real-world application being modelled. You can use technolog to create the graph modelling the application. Use this graph to interpolate or etrapolate information required to solve the problem. You can solve trigonometric equations graphicall. Use the graph of a function to determine the -intercepts or the points of intersection with a given line. You can epress our solutions over a specified interval or as a general solution. 7 MHR Chapter 5

56 Check Your Understanding Practise. a) Use the graph of = sin to determine the solutions to the equation sin = for the interval π. -π - = sin π π b) Determine the general solution for sin =. c) Determine the solutions for sin = in the interval π.. The partial sinusoidal graphs shown below are intersected b the line = 6. Each point of intersection corresponds to a value of where = 6. For each graph shown determine the approimate value of where = 6. a) b) The partial graph of a sinusoidal function = cos (( - 6 )) + 6 and the line = are shown below. From the graph determine the approimate solutions to the equation cos (( - 6 )) + 6 =. 8 6 = = cos (( - 6 )) Solve each of the following equations graphicall. a) -.8 sin ( _ π 6 ( - ) ) + 6 = 6, π b) cos (( - 5 )) + 8 =, 6 c) 7 cos ( - 8) =, π d) 6. sin (( + 8 )) - =, 6 5. Solve each of the following equations. a) sin ( _ π ( - 6) ) =.5, π b) cos ( - 5 ) + 7 =, 6 c) 8 cos ( - 5) =, general solution in radians d) 5. sin (5( + 8 )) - = -, general solution in degrees 5. Equations and Graphs of Trigonometric Functions MHR 75

57 6. State a possible domain and range for the given functions, which represent real-world applications. a) The population of a lakeside town with large numbers of seasonal residents is modelled b the function P(t) = 6 sin (t - 8) + 8. b) The height of the tide on a given da can be modelled using the function h(t) = 6 sin (t - 5) + 7. c) The height above the ground of a rider on a Ferris wheel can be modelled b h(t) = 6 sin (t - ) +. d) The average dail temperature ma be modelled b the function h(t) = 9 cos _ π (t - ) A trick from Victorian times was to listen to the pitch of a fl s buzz, reproduce the musical note on the piano, and sa how man times the fl s wings had flapped in s. If the fl s wings flap times in one second, determine the period of the musical note. 8. Determine the period, the sinusoidal ais, and the amplitude for each of the following. a) The first maimum of a sine function occurs at the point (, ), and the first minimum to the right of the maimum occurs at the point (8, 6). b) The first maimum of a cosine function occurs at (, ), and the first minimum to the right of the maimum occurs at ( π _, -6 ). c) An electron oscillates back and forth 5 times per second, and the maimum and minimum values occur at + and -, respectivel. Appl 9. A point on an industrial flwheel eperiences a motion described b the function h(t) = cos ( π _.7 t ) + 5, where h is the height, in metres, and t is the time, in minutes. a) What is the maimum height of the point? b) After how man minutes is the maimum height reached? c) What is the minimum height of the point? d) After how man minutes is the minimum height reached? e) For how long, within one ccle, is the point less than 6 m above the ground? f) Determine the height of the point if the wheel is allowed to turn for h min.. Michelle is balancing the wheel on her biccle. She has marked a point on the tire that when rotated can be modelled b the function h(t) = 59 + sin 5t, where h is the height, in centimetres, and t is the time, in seconds. Determine the height of the mark, to the nearest tenth of a centimetre, when t = 7.5 s.. The tpical voltage, V, in volts (V), supplied b an electrical outlet in Cuba is a sinusoidal function that oscillates between -55 V and +55 V and makes 6 complete ccles each second. Determine an equation for the voltage as a function of time, t. 76 MHR Chapter 5

58 . The Universit of Calgar s Institute for Space Research is leading a project to launch Cassiope, a hbrid space satellite. Cassiope will follow a path that ma be modelled b the function h(t) = 5 sin 8π(t - 5) +, where h is the height, in kilometres, of the satellite above Earth and t is the time, in das. a) Determine the period of the satellite. b) How man minutes will it take the satellite to orbit Earth? c) How man orbits per da will the satellite make? b) One of the main food sources for the Arctic fo is the lemming. Suppose the population, L, of lemmings in the region is modelled b the function L(t) = 5 sin _ π (t - ) +. Graph the function L(t) using the same set of aes as for F(t).. The Arctic fo is common throughout the Arctic tundra. Suppose the population, F, of foes in a region of northern Manitoba is modelled b the function F(t) = 5 sin _ π t +, where t is the time, in months. a) How man months would it take for the fo population to drop to 65? Round our answer to the nearest month. c) From the graph, determine the maimum and minimum numbers of foes and lemmings and the months in which these occur. d) Describe the relationships between the maimum, minimum, and mean points of the two curves in terms of the lifestles of the foes and lemmings. List possible causes for the fluctuation in populations.. Office towers are designed to swa with the wind blowing from a particular direction. In one situation, the horizontal swa, h, in centimetres, from vertical can be approimated b the function h = sin.56t, where t is the time, in seconds. a) Graph the function using graphing technolog. Use the following window settings: : [,, ], : [-,, 5]. b) If a guest arrives on the top floor at t =, how far will the guest have swaed from the vertical after. s? c) If a guest arrives on the top floor at t =, how man seconds will have elapsed before the guest has swaed cm from the vertical? 5. Equations and Graphs of Trigonometric Functions MHR 77

