19.1 The Sine and Cosine Functions: Definitions and Basic Properties 601 x 6 2 –6 –ππ f(x) f(x) = √4 –4 cos 2 (x-π) = 2 |sin x| Figure 19.9 f(x)is minimum when sin x = 0, that is, when x =±πn, where n is any integer. The minimum value of f is 0. f(x)is maximum where sin x takes on the values of ±1. f is maximum at x = −π 2 , π 2 , 3π 2 , 5π 2 , Inother words, f(x)is maximum at x = π 2 + πn,where n is any integer. The maximum value of f is 2. ◆ Question: Let f(x)= 4 − 4 cos 2 (x − π),asinExample 19.2. Do you think that f (x) = 0 at the local maxima of f ? At the local minima of f ? PROBLEMS FOR SECTION 19.1 1. Use a straightedge and the calibrated unit circle drawn below to estimate each of the following values. (You can check your answers with a calculator set in radian mode.) (a) cos(1.1) (b) sin(1.1) (c) sin(3.5) (d) cos(4.2) (e) cos(5.9) (f) sin(2.2) (g) sin(5.7) 3 2 4 .1 .2 .3 .4 .5 .6 .7 .8 .9 5 6 1 1 0.5 0.4 0.3 0.2 0.1 v u 2. Use the calibrated unit circle to estimate all t-values between 0 and 6 such that (a) cos t = 0.3. (b) sin t = 0.7. (c) sin t =−0.7. 602 CHAPTER 19 Trigonometry: Introducing Periodic Functions 3. P(w) is indicated in the figure below. v u 5 13, 12 13 ) ) P(w) = 1 1 –1 –1 Find the following. (a) sin w (b) cos w (c) sin(−w) (d) cos(−w) (e) sin(w + 6π) (f) cos(w − 2π) (g) Is cos(2w) positive, negative, or zero? Explain briefly. 4. Beginning at point (1, 0) and traveling a distance t counterclockwise along the unit circle, we arrive at a point with coordinates −1 3 , 2 √ 2 3 . Find the following. (a) cos t (b) sin t (c) sin(−t) (d) cos(−t) (e) sin(t − π) (f) sin(t − 10π) (g) Is sin(t + π 2 ) positive, negative, or zero? Explain. 5. Which of the following equations hold for all x? Explain your answers in terms of the unit circle. (a) sin x = sin(−x) (b) sin x =−sin(−x) (c) cos x = cos(−x) (d) cos x =−cos(−x) 6. Evaluate the following limits. Explain your reasoning. (a) lim x→∞ sin x (b) lim x→∞ sin x x 19.2 Modifying the Graphs of Sine and Cosine 603 7. Evaluate the following limits. (a) lim x→∞ cos x (b) lim x→0 sin 1 x (c) lim x→0 + 1 sin x (d) lim x→∞ sin x 2 x+1 (e) lim x→∞ cos πx 3 −99 x 3 −x 2 +7 19.2 MODIFYING THE GRAPHS OF SINE AND COSINE We began our discussion of trigonometric functions by suggesting that they would be useful for modeling periodic phenomena. In order for the sine and cosine functions to be useful to us we must be able to alter their periods and adjust the levels of peaks and valleys. We introduce the following terminology. Definitions The balance line of a sine or cosine function (or modifications of them through shifting, flipping, stretching and shrinking) is the centrally located horizontal line about which the values of the function oscillate. The functions f(x)=sin x and g(x) = cos x have balance lines at y = 0. Shifting the graph vertically by k units shifts the balance line by k units. Stretching the sin x or cos x graphs, whether vertically or horizontally, does not alter the balance line. The balance value (also called the average value of the function over a complete cycle) is the y-value of the balance line. The amplitude of a sine or cosine function (or modifications of them as specified above) is the positive number indicating the maximum vertical distance between the graph and its balance line. We have already defined the period to be the smallest positive constant k such that f(x +k) = f(x)for all x. amplitude balance line period Figure 19.10 The term sinusoidal function is used to refer to sine functions, cosine functions, and modifications of them obtained by shifting, flipping, stretching, and shrinking. If you are wondering why the word sinusoidal can refer to the cosine function, recall that cos x is simply a horizontal translate of sin x. Knowing the maximum and minimum values of a sinusoidal function, we can calculate its amplitude and balance value. The amplitude is half of the vertical distance between the maximum and minimum values: 604 CHAPTER 19 Trigonometry: Introducing Periodic Functions amplitude = maximum value − minimum value 2 . The balance value is the average of the maximum and minimum values; add them and divide by 2: balance value = maximum value + minimum value 2 . Examples are given in Figure 19.11. From the pictures, determine the amplitude, period, and balance value. Check your answers with those given. 