18.5 Applications of Geometric Sums and Series 591 22. Suppose you borrow $18,000 at an interest rate of 8% compounded annually. You begin paying back money four years from today and make fixed payments annually. You pay back the entire debt after six payments. What are your annual payments? Begin by figuring out the ballpark figures. Will you pay more than $3000 each year? What is an upper bound for the amount of money you will pay each year? 23. A prince takes out a loan of $200,000 in order to finance his castle. The interest rate is 12% per year compounded monthly and he has a 15-year mortgage. He will pay back the loan by paying a fixed amount, F dollars, every month beginning one month from today and continuing for the next 15 years. What is F ? Note that the sum of the present values of his payments (pulled back to the present using an interest rate of 12%) should equal his loan. 24. Lithium, a drug that is used to treat manic depression, or bipolar disorder, has a half-life of 24 hours. Suppose a patient begins taking a pill of M mg every 12 hours. What is the level of the drug in the patient’s body two weeks into treatment, immediately after taking the 28th pill? 25. A physician prescribes a pill to be taken daily. Suppose that the half-life of the medica- tion in the patient’s bloodstream is 10 days. How many milligrams of medicine should the doctor prescribe if she wants the maximum level of the drug in the bloodstream to reach L mg, but not to surpass it. Assume that the drug is to be taken indefinitely. Your answer will be in terms of L. 26. At the beginning of each month a medical research center buries its refuse in its refuse dump. The monthly refuse deposit contains 40 grams of radioactive material. The radioactive material decays at a rate proportional to itself, with proportionality constant −0.2. (a) How much of the radioactive material buried at the beginning of the month is radioactive t months later? (b) Immediately after the 60th monthly dump, how much radioactive material is in the refuse site? (c) If the situation goes on indefinitely, how much radioactive material will the site contain? 27. You take out a loan of $3000 at an interest rate of 6% compounded monthly. You start paying back the loan exactly one year later. How much should each payment be if the loan is paid off after 24 equal monthly payments? Give an exact answer and an approximation correct to the nearest penny. PART VII Trigonometric Functions 19 CHAPTER Trigonometry: Introducing Periodic Functions Transition to Trigonometry Familiarity with a variety of families of functions provides us with tools necessary for modeling phenomena in the world around us. In the next few chapters we will work with functions that are particularly useful in modeling cyclic or repeating phenomena because the functions themselves are cyclic. Examples of cyclic behavior abound in nature; the rhythm of a heartbeat, the length of a day, the height of the sun in the sky, the path of a sound wave, and the motion of the planets all feature repeating patterns. Wheels are spinning all around us—wheels of bikes, trucks, cars—gears of vehicles, watches, and other machines. Think about the motion of a spot on a steadily rotating gear, or of a seat on a steadily spinning Ferris wheel. The height of the seat is a cyclic function of time; it rises and falls in a smooth, repeating manner. In this chapter we introduce trigonometric functions, 1 cyclic functions that exhibit and help us explore behaviors we observe around us. 1 The word trigonometry refers to triangles, not circles. Trigonometry can be viewed in two distinct ways. Historically it developed in the context of triangles, and hence the name of this family of functions refers to triangles. We will take a triangle perspective in Chapter 20. 