19.2 Modifying the Graphs of Sine and Cosine 611 7. Find equations that fit the curves below: 1 y x 12345 y x (b)(a) 2π –π (10, –π) 8. Graph the following. (a) y =|sin x| (b) y = sin |x| 9. Find the period, amplitude, and balance value of each of the following functions. (a) f(x)=0.5 sin(3x) (b) g(x) =−4cos(x/3) (c) h(x) = sin(0.2x)+π π (d) j(x)=4[sin(πx) − 1] (e) k(x) =4 sin(πx − 1) 10. (a) A Ferris wheel with diameter 20 feet makes one revolution every 8 minutes. Graph the height of a point on the Ferris wheel versus time, assuming that at t = 0 the point is at height 0. Give an equation whose graph is the picture you’ve drawn. (b) The Ferris wheel slows down so that it makes one revolution every 10 minutes. Adjust both your picture and your equation. 11. (a) What is the domain of g(x) = √ sin x? (b) On the same set of axes graph f(x)=sin x and g(x) = √ sin x. (Use different colors if you have them at your disposal.) 12. Suppose we have the equation of a sine curve, y =sin  πx 2  , with period 4, amplitude 1. (a) We wish to shift the graph over 1 unit to the left. Which of the following will accomplish this? i. y = sin  π 2 x − 1  ii. y = sin  π 2 (x − 1)  = sin  π 2 x − π 2  iii. y = sin  π 2 x + 1  iv. y = sin  π 2 (x + 1)  = sin  π 2 x + π 2  Be sure that you get this right. Draw a picture of what you are aiming for, and then try a point or two to verify. For example, you might run a test on x = 0. (b) For each of the remaining three choices in part (a), describe in words what happens to the original graph. (c) In order to obtain the desired result in part (a), it is probably simplest to use a cosine function. What cosine function will give the desired result? (d) Suppose A, B, and C are positive constants and y =A sin(Bx +C). What are the period and amplitude of the sine graph? Describe the horizontal shift. 612 CHAPTER 19 Trigonometry: Introducing Periodic Functions 13. Match the equations with the graphs. Do this first without consulting a graphing calculator. (Think about spacing of intercepts, sign, relative heights of peaks, etc.) (a) y = x sin x (b) y = x 2 sin x (c) y = sin(x 2 ) (d) y = e −0.2x sin x (e) y = e 0.2x sin x (f) y = 0.5x + sin x –10 1 2 10 1 2 1 2 1 2 y y x y yy y x xx x x (i) (iii) (iv) (v) (vi) (ii) –1–2 1 2 1 2 1 2 10 20 60 –1–2 1 2 1 –1 –2 1 2 1 2 14. The region of Bogor in Java has a rainforest climate with 450 mm of rain falling in the rainiest month (February) and 230 mm of rain falling in the driest month. Model the amount of rain per month using a sinusoidal function. Let R(t) be the number of millimeters of rain per month where t = 0 denotes the height of the rainy season and t is measured in months. 19.2 Modifying the Graphs of Sine and Cosine 613 (a) Give an expression for R(t). (b) According to your model, on average how many millimeters of rain fall per month each year? Recall that the balance value of the function is the average value. (c) How many millimeters of rain does your model predict each year? Compare this with the figure of 4370 mm recorded. What is the percent error involved in your model? Facts from Vegetation of the Earth in Relation to Climate and the Eco-physiological Condi- tions by Heinrich Walter (translated from the second German edition by Joy Wieser), Springer- Verlag, 1973, p. 39. 15. Studies conducted over a nine-year period indicate that in the alpine belt of the tropics of Venezuela, in P ´ aramo de Mucuch ´ ies, the number of rainy days per month varies from an average low of 4 per month in the dry season to a high of 23 per month in the wet season, half a year later. (a) Model the number of rainy days per month using a sinusoidal function. Let t = 0 correspond to the driest month. (b) On average, how many rainy days does your model predict per year? Compare this with the recorded average number of rainy days per year: 181. (Your estimate will be a bit low, because in fact the rainy season is slightly longer than the dry season.) Source: Vegetation of the Earth in Relation to Climate and the Eco-physiological Conditions by Heinrich Walter (translated from the second German edition by Joy Wieser), Springer-Verlag, 1973, p. 55. 16. A wave has amplitude 3 and frequency 10. Give a possible formula for the wave. (There are infinitely many correct answers.) 17. The gravitational pull of the sun and moon on large bodies of water produces tides. Tides generally rise and fall twice every 25 hours. (The length of a cycle is 12.5 hours as opposed to 12 hours due to the moon’s revolution around the earth.) The range between high and low tide varies greatly with location. On the Pacific coast of America this range can be as much as 15 feet. The Bay of Fundy in New Brunswick has an extremely dramatic range of about 45 feet. (a) Model the tidal fluctuations on the Pacific coast using a sinusoidal function. Let H(t) give the height (in feet) above and below the average level of the ocean, where t is time in hours. Let t = 0 correspond to high tide. (b) Model the tidal fluctuations in the Bay of Fundy using a sinusoidal function. Use the same conventions as in part (a). Source: Ecology by Robert E. Ricklefs, Nelson, 1973, p. 124. 