12 –π 2 ( , 2) π 2 ( , 2) π 2 π 2 (2, ) yy y tw x (a) (b) (c) –2 6 –1 5 10 –2π balance value: 12 + 2 2 = 7 amplitude: 12 – 2 2 = 5 period: π balance value: amplitude: period: 5 5 8 balance value: + (–2π) 2 = –3π/4 π 2 – (–2π) 2 = 5π/4 amplitude: period: 4 Figure 19.11 How Are Modifications of the Graphs Reflected by Modifications of the Functions, and Vice Versa? To change the balance line of a sine or cosine function is to shift the function vertically (and vice versa). To change the amplitude of a sine or cosine function is to stretch or shrink the function vertically (and vice versa). To change the period of a sine or cosine function is to stretch or shrink the function horizontally (and vice versa). We know how to shift a function vertically and how to stretch and shrink both hori- zontally and vertically, so we simply apply our knowledge to these new functions. We also know how to flip a graph about a horizontal line. For the time being we will try to write equations for graphs without involving horizontal shifts of the sine and cosine functions, if possible. 19.2 Modifying the Graphs of Sine and Cosine 605 EXERCISE 19.6 Graph the following. Check your answers using a calculator (or computer), making sure the calculator is in radian mode. Do not rely completely on your calculator; blind faith in it can lead you dreadfully astray. 6 (a) i. y = 3 sin x ii. y =−2cos x + 2 iii. y =−(1/2)sin x − 1 (b) i. y = sin 2x ii. y = sin x 2 iii. y = 3 cos(2πx) OBSERVATION After doing part (b) of Exercise 19.6, you’ll see that sin x has period 2π , sin 2x has period π (one cycle is completed in π units) corresponding to horizontal compression by a factor of 2, and sin(x/2) has period 4π (one cycle is completed in 4π units) corresponding to horizontal stretching by a factor of 2. This is in keeping with what we know about horizontal compression and stretching; in the case of trigonometric functions it is simpler to understand why replacing x by 2x will compress the graph and replacing x by x 2 will stretch it. The point P(x)makes a complete spin around the unit circle as x varies from 0 to 2π . If we replace x by 2x, the point P(2x) makes a complete spin around the unit circle as x varies from 0 to π , that is, in an interval of half the length. Analogously, the point P( x 2 ) makes one complete spin around the unit circle as x varies from 0 to 4π , that is, in an interval of double the original length. We can generalize to say that if we replace x by Bx, the point P(Bx)makes a complete spin around the unit circle as x varies from 0 to 2π |B| , so the period of sin Bx is 2π |B| . How must we modify the sine or cosine functions to reflect either the Ferris wheel problem from Exercise 19.4 or to model the height or horizontal position of a particular point on the rim of any spinning wheel as a function of time? The balance value corresponds to the height of the center of the wheel, the amplitude corresponds to the radius of the wheel, and the period corresponds to the amount of time it takes to complete one full revolution. Replacing t by (2t) doubles the speed of the wheel, halving the time to complete one trip around. Similarly, replacing t by t 2 cuts the speed of the wheel in half, thereby doubling the time it takes for a point on the rim to return to its original position. If time is measured in minutes, then the period gives a measure of minutes/revolution. The number of revolutions per unit time is called the frequency. The frequency is defined to be the reciprocal of the period. 7 frequency = 1 period EXERCISE 19.7 (a) A Ferris wheel with diameter 30 meters makes one revolution every 2 minutes and rotates counterclockwise. Consider a seat that starts at ground level, a height of 0 meters at time t = 0. Express the vertical height of the seat as a function of time t. 6 Your calculator can be very misleading when graphing trigonometric functions. For example, try graphing y = sin(50x) and y = sin(20x) with domain [−10, 10] on your calculator. You will be appalled! 