text stringlengths 235 3.08k |
|---|
that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution. y y = 3x + 1 β5 β4 β3 β2 5 4 3 2 1 β1β1 β2 β3 β4 β5 x y = β2 1 2 3 4 5 They do not cross. Figure 2 Try It #4 Solve 2x = β100. Solving exponential equations Using logarithms Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since log(a) = log(b) is equivalent to a = b, we may apply logarithms with the same base on both sides of an exponential equation. SECTION 4.6 exponential and logarithmic eQuations 393 How Toβ¦ Given an exponential equation in which a common base cannot be found, solve for the unknown. 1. Apply the logarithm of both sides of the equation. a. If one of the terms in the equation has base 10, use the common logarithm. b. If none of the terms in the equation has base 10, use the natural logarithm. 2. Use the rules of logarithms to solve for the unknown. Example 5 Solve 5x + 2 = 4x. Solution Solving an Equation Containing Powers of Different Bases Use laws of logs. Take ln of both sides. Use the distributive law. 5x + 2 = 4x ln(5x + 2) = ln(4x) (x + 2)ln(5) = xln(4) xln(5) + 2ln(5) = xln(4) xln(5) β xln(4) = β 2ln(5) x(ln(5) β ln(4)) = β 2ln(5) 5 1 __ __ ξͺ ξͺ = ln ξ’ xln ξ’ 25 4 1 __ ln ξ’ ξͺ 25 _ Divide by the coefficient of x. 5 __ ξͺ ln ξ’ 4 Use the laws of logs. x = There is no easy way to get the powers to have the same base. Get terms containing x on one side, terms without x on the other. On the left hand side, factor out an |
x. Try It #5 Solve 2x = 3x + 1. Q & Aβ¦ Is there any way to solve 2x = 3x? Yes. The solution is 0. Equations Containing e One common type of exponential equations are those with base e. This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base e on either side, we can use the natural logarithm to solve it. How Toβ¦ Given an equation of the form y = Aekt, solve for t. 1. Divide both sides of the equation by A. 2. Apply the natural logarithm of both sides of the equation. 3. Divide both sides of the equation by k. Example 6 Solve an Equation of the Form y = Ae k t Solve 100 = 20e 2t. Solution 100 = 20e 2t 5 = e 2t ln(5) = 2t t = ln(5) ___ 2 Divide by the coefficient of the power. Take ln of both sides. Use the fact that ln(x) and e x are inverse functions. Divide by the coefficient of t. 39 4 CHAPTER 4 exponential and logarithmic Functions Analysis Using laws of logs, we can also write this answer in the form t = ln β of the answer, we use a calculator. β 5. If we want a decimal approximation Try It #6 Solve 3e 0.5t = 11. Q & Aβ¦ Does every equation of the form y = Aekt have a solution? No. There is a solution when k β 0, and when y and A are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is 2 = β3et. Example 7 Solving an Equation That Can Be Simplified to the Form y = Ae k t Solve 4e2x + 5 = 12. Solution 4e2x + 5 = 12 4e2x = 7 7 __ e2x = 4 7 __ ξͺ 2x = ln ξ’ 4 Combine like terms. Divide by the coefficient of the power. Take ln of both sides. 7 1 __ __ ξͺ ln ξ’ x = 4 2 Solve for x. Try It #7 Solve 3 + e2t = 7e2t |
. Extraneous Solutions Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output. Example 8 Solving Exponential Functions in Quadratic Form Solve e2x β e x = 56. Solution e 2x β e x = 56 e 2x β e x β 56 = 0 (e x + 7)(e x β 8) = 0 Get one side of the equation equal to zero. Factor by the FOIL method. e x + 7 = 0 or e x β 8 = 0 If a product is zero, then one factor must be zero. e x = β7 or e x = 8 Isolate the exponentials. e x = 8 x = ln(8) Reject the equation in which the power equals a negative number. Solve the equation in which the power equals a positive number. Analysis When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation e x = β7 