593 594 CHAPTER 19 Trigonometry: Introducing Periodic Functions 19.1 THE SINE AND COSINE FUNCTIONS: DEFINITIONS AND BASIC PROPERTIES Definition Below is a circle of radius 1 centered at the origin. This is referred to as the unit circle. We’ll define trigonometric functions with reference to a point P = (u, v) on the unit circle. We locate P using a real number x as follows. Start at (1, 0). If x ≥ 0, travel along the circle in a counterclockwise direction a distance x units to arrive at P(x). If x<0,travel along the circle in a clockwise direction a distance |x| units to arrive at P(x). In other words, x indicates a directed distance around the unit circle. Equivalently, x indicates a directed arc length from (1, 0), where the term arc length means a distance along a circle. As x varies, the point P moves around the unit circle. As the position of P varies, so do its u- and v-coordinates. 1 sin x cos x P(x) = (cos x, sin x) v 1 u x directed distance along the circle u 2 + v 2 = 1 Figure 19.1 sin x = the v-coordinate of P(x) (the vertical coordinate, the second coordinate, the (signed) height of P ) cos x = the u-coordinate of P(x) (the horizontal coordinate, the first coordinate) 2 These functions are called the sine and cosine functions, respectively. From these definitions all the properties of the trigonometric functions follow. 2 Here’s a way to remember which is which: put cosine and sine in alphabetical order. Cosine is first; it corresponds to the first coordinate of P . Sine is second, and corresponds to the second coordinate of P . We have labeled our coordinate axes u and v because we want x to determine the point P , making sine and cosine functions of x. 19.1 The Sine and Cosine Functions: Definitions and Basic Properties 595 Our method for locating P essentially involves wrapping the real number line around the unit circle, with zero glued to the point (1, 0) on the circle. Figure 19.2 shows the unit circle with the portion of the number line from 0 to 2π wrapped around it like measuring tape. We’ll refer to this as the calibrated unit circle. ◆ EXAMPLE 19.1 Use the calibrated unit circle to approximate sin(1.1) and cos(1.1). SOLUTION Locate P(1.1) by moving along the unit circle a distance 1.1 counterclockwise from the point (1.0). Approximate the coordinates of P(1.1). P(1.1) ≈ (0.45, 0.9). Therefore sin(1.1) ≈ 0.9 and cos(1.1) ≈ 0.45. v u 3 2 4 .3 .2 .1 .1 .2 .3 5 6 1 0.5 0.4 0.3 0.2 0.1 P(1.1) ≈ (.45, .9) Figure 19.2 ◆ EXERCISE 19.1 (a) Find values of x between 0 and 2π such that the coordinates of P(x) are (i) (1, 0). (ii) (0, 1). (iii) (−1, 0).(iv)(0, −1). (b) Use the definitions of sine and cosine to evaluate the following. (i) sin π (ii) cos π (iii) sin π 2 (iv) cos π 2 (v) sin −3π 2 (vi) cos 3π 2 EXERCISE 19.2 Use the calibrated unit circle shown to approximate the following. (You can check your answers using a calculator, but be sure that the calculator is in radian mode as opposed to degree mode.) 3 (i) sin 0.3 (ii) sin 2 (iii) sin 3 (iv) sin 4 (v) sin 5 (vi) cos 5 EXERCISE 19.3 Use the calibrated unit circle shown to approximate all x-values between 0 and 2π such that (i) sin x = 0.8. (ii) sin x =−0.4. (iii) cos x = 0.8. (iv) cos x =−0.2. 3 Radians and degrees will be discussed in Section 19.4. To check whether your calculator is in radian mode, try to evaluate cos π. You will get −1 only if your calculator is in radian mode. 596 CHAPTER 19 Trigonometry: Introducing Periodic Functions (Again, use a calculator to check the accuracy of your answers. There are two answers to each question.) The calibrated unit circle displayed in Figure 19.2 shows only the interval [0, 2π ] wrapped around the unit circle, but in fact we want to wrap the entire number line around the circle. The circumference of the unit circle is 2π , so every directed distance of 2π (positive or negative) brings us back to the same point P . Therefore there are infinitely many x-values corresponding to any point P .Ifx 0 corresponds to the point P ,sodox 0 +2πand x 0 − 2π. In fact, P(x 0 )=P(x 0 +2πn), where n is any integer, because circumnavigating the circle any integer number of times (in either direction) has no impact on the terminal point P . Periodicity One of the most striking characteristics of trigonometric functions is their cyclic nature. A periodic function is marked by repeated cycles. Definition A function f is periodic if there is a positive