7 When talking about radio waves, for instance, it is more common to talk of the frequency than it is to talk of the period. 606 CHAPTER 19 Trigonometry: Introducing Periodic Functions (b) If the Ferris wheel slows down to one revolution every 3 minutes, what is the new equation? (c) A Ferris wheel with diameter 25 meters makes one revolution every minute and rotates clockwise. Write an equation for the vertical height of a seat that starts at ground level at time t = 0. Changing the Amplitude, Period, and Balance Value: A Summary If y = A sin Bx + K, the balance value is K, the amplitude is |A|, the period is 2π |B| . If y = A cos Bx + K, the balance value is K, the amplitude is |A|, the period is 2π |B| . We can always write a trigonometric equation in such a way that B is positive since sin(−x) =−sin x and cos(−x) = cos x. When constructing equations to match graphs in the examples below, we will choose B to be positive. Finding an Equation to Fit a Sinusoidal Graph Below we fit an equation to a sinusoidal graph that at x = 0 is either at an extreme value or at its balance value. If neither of these were the case, we could either shift the graph horizontally or establish a new benchmark for x = 0 to ensure one of these cases. If the graph has a maximum or minimum at x = 0, it is simplest to use an equation of the form y = A cos Bx + K, whereas if balance value is attained at x = 0, it is simplest to use an equation of the form y = A sin Bx + K. Determine the period. The period is 2π |B| and we can choose B to be positive, so B = 2π the period . Determine the amplitude. The amplitude is |A|. Choose A or −A by comparing the graph in question to the standard sine and cosine graphs. Determine the balance value. This is K. ◆ EXAMPLE 19.3 Find possible equations to match the graphs displayed earlier (in Figure 19.11) and repro- duced on the next page. 19.2 Modifying the Graphs of Sine and Cosine 607 1 –1 1 2 12 –π 2 ( , 2) π 2 ( , 2) π 2 π 2 (2, ) yy y tw x (a) (b) (c) –2 6 –1 5 10 –3 –2 –1 12 –2π 3 –1 1 balance value: amplitude: 7 5 π period: balance value: amplitude: period: 5 5 8 balance value: –3π/4 5π/4 4 amplitude: period: Figure 19.12 SOLUTION (a) When t = 0 the graph has a maximum, so we model it with the cosine function. y = A cos Bt + K The period is 2π |B| = π,so 2 |B| = 1 and |B|=2. We’ll choose B = 2. There is no vertical flipping, so A is positive. A = 5 Knowing the balance value is 7 gives y = 5 cos(2t) + 7. (b) When w = 0 this graph is at its balance value, so we model it with the sine function. y = A sin Bw + K The period is 2π |B| = 8, so 8|B|=2π.Let B = π/4. There is no vertical flipping, so A is positive. A = 5 Knowing the balance value is 5 gives us y = 5 sin π 4 w + 5. (c) When x = 0 this graph is at a minimum, so we model it with the cosine function. y = A cos Bx + K The period is 2π |B| = 4, so 4|B|=2π and |B|=π/2. Choose B = π/2. The cosine function must be flipped over a horizontal line, so A is negative. A = −5π 4 Knowing the balance value is −3π 4 gives us y =− 5π 4 cos π 2 x − 3π 4 . ◆ Horizontal Shifting We shift the sine and cosine graphs horizontally k units just as we would any other graph. For instance, replacing x by x − 3 shifts the graph to the right 3 units. We can express any cosine graph as a sine graph by making an appropriate horizontal shift. For instance, cos x = sin(x + π/2). More generally, y = A sin[B(x − C)] is the graph of A sin Bx shifted C units right. y = A cos[B(x − C)] is the graph of A cos Bx shifted C units right. 608 CHAPTER 19 Trigonometry: Introducing Periodic Functions A bit of care must be taken when simultaneously shifting horizontally and chang- ing the period. Suppose we were to graph y = sin t 2 + π . Rewriting sin t 2 + π as sin 1 2 (t + 2π) helps us see that this is the graph of sin t 2 shifted 2π units to the left. tx (a) y = sin (t/2) – 4π –3π –2π 2π 2π 3π 4π–ππ π π t (b) y = sin ( (t+ 2π)) – 4π –3π –2π 2π 3π 4π–ππ shift sin t/2 left 2π units 2 y = sin x 1 1 1 1 2 Figure 19.13 Notice that first changing the period of sine to 4π and then shifting 2π units to the left is different from shifting first (which in this case has no effect) and then changing the period. The