59 5. In Inuvik, Northwest Territories (latitude 68. N), the Sun does not set for 56 das during the summer. The midnight Sun sequence below illustrates the rise and fall of the polar Sun during a da in the summer. 6. The table shows the average monthl temperature in Winnipeg, Manitoba, in degrees Celsius. Average Monthl Temperatures for Winnipeg, Manitoba ( C) Jan Feb Mar Apr Ma Jun Average Monthl Temperatures for Winnipeg, Manitoba ( C) Jul Aug Sep Oct Nov Dec Height of Sun Above the Horizon (Sun widths) Elapsed Time (h) a) Determine the maimum and minimum heights of the Sun above the horizon in terms of Sun widths. b) What is the period? c) Determine the sinusoidal equation that models the midnight Sun. Did You Know? In, a stud showed that the Sun s width, or diameter, is a stead 5 km. The researchers discovered over a -ear period that the diameter changed b less than km. a) Plot the data on a scatter plot. b) Determine the temperature that is halfwa between the maimum average monthl temperature and the minimum average monthl temperature for Winnipeg. c) Determine a sinusoidal function to model the temperature for Winnipeg. d) Graph our model. How well does our model fit the data? e) For how long in a -month period does Winnipeg have a temperature greater than or equal to 6 C? 7. An electric heater turns on and off on a cclic basis as it heats the water in a hot tub. The water temperature, T, in degrees Celsius, varies sinusoidall with time, t, in minutes. The heater turns on when the temperature of the water reaches C and turns off when the water temperature is C. Suppose the water temperature drops to C and the heater turns on. After another min the heater turns off, and then after another min the heater starts again. a) Write the equation that epresses temperature as a function of time. b) Determine the temperature min after the heater first turns on. 78 MHR Chapter 5

60 8. A mass attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the floor varies sinusoidall with time. When the mass is released, it takes. s to reach a high point of 6 cm above the floor. It takes.8 s for the mass to reach the first low point of cm above the floor. 6 cm cm a) Sketch the graph of this sinusoidal function. b) Determine the equation for the distance from the floor as a function of time. c) What is the distance from the floor when the stopwatch reads 7. s? d) What is the first positive value of time when the mass is 59 cm above the floor? 9. A Ferris wheel with a radius of m rotates once ever 6 s. Passengers get on board at a point m above the ground at the bottom of the Ferris wheel. A sketch for the first 5 s is shown. c) Determine the amount of time that passes before a rider reaches a height of 8 m for the first time. Determine one other time the rider will be at that height within the first ccle.. The Canadian National Historic Windpower Centre, at Etzikom, Alberta, has various stles of windmills on displa. The tip of the blade of one windmill reaches its minimum height of 8 m above the ground at a time of s. Its maimum height is m above the ground. The tip of the blade rotates times per minute. a) Write a sine or a cosine function to model the rotation of the tip of the blade. b) What is the height of the tip of the blade after s? c) For how long is the tip of the blade above a height of 7 m in the first s? Height Time a) Write an equation to model the path of a passenger on the Ferris wheel, where the height is a function of time. b) If Emil is at the bottom of the Ferris wheel when it begins to move, determine her height above the ground, to the nearest tenth of a metre, when the wheel has been in motion for. min. 5. Equations and Graphs of Trigonometric Functions MHR 79