because a positive number never equals a negative number. The solution ln(β7) is not a real number, and in the real number system this solution is rejected as an extraneous solution. SECTION 4.6 exponential and logarithmic eQuations 395 Try It #8 Solve e2x = e x + 2. Q & Aβ¦ Does every logarithmic equation have a solution? No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions. Using the Definition of a logarithm to Solve logarithmic equations We have already seen that every logarithmic equation logb(x) = y is equivalent to the exponential equation b y = x. We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression. For example, consider the equation log2(2) + log2( |
3x β 5) = 3. To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for x: log2(2) + log2(3x β 5) = 3 log2(2(3x β 5)) = 3 log2(6x β 10) = 3 23 = 6x β 10 8 = 6x β 10 18 = 6x x = 3 Apply the product rule of logarithms. Distribute. Apply the definition of a logarithm. Calculate 23. Add 10 to both sides. Divide by 6. using the definition of a logarithm to solve logarithmic equations For any algebraic expression S and real numbers b and c, where b > 0, b β 1, logb(S) = c if and only if b c = S Example 9 Using Algebra to Solve a Logarithmic Equation Solve 2ln(x) + 3 = 7. Solution Try It #9 Solve 6 + ln(x) = 10. 2ln(x) + 3 = 7 2ln(x) = 4 ln(x) = 2 x = e2 Subtract 3. Divide by 2. Rewrite in exponential form. Example 10 Using Algebra Before and After Using the Definition of the Natural Logarithm Solve 2ln(6x) = 7. Solution 2ln(6x) = 7 7 __ ln(6x) = 2 7 __ 6x = e 2 7 1 __ __ e x = 2 6 Divide by 2. Use the definition of ln. Divide by 6. 39 6 CHAPTER 4 exponential and logarithmic Functions Try It #10 Solve 2ln(x + 1) = 10. Example 11 Solve ln(x) = 3. Solution Using a Graph to Understand the Solution to a Logarithmic Equation ln(x) = 3 x = e 3 Use the definition of the natural logarithm. Figure 3 represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20. In other words e3 β 20. A calculator gives a better approximation: e3 β 20.0855. y = 1n(x) y = 3 (e3, 3) β (20. |
0855, 3) 4 8 12 16 20 24 28 x y 4 2 1 β1 β2 Figure 3 The graphs of y = ln(x ) and y = 3 cross at the point (e 3, 3), which is approximately (20.0855, 3). Try It #11 Use a graphing calculator to estimate the approximate solution to the logarithmic equation 2x = 1000 to 2 decimal places. Using the One-to-One Property of logarithms to Solve logarithmic equations As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers x > 0, S > 0, T > 0 and any positive real number b, where b β 1, logb(S) = logb(T) if and only if S = T. For example, If log2(x β 1) = log2(8), then x β 1 = 8. So, if x β 1 = 8, then we can solve for x, and we get x = 9. To check, we can substitute x = 9 into the original equation: log2(9 β 1) = log2(8) = 3. In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown. For example, consider the equation log(3x β 2) β log(2) = log(x + 4). To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for x: log(3x β 2) β log(2) = log(x + 4) log ξ’ ξͺ = log(x + 4) 3x β 2 ______ 2 3x β 2 ______ 2 = x + 4 3x β 2 = 2x + 8 Apply the quotient rule of log |