18. The average rental price for a two-bedroom apartment in Malden was $800 in 1990 and was $1000 in 2000. The price has been increasing over the past decade. We want to model the price of a two-bedroom apartment in Malden as a function of time and use our model to predict the price in the year 2020. Alex thinks that prices are increasing at a constant rate, so he models the price with a linear function, L(t). 614 CHAPTER 19 Trigonometry: Introducing Periodic Functions Jamey thinks that the percent change in price is constant, so he models the price with an exponential function, E(t). Mike, an optimist who loves trigonometry, thinks that price is a sinusoidal function of time. He thinks that $800 is an all-time low and $1000 is an all-time high. He models the price with a sine or cosine function, T(t). (a) Suppose we let t = 0 correspond to the year 1990 and measure time in years. Find a formula for each of the following. Accompany your formula with a sketch. i. L(t) ii. E(t) iii. T(t) (b) Which model predicts the highest price for the year 2003? Which model predicts the lowest price for the year 2003? (c) What prices will Alex, Jamey, and Mike predict for the year 2020? (d) Alex, Jamey, and Mike are combing the newspapers for informationthat might lend credence to one model over the other two. Which model, the linear, exponential, or trigonometric, is best supported by each of the following statements? i. “Prices in Malden have been growing at an increasing rate over the past decade.” ii. “In the early 1990s prices in Malden were increasing at an increasing rate. After 1995 the rate of increase began to drop off.” iii. “Prices of apartments in Malden have been increasing very steadily over the past decade.” 19. Determine whether or not each function is periodic. If a function is periodic, determine its period. (a) f(x)=cos |x| (b) g(x) = sin |x| (c) h(x) =|cos x| (d) j(x)=|sin x| (e) k(x) = sin(x 2 ) (f) l(x) = sin 2 x 20. Graph f(x)= 1 cos x on [−π,2π]. 21. Let f(x)= sin x x . This function will be quite important when we are interested in the derivative of sine and cosine. (a) What is the domain of f(x)? (b) Use a graphing calculator or computer to help you sketch the graph of f(x). (c) Although f(x) is undefined at x = 0, lim x→0 f(x) exists. What do you think this limit might be? Check out your conjecture numerically. Observe that if lim x→0 sin x x = L, then for x very close to zero, sin x x ≈ L, or, equivalently, sin x ≈ Lx for x close to zero. 22. A typical person might have a pulse of 70 heartbeats per minute and a blood pressure reading of 120 over 80, where 120 is the high pressure and 80 is the low. Model blood pressure as a function of time using a sinusoidal function B(t), where t is time in minutes. (a) What is the amplitude of B(t)? (b) What is the period of B(t)? Notice that you have been given the frequency and from this must find the period. (c) Write a possible formula for B(t). 19.3 The Function f(x)=tan x 615 19.3 THE FUNCTION f(x)=tan x Definition The function tan x, called the tangent function, is defined as tan x = sin x cos x . Geometric Interpretation of tan x As before, locate the point P(x)=(u, v) = (cos x, sin x) on the unit circle by traveling a directed distance x around the circle from the starting point (1, 0). Draw a line from the origin to the point P(x). The slope of this line is given by rise run = v u = sin x cos x . Thus, tan x is the slope of the line from the origin to the point P(x). Let OP denote the line segment from the origin to P(x). The slope of OP is 0 when P = (1, 0) and (−1, 0), that is, when x = nπ , where n is any integer. The slope of OP is undefined when P = (0, 1) and (0, −1), that is, when x = π/2 + nπ , where n is any integer. v v u u x P(x) = (u,v) = (cos x, sin x) tan x = slope of OP — O Figure 19.15 EXERCISE 19.10 Use your knowledge of the unit circle to show that the following are true. (a) tan π 4 = 1 (b) tan 3π 4 =−1 (c) tan 19π = 0 (d) tan −19π 2 is undefined The Graph of tan x As x goes from 0 to π/2, the slope of OP increases without bound. As x increases from 0 to π/4, tan x increases from 0 to 1. As x increases from π/4toπ/2, tan x starts at 1 and increases without bound. For x in the interval  π 2 , π  , the slope of OP is negative but gets less and less negative as x increases. lim x→π/2 + tan x =−∞ and lim x→π/2 − tan x =∞ Wecan see that tan x has no local extrema. Figure 19.16, on the following pages, shows the graph of tan x on the interval [0, π]. 