constant k such that for all x in the domain of f , f(x +k)= f(x). The smallest of such constant k is called the period of f . If a function has period k, then we can select any single interval of length k in the domain, graph the function over this domain, and from this construct the entire function by horizontally shifting this fundamental block k units (left and right) repeatedly. Below are graphs of some periodic functions. – 6 – 336 y x – 6 –1 4 92–2414 y x y x period = 3 period = 10 period = 4 Figure 19.3 From their definitions we observe that the values of the output of sine and cosine repeat every 2π units; sin x and cos x both are periodic with period 2π. Therefore, by graphing the functions on any interval of length 2π we know how they behave everywhere. 19.1 The Sine and Cosine Functions: Definitions and Basic Properties 597 The Graphs of sin x and cos x Work through the following exercise. EXERCISE 19.4 A steadily spinning Ferris wheel with a radius of 10 meters makes one counterclockwise revolution every 2 minutes. Placing the origin of a coordinate system at the center of the vertical wheel, consider the position of a seat that is at the point (10, 0) at time t = 0. (a) Plot the vertical position (height) of the seat as a function of time t, t in minutes. (b) Plot the horizontal position of the seat as a function of time t, t in minutes. The graphs of sin x and cos x are closely related to those of Exercise 19.4. To see the connection, think of a Ferris wheel with a radius of 1 unit and focus on the point on the rim that starts at position (1, 0). Then, instead of plotting height versus time, plot height versus the distance the point travels. This will give the graph of sin x. The graph of cos x is obtained by looking at the horizontal position of the same point. The graphs are sketched below. y xx y – 6 3 3 – 6 –5 –5 – 4 – 4 –2 –2 –3 –3 –1 –1 1 12 456 2567 7 8 89 4 –1 1 –1 1 f(x) = sin x (a) f(x) = cos x (b) Figure 19.4 In Example 19.2 and Exercise 19.3 we approximated the values of sin x and cos x for various values of x.Ifxis any integer multiple of π/2, we can evaluate sin x and cos x exactly because we can find the coordinates of P(x)exactly. The circumference of the unit circle is 2π, therefore we know the following. P(x)is (1, 0) for x = 0, ±2π, ±4π, ,or −4π,−2π,0,2π,4π, i.e., for x = 2πn, where n is an integer. P(x)is (−1, 0) for x = π, π ± 2π, π ± 4π, ,or−3π,−π,π,3π,5π, i.e., for x = π + 2πn, where n is an integer. P(x)is (0, 1) for x = π 2 , π 2 ± 2π, π 2 ± 4π, i.e., for x = π 2 + 2πn, where n is an integer. P(x)is (0, −1) for x = 3π 2 , 3π 2 ± 2π, 3π 2 ± 4π, i.e., for x = 3π 2 + 2πn, where n is an integer. 598 CHAPTER 19 Trigonometry: Introducing Periodic Functions 3 2 4 .3 .2 .1 .1 .2 .3 5 6 1 (–1, 0) (1, 0) (0, 1) (0, –1) u v Figure 19.5 From the definitions of sine and cosine we can see that the zeros of these functions as well as all the local maxima and local minima occur at integer multiples of π/2. Therefore, when sketching sin x and cos x, the x-axis is frequently labeled just in multiples of π/2. This labeling has been known to trap dozing students into assuming, incorrectly, that the trigonometric functions are defined only for x-values with the number π explicitly written as a factor. This notion is wrong! Returning to the original definitions of sine and cosine makes it clear that the domain of the functions is all real numbers since the entire real number line is wrapped around the unit circle. The graphs in Figure 19.6 highlight x-values for which the sine and cosine graphs have zeros, local maxima, and local minima. y x –2π 2π–π π –1 1 (a) f(x) = sin x –π –3π 2 3π 2 2 π 2 y x –2π 2π–π π –1 1 (b) f(x) = cos x Figure 19.6 Domain and Range The domain of sin x is (−∞, ∞). The range of sin x is [−1, 1]. The domain of cos x is (−∞, ∞). The range of cos x is [−1, 1]. Symmetry Properties of sin x and cos x EXERCISE 19.5 Using the unit circle, show that cos x = cos(−x) and that sin x =−sin(−x). In Exercise 19.5 you have shown that 19.1 The Sine and Cosine Functions: Definitions and Basic Properties 599 cos x is an even