order in which we execute the transformations matters: The transforma- tion corresponding to multiplication/division takes precedent over that corresponding to addition/subtraction. When faced with an expression like cos(2x + 3π), rewrite it as cos 2 x + 3π 2 to analyze the horizontal shift correctly. It is the graph of cos 2x shifted left 3π 2 units. EXERCISE 19.8 Graph the functions given below. Use a domain that allows you to display one full period. Check your answers with the help of a graphing calculator. (a) y = 5 cos 2 x + π 2 (b) y =−3cos x 2 + π (c) y = 5 sin(2x − π) − 1 Testing the Limits of Your Technological Tools When using a graphing calculator or computer to give the graph of a trigonometric function, a thoughtful choice of viewing window and a healthy dose of skepticism can be invaluable. Consider the function f(x)=sin 100x. Use the technology available to you to graph this function on the intervals [−5, 5], [−4.5, 4.5], [−3, 3], and [−1, 1]. We know that the graph is a sine function with a period of 2π 100 = π 50 ≈ 0.063. Therefore, on the interval [−1, 1] there ought to be more than 30 cycles of the sine function. Depending upon the technology you have used to graph, the graphs will vary, but in all probability they will be highly misleading. A graphing calculator or computer generally plots points and connects the dots, but the oscillations of the function f(x)=sin 100x are so numerous these plotted points do not give a good reading of the behavior of the function overall. Theme and Variation We know that for constants A and B: If y = A cos Bx, the amplitude is |A|, the period is 2π |B| . 19.2 Modifying the Graphs of Sine and Cosine 609 Suppose that the positions of A or of B were not filled by constants. The resulting functions would not be periodic, yet they would retain some of the oscillation of the cosine function. EXERCISE 19.9 Match each of the five functions below with the appropriate graph from Figure 19.14. Use a graphing calculator or computer to check your work. (a) y = cos(π x) (b) y = e −.2x cos(πx) (c) y = x cos(π x) (d) y = x 2 cos(πx) (e) y = cos(x 2 ) –6 1 2 3 4 5 6 –1 1 –6 1 2 3 4 5 6 –1 1 2 3 – 6 1 2 3 4 5 6 –1 1 2 3 – 6 1 2 3 4 5 6 –6 1 2 3 4 5 6 –1 1 yy y y y x x x x x I II III V IV Figure 19.14 PROBLEMS FOR SECTION 19.2 1. (a) On the same set of axes graph the following. Set the domain to show at least one complete cycle of the function. (Colored pens/pencils can be helpful in identifying which graph is which.) (i) y = sin x (ii) y = 2 sin x (iii) y =−3sin x 610 CHAPTER 19 Trigonometry: Introducing Periodic Functions (b) Describe in words the effect of the parameter A in y = A sin(x). 2. (a) On the same set of axes graph the following. (i) y = sin x (ii) y =sin(2x) (iii) y = sin(x/2) (iv) y =sin(4x) (v) y = sin(−2x) (b) Describe in words the effect of the parameter B in y = sin(Bx). 3. (a) Describe in words the effect of the parameter C in y = sin(x) + C. (b) Describe in words the effect of the parameter D in y = sin(x + D). 4. (a) Using the results of the previous three problems, without using a calculator graph the function y =−3sin(2x) +3. (b) What is the amplitude of the curve in part (a)? The period? (c) Check your answer to part (a) using a graphing calculator or computer. 5. Find the domain and range of each of the following functions. (a) f(x)=3sin(2x + 1) (b) g(x) =|2cos x| (c) h(x) = cos |x| (d) j(x)=2cos x −1 (e) k(x) = √ sin x 6. Find an equation to fit each of the sinusoidal graphs below. y t (a) y x (b) (b, k) y w (c) (–1, ) (1, ) π 4 –5π 4 (3, 3) y t (d) (2, 3) (14, 9) . Modifications of the Functions, and Vice Versa? To change the balance line of a sine or cosine function is to shift the function vertically (and vice versa). To change the amplitude of a sine or cosine. versa). We know how to shift a function vertically and how to stretch and shrink both hori- zontally and vertically, so we simply apply our knowledge to these new functions. We also know how to flip a graph. time? The balance value corresponds to the height of the center of the wheel, the amplitude corresponds to the radius of the wheel, and the period corresponds to the amount of time it takes to complete