616 CHAPTER 19 Trigonometry: Introducing Periodic Functions y = tan x π π 2 x Figure 19.16 The Period of tan x Notice that when x increases from π to 3π 2 , the line through the origin and P(x)lies right on top of the lines we were looking at as x increased from 0 to π 2 . We can conclude that the period of tan x is π . Do the exercise below to make a more rigorous argument. EXERCISE 19.11 Show that tan x has period π by using the definition of tan x and your knowledge of sine and cosine. In other words, show that tan(x + π) = tan x for all x by using the fact that tan(x + π) = sin(x+π) cos(x+π) . Putting this information together gives a rough graph of tan x. y y = tan x x π –π –2π 2π π 2 –π 2 –3π 2 –3π 2 Figure 19.17 An Alternative Approach to Graphing tan x To graph tan x = sin x cos x , we can use some of the same strategies used in graphing rational functions. If a function f is a fraction whose numerator and denominator are defined, continuous, and have no common zeros, then f is zero wherever the numerator is zero; f has a vertical asymptote wherever the denominator is zero; f is positive wherever the numerator anddenominator have the same signs, and negative wherever the signs are opposite. 19.3 The Function f(x)=tan x 617 The Zeros of tan x. Since sin x and cos x have no common zeros, the zeros of tan x = sin x cos x will occur at the zeros of sin x. Thus, the zeros of tan x are at x = nπ, where n is any integer. The zeros are −3π,−2π,−π,0,π,2π,3π x sin x –π–2ππ2π Figure 19.18 The Vertical Asymptotes of tan x. tan x is undefined where the denominator, cos x, equals zero. tan x will have vertical asymptotes at x = π 2 + nπ, where n is any integer. In other words, the asymptotes are at −5π 2 , −3π 2 , −π 2 , π 2 , 3π 2 , 5π 2 . x cos x –π–2ππ π 2π 2 –π 2 3π 2 –3π 2 Figure 19.19 The Sign of tan x. tan x = sin x cos x , so tan x is positive where sin x and cos x have the same sign, and tan x is negative where they have opposite signs. x –2π –π 2π π π 2 3π 2 –3π 2 –π 2 tan x Figure 19.20 Between x = 0 and x = π/2, the numerator and denominator are positive. The numerator is increasing and the denominator is approaching zero, so tan x grows without bound. tan(−x) = sin(−x) cos(−x) = − sin x cos x =−tan x; therefore tan x is an odd function. Thus, know- ing the graph on [0, π 2 ), we can produce the graph on (− π 2 , 0]. Now we’ve got an entire period graphed, so we are done. 618 CHAPTER 19 Trigonometry: Introducing Periodic Functions Once we know the graph of tan x, we can use our knowledge of shifts, stretches, shrinks, and flips to graph any function of the form A tan(Bx) + K. PROBLEMS FOR SECTION 19.3 1. (a) Using what you know about the properties of polynomial functions, explain how the graph of f(x)=sin x tells you that it is not a polynomial. (Think about the number of roots and the long-term behavior.) (b) Using what you know about the properties of rational functions, explain how the graph of f(x)=tan x tells you that it is not a rational function. (Think about the number of roots and vertical asymptotes.) (c) What are characteristics of trigonometric functions that distinguish them from other functions we’ve studied? 2. Evaluate the following limits. (a) lim x→−π/2 + tan x (b) lim x→−π/2 − tan x (c) lim x→−π/2 tan x 3. Find equations to fit each of the periodic functions drawn below. y x 24 x = –1 x = 1 x = 3 x = 5 y x 2 –26 x = – 4 x = 4 x = 8 (a) (b) 4. Suppose tan α = b. Find the following. Explain your reasoning. (a) tan(α + π) (b) tan(−α) (c) tan(π − α) 5. Graph the following. (a) g(x) =−tan x (b) h(x) =| tan x | 6. Find all x such that tan x = 0. 7. Find all x such that (a) tan x = 1. (b) tan x =−1. Try to do this using the unit circle definitions. 8. Sketch the graph of g(x) = tan 2x on [0, 2π]. 9. Sketch the graph of f(x)=3tan( x 2 ) on the interval [−2π ,2π]. 10. Consider the function f(x)= cos x sin x . 19.4 Angles and Arc Lengths 619 (a) Where is f undefined? (b) Where are the zeros of f ? (c) What is the period of f ? (d) Sketch the graph of f on the interval [0, 2π]. 11. Sketch the graph of f(x)= 1 sin x on [0, 2π]. What is the period of f ? 12. Suppose tan β = 7. (a) Find all x such that tan x = 7. (b) Find all x such that tan x =−7. 