61 . In a 66-da ear, the average dail maimum temperature in Vancouver, British Columbia, follows a sinusoidal pattern with the highest value of.6 C on da 8, Jul 6, and the lowest value of. C on da 6, Januar 6. a) Use a sine or a cosine function to model the temperatures as a function of time, in das. b) From our model, determine the temperature for da 7, Ma 6. c) How man das will have an epected maimum temperature of. C or higher? Etend. An investment compan invests the mone it receives from investors on a collective basis, and each investor shares in the profits and losses. One compan has an annual cash flow that has fluctuated in ccles of approimatel ears since 9, when it was at a high point. The highs were approimatel +% of the total assets, while the lows were approimatel -% of the total assets. a) Model this cash flow as a cosine function of the time, in ears, with t = representing 9. b) Graph the function from part a). c) Determine the cash flow for the compan in 8. d) Based on our model, do ou feel that this is a compan ou would invest with? Eplain.. Golden, British Columbia, is one of the man locations for heliskiing in Western Canada. When skiing the open powder, the skier leaves behind a trail, with two turns creating one ccle of the sinusoidal curve. On one section of the slope, a skier makes a total of turns over a -s interval. a) If the distance for a turn, to the left or to the right, from the midline is. m, determine the function that models the path of the skier. b) How would the function change if the skier made onl eight turns in the same -s interval? Create Connections C a) When is it best to use a sine function as a model? b) When is it best to use a cosine function as a model? C a) Which of the parameters in = a sin b( - c) + d has the greatest influence on the graph of the function? Eplain our reasoning. b) Which of the parameters in = a cos b( - c) + d has the greatest influence on the graph of the function? Eplain our reasoning. 8 MHR Chapter 5

62 C The sinusoidal door b the architectural firm Matharoo Associates is in the home of a diamond merchant in Surat, India. The door measures 5. m high and.7 m wide. It is constructed from sections of 5-mm-thick Burma teak. Each section is carved so that the door integrates 6 pulles, 8 ball bearings, a wire rope, and a counterweight hidden within the single pivot. When the door is in an open position, the shape of it ma be modelled b a sinusoidal function. a) Assuming the amplitude is half the width of the door and there is one ccle created within the height of the door, determine a sinusoidal function that could model the shape of the open door. b) Sketch the graph of our model over one period. Project Corner Broadcasting Radio broadcasts, television productions, and cell phone calls are eamples of electronic communication. A carrier waveform is used in broadcasting the music and voices we hear on the radio. The wave form, which is tpicall sinusoidal, carries another electrical waveform or message. In the case of AM radio, the sounds (messages) are broadcast through amplitude modulation. An NTSC (National Television Sstem Committee) mittee) television transmission is comprised of video and sound signals broadcast using carrier waveforms. The video signal is amplitude modulated, while the sound signal is frequenc modulated. Eplain the difference between amplitude modulation and frequenc modulation with respect to transformations of functions. How are periodic functions involved in satellite radio broadcasting, satellite television broadcasting, or cell phone transmissions? 5. Equations and Graphs of Trigonometric Functions MHR 8

63 Chapter 5 Review 5. Graphing Sine and Cosine Functions, pages 7. Sketch the graph of = sin for a) What are the -intercepts? b) What is the -intercept? c) State the domain, range, and period of the function. d) What is the greatest value of = sin?. Sketch the graph of = cos for a) What are the -intercepts? b) What is the -intercept? c) State the domain, range, and period of the function. d) What is the greatest value of = cos?. Match each function with its correct graph. a) = sin b) = sin c) = -sin d) = _ sin A B π π π π C D π π. Without graphing, determine the amplitude and period, in radians and degrees, of each function. a) = - sin b) = cos.5 c) = _ sin 5 _ 6 d) =-5 cos _ 5. a) Describe how ou could distinguish between the graphs of = sin, = sin, and = sin. Graph each function to check our predictions. b) Describe how ou could distinguish between the graphs of = sin, =-sin, and = sin (-). Graph each function to check our predictions. c) Describe how ou could distinguish between the graphs of = cos, =-cos, and = cos (-). Graph each function to check our predictions. 6. Write the equation of the cosine function in the form = a cos b with the given characteristics. a) amplitude, period π b) amplitude, period 5 π π c) amplitude _, period 7 d) amplitude _, period π _ 6 8 MHR Chapter 5