function; its graph is symmetric about the y-axis. sin x is an odd function; its graph is symmetric about the origin. Some Trigonometric Identities (cos x, sin x) is a point on the unit circle u 2 + v 2 = 1; therefore, (sin x) 2 + (cos x) 2 = 1 for all values of x. We can also see this by using the Pythagorean Theorem and the triangle drawn in Figure 19.7. For this reason it is sometimes called a Pythagorean identity. The equation (sin x) 2 + (cos x) 2 = 1 is called a trigonometric identity because the left- and right-hand sides of the equation are identically equal for all x. 4 v u |u 1 | u v v 1 1 1 (u 1 , v 1 ) = P(x 1 ) P(x) (cos x 1 , sin x 1 ) (u, v) = (cos x, sin x) Figure 19.7 NOTATION The conventional notation for (sin x) 2 is sin 2 x. In other words, sin 2 x means (sin x) 2 . By contrast, sin x 2 means sin(x 2 ). The Pythagorean identity given above is usually written as sin 2 x + cos 2 x = 1. More generally, cos n x means (cos x) n for all n except n =−1. sin n x means (sin x) n for all n except n =−1. The one important exception is n =−1. cos −1 x denotes the inverse of the cosine function, and not (cos x) −1 . That’s just the convention. To refer to the reciprocal of cos x, we write (cos x) −1 = 1 cos x ; this expression is generally referred to as sec x. In the next two chapters we will define trigonometric functions (like sec x) constructed from sine and cosine as 4 An identity is an equation that holds for all possible values of the variable. Note the difference between an equation such as x 2 = 4, which holds only for x = 2 and x =−2, and the identity (sin x) 2 + (cos x) 2 = 1, which holds for all x, or the identity (x + y) 2 = x 2 + 2xy + y 2 , which holds for all x and all y . 600 CHAPTER 19 Trigonometry: Introducing Periodic Functions well as defining the inverse trigonometric functions. These notational conventions are very standard; in order to communicate in the language of mathematics you need to learn them. A second set of trigonometric identities comes from the periodic nature of the functions. sin(x + 2πn)= sin x for any integer n cos(x + 2πn) = cos x for any integer n A third set of trigonometric identities is suggested by looking at the relationship between the graphs of sin x and cos x. These graphs are horizontal translates. 5 We can express this in innumerable ways. For instance, if the graph of cos x is shifted to the right π/2 units, we obtain the graph of sin x. This observation is equivalent to sin x = cos(x − π/2). Shifting the graph of sin x horizontally π units produces the graph of − sin x. Similarly, shifting the graph of cos x horizontally π units produces the graph of − cos x. sin(x ± π) =−sin x cos(x ± π) =−cos x yy x x y = sin xy = cos x 2π 2π ππ –π –π 3π 2 3π 2 2 π 2 π 2 –π –π 2 Figure 19.8 Trigonometric identities can be useful in simplifying expressions. ◆ EXAMPLE 19.2 Let f(x)=  4 − 4 cos 2 (x − π). For what values of x is f minimum? maximum? SOLUTION It is not necessary to differentiate f in order to answer this question. We’ll begin by simplifying f(x). Weknow cos(x − π) =−cos x; therefore, 4 − 4 cos 2 (x − π) = 4 − 4[− cos x] 2 = 4 − 4 cos 2 x = 4(1 − cos 2 x) = 4 sin 2 x. Thus, f(x)=  4 sin 2 x. You might, at first, think that  4 sin 2 x = 2 sin x, but that is not always so.  4 sin 2 x must be nonnegative, and sin x can be negative;  4 sin 2 x = 2| sin x|. f(x)=2|sin x| 5 This is true, but we have not proven it. The proof is left as an exercise in the next chapter, where it is more easily approached. . labeled our coordinate axes u and v because we want x to determine the point P , making sine and cosine functions of x. 19.1 The Sine and Cosine Functions: Definitions and Basic Properties 595 Our. Example 19.2 and Exercise 19.3 we approximated the values of sin x and cos x for various values of x.Ifxis any integer multiple of π/2, we can evaluate sin x and cos x exactly because we can find the. payments? Give an exact answer and an approximation correct to the nearest penny. PART VII Trigonometric Functions 19 CHAPTER Trigonometry: Introducing Periodic Functions Transition to Trigonometry Familiarity