13. Decide whether each of the following functions is even, odd, or neither. (a) f(x)=1+cos x (b) g(x) = 1 + sin x (c) h(x) = sin 2x + tan x (d) j(x)=|sin x| (e) k(x) = sin x + cos x 19.4 ANGLES AND ARC LENGTHS The geometric interpretation of tan x as the slope of the line from the origin to P(x)suggests an alternative way of thinking of the trigonometric functions. Rather than thinking of the point P(x) as being determined by x, where x is a directed distance measured along the unit circle from the point (1, 0), we can think of the point P as being determined by the intersection of the unit circle and a ray drawn from the origin and making a specific angle with the positive horizontal axis. Thus, we can think of the input of any trigonometric function as either a directed distance along the unit circle or as an angle. How Should We Measure the Angle? An angle can be thought of as being actively constructed by two rays (half-lines) sharing a common endpoint. Begin by having the rays coinciding. One of these rays is designated as the initial side of the angle. The other ray is rotated about its endpoint and in its terminal position is designated as the terminal side of the angle. The common endpoint of the rays is called the vertex of the angle. A positive angle measurement indicates that the angle was constructed by rotating the terminal side in a counterclockwise direction; a negative angle measurement indicates construction by rotating the terminal side in a clockwise direction. The Greek letter θ, pronounced “theta”, is often used to refer to an angle. terminal side initial side positive angle measurement θ For the angle to determine a unique point P on the unit circle we place its initial side along the positive u-axis, the vertex at the origin, and let the terminal side (corresponding 620 CHAPTER 19 Trigonometry: Introducing Periodic Functions to OP ) cut the circle at point P . An angle positioned this way is said to be in standard position. Two angles in standard position are said to be coterminal if their sides coincide. The trigonometric functions of coterminal angles are identical because coterminal angles determine the same point P on the unit circle. In ancient Babylonia, a convention was developed to measure angles in degrees (sym- bolized by ◦ ) with a full revolution corresponding to 360 ◦ , a quarter revolution (a right angle) corresponding to 90 ◦ , and so on. This way of measuring continues to be commonly used today. 8 A very convenient alternative for measuring an angle is to put it in standard position and use the very same x that we’ve been using for the directed distance along the unit circle to measure the angle. The circumference of the unit circle is 2π units, so we use 2π to represent the angle of one complete revolution, π/2 for a quarter of a revolution, and so on. We give this angular measure the name radians, with a full revolution consisting of 2π radians. Many formulas are simplest when using this measure. 9 It is a good idea to put your calculator in radian mode and leave it there. Naturally, an angle need not be in standard position to be measured in radians. We give a general definition below. Definition An angle of x radians is the angle that subtends 10 an arc (a segment of a circle) of length x on the unit circle. Just as x indicates directed distance along the circle, so does it indicate direction for an angle, positive corresponding to counterclockwise revolution from initial to terminal side and negative corresponding to clockwise revolution. v u P(1.1) 1.1 1 1 1 x = 1.1 radians Figure 19.21 We can consider the point P(x)on the unit circle as being determined by an angle of x radians measured from the positive horizontal axis or by a directed distance of x units along the unit circle from (1, 0). For example, the point (0, 1) is π 2 units along the unit circle from 8 The choice of 360 may be due to there being close to 360 days in a year. There are other hypotheses as to the origin of the 360-degree set-up. For more information, see either Howard Eves’s An Introduction to the History of Mathematics, Sixth Edition, Sauders College Publishing, 1990, p. 42, or Eli Maor’s Trigonometric Delights, Princeton University Press, 1998, p. 15 and p. 18, footnote 2. 9 It will make differentiation of trigonometric functions much cleaner later on. 10 For an angle to subtend an arc of a circle, the vertex of the angle and the center of the circle should be made to coincide. The arc swept out from the initial ray to the terminal ray is the arc subtended by the angle. . differentiation of trigonometric functions much cleaner later on. 10 For an angle to subtend an arc of a circle, the vertex of the angle and the center of the circle should be made to coincide. The. 19.20 Between x = 0 and x = π/2, the numerator and denominator are positive. The numerator is increasing and the denominator is approaching zero, so tan x grows without bound. tan(−x) = sin(−x) cos(−x) = −. Periodic Functions to OP ) cut the circle at point P . An angle positioned this way is said to be in standard position. Two angles in standard position are said to be coterminal if their sides