64 7. Write the equation of the sine function in the form = a sin b with the given characteristics. a) amplitude 8, period 8 b) amplitude., period 6 c) amplitude _, period π d) amplitude, period _ π 5. Transformations of Sinusoidal Functions, pages Determine the amplitude, period, phase shift, and vertical displacement with respect to = sin or = cos for each function. Sketch the graph of each function for two ccles. a) = cos ( - π _ ) - 8 b) = sin _ ( - π _ ) + c) = - cos ( - ) + 7 d) = _ sin _ ( - 6 ) - 9. Sketch graphs of the functions f() = cos ( - π _ ) and g() = cos ( - _ π ) on the same set of aes for π. a) State the period of each function. b) State the phase shift for each function. c) State the phase shift of the function = cos b( - π). d) State the phase shift of the function = cos (b - π).. Write the equation for each graph in the form = a sin b( - c) + d and in the form = a cos b( - c) + d. a) -9-9 b) c) π _ π_ π π π 5π π - - d) -π π π π π 5π 6π -. a) Write the equation of the sine function with amplitude, period π, phase shift _ π units to the right, and vertical displacement 5 units down. b) Write the equation of the cosine function with amplitude.5, period π, phase shift _ π units to the left, and 6 vertical displacement unit up. c) Write the equation of the sine function with amplitude _, period 5, no phase shift, and vertical displacement 5 units down. Chapter 5 Review MHR 8

65 . Graph each function. State the domain, the range, the maimum and minimum values, and the -intercepts and -intercept. a) = cos ( - 5 ) + b) = sin ( - π _ ) +. Using the language of transformations, describe how to obtain the graph of each function from the graph of = sin or = cos. a) = sin ( - π _ ) + 6 b) = - cos _ ( + _ π ) - c) = _ cos ( - ) + d) = -sin ( + 5 ) - 8. The sound that the horn of a cruise ship makes as it approaches the dock is different from the sound it makes when it departs. The equation of the sound wave as the ship approaches is = sin θ, while the equation of the sound wave as it departs is = sin _ θ. a) Compare the two sounds b sketching the graphs of the sound waves as the ship approaches and departs for the interval θ π. b) How do the two graphs compare to the graph of = sin θ? 5. The Tangent Function, pages a) Graph = tan θ for -π θ π and for -6 θ 6. b) Determine the following characteristics. i) domain ii) range iii) -intercept iv) -intercepts v) equations of the asmptotes 6. A point on the unit circle has coordinates P (, _ ). a) Determine the eact coordinates of point Q. b) Describe the relationship between sin θ, cos θ, and tan θ. c) Using the diagram, eplain what happens to tan θ as θ approaches 9. Q P θ A d) What happens to tan θ when θ = 9? 7. a) Eplain how cos θ relates to the asmptotes of the graph of = tan θ. b) Eplain how sin θ relates to the -intercepts of the graph of = tan θ. 8. Tan θ is sometimes used to measure the lengths of shadows given the angle of elevation of the Sun and the height of a tree. Eplain what happens to the shadow of the tree when the Sun is directl overhead. How does this relate to the graph of = tan θ? 9. What is a vertical asmptote? How can ou tell when a trigonometric function will have a vertical asmptote? 5. Equations and Graphs of Trigonometric Functions, pages Solve each of the following equations graphicall. a) sin - =, π b) = cos ( - ) + 5, 6 c) sin ( _ π ( - 6) ) =.5, general solution in radians d) cos ( - 5 ) + 7 =, general solution in degrees 8 MHR Chapter 5

66 . The Roal British Columbia Museum, home to the First Peoples Ehibit, located in Victoria, British Columbia, was founded in 886. To preserve the man artifacts, the air-conditioning sstem in the building operates when the temperature in the building is greater than C. In the summer, the building s temperature varies with the time of da and is modelled b the function T = cos t + 9, where T represents the temperature in degrees Celsius and t represents the time, in hours. a) Graph the function. b) Determine, to the nearest tenth of an hour, the amount of time in one da that the air conditioning will operate. c) Wh is a model for temperature variance important in this situation?. The height, h, in metres, above the ground of a rider on a Ferris wheel after t seconds can be modelled b the sine function h(t) = sin _ π (t - ) a) Graph the function using graphing technolog. b) Determine the maimum and minimum heights of the rider above the ground. c) Determine the time required for the Ferris wheel to complete one revolution. d) Determine the height of the rider above the ground after 5 s.. The number of hours of dalight, L, in Lethbridge, Alberta, ma be modelled b a sinusoidal function of time, t. The longest da of the ear is June, with 5.7 h of dalight, and the shortest da is December, with 8. h of dalight. a) Determine a sinusoidal function to model this situation. b) How man hours of dalight are there on April?. For several hundred ears, astronomers have kept track of the number of solar flares, or sunspots, that occur on the surface of the Sun. The number of sunspots counted in a given ear varies periodicall from a minimum of per ear to a maimum of per ear. There have been 8 complete ccles between the ears 75 and 98. Assume that a maimum number of sunspots occurred in the ear 75. a) How man sunspots would ou epect there were in the ear? b) What is the first ear after in which the number of sunspots will be about 5? c) What is the first ear after in which the number of sunspots will be a maimum? Chapter 5 Review MHR 85