arithms. Apply the one to one property of a logarithm. Multiply both sides of the equation by 2. x = 10 Subtract 2x and add 2. SECTION 4.6 exponential and logarithmic eQuations 397 To check the result, substitute x = 10 into log(3x β 2) β log(2) = log(x + 4). log(3(10) β 2) β log(2) = log((10) + 4) log(28) β log(2) = log(14) 28 __ ξͺ = log(14) 2 log ξ’ The solution checks. using the one-to-one property of logarithms to solve logarithmic equations For any algebraic expressions S and T and any positive real number b, where b β 1, logb(S) = logb(T) if and only if S = T Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution. How Toβ¦ Given an equation containing logarithms, solve it using the one-to-one property. 1. Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form logbS = logbT. 2. Use the one-to-one property to set the arguments equal. 3. Solve the resulting equation, S = T, for the unknown. Example 12 Solving an Equation Using the One-to-One Property of Logarithms Solve ln(x2) = ln(2x + 3). Solution ln(x2) = ln(2x + 3) x2 = 2x + 3 x2 β 2x β 3 = 0 (x β 3)(x + 1) = 0 Use the one-to-one property of the logarithm. Get zero on one side before factoring. Factor using FOIL. x β 3 = 0 or x + 1 = 0 If a product is zero, one of the factors must be zero. x = 3 or x = β1 Solve for x. Analysis There are two solutions: 3 or β1. The solution β1 is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive. Try It #12 Solve |
ln(x2) = ln(1). Solving Applied Problems Using exponential and logarithmic equations In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm. One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. Table 1 lists the half-life for several of the more common radioactive substances. 39 8 CHAPTER 4 exponential and logarithmic Functions Substance gallium-67 cobalt-60 Use nuclear medicine manufacturing technetium-99m nuclear medicine americium-241 construction Half-life 80 hours 5.3 years 6 hours 432 years carbon-14 uranium-235 archeological dating 5,715 years atomic power 703,800,000 years Table 1 We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay: A(t) = A0 e ln(0.5) _____ T t t __ T A(t) = A0e ln(0.5) t _ A(t) = A0 (e ln(0.5)) T t _ 1 __ ξͺ A(t) = A0 ξ’ T 2 where β’ A0 is the amount initially present β’ T is the half-life of the substance β’ t is the time period over which the substance is studied β’ y is the amount of the substance present after time t Example 13 Using the Formula for Radioactive Decay to Find the Quantity of a Substance How long will it take for ten percent of a 1,000-gram sample of uranium-235 to decay? Solution y = 1000 e ln(0.5) __________ t 703,800,000 900 = 1000 e ln(0.5) __________ t 703,800,000 0.9 = e ln(0.5) __________ t 7 |
03,800,000 ln(0.9) = ln ξ’ e ln(0.5) __________ 703,800,000 t ξͺ ln(0.9) = ln(0.5) __ t 703,800,000 t = 703,800,000 Γ ln(0.9) _ ln(0.5) years After 10% decays, 900 grams are left. Divide by 1000. Take ln of both sides. ln(eM) = M Solve for t. t β 106,979,777 years Analysis Ten percent of 1,000 grams is 100 grams. If 100 grams decay, the amount of uranium-235 remaining is 900 grams. Try It #13 How long will it take before twenty percent of our 1,000-gram sample of uranium-235 has decayed? Access these online resources for additional instruction and practice with exponential and logarithmic equations. β’ Solving logarithmic equations (http://openstaxcollege.org/l/solvelogeq) β’ Solving exponential equations with logarithms (http://openstaxcollege.org/l/solveexplog) SECTION 4.6 section exercises 399 4.6 SeCTIOn exeRCISeS VeRBAl 1. How can an exponential equation be solved? 2. When does an extraneous solution occur? How can 3. When can the one-to-one property of logarithms be used to solve an equation? When can it not be used? an extraneous solution be recognized? AlGeBRAIC For the following exercises, use like bases to solve the exponential equation. 5. 64 β
43x = 16 8. 625 β
53x + 3 = 125 6. 32x + 1 β
3x = 243 363b _ 362b = 216 2 β b 9. 4. 4β3v β 2 = 4βv 1 _ = 2n + 2 7. 2β3n β
4 3n 1 ξͺ _ 64 β
8 = 26 10. ξ’ For the following exercises, use logarithms to solve. 11. 9x β 10 = 1 12. 2e 6x = 13 14. 2 β
109a = 29 17. e β3k + 6 = 44 20. 2x + 1 |
= 52x β 1 23. 10e 8x + 3 + 2 = 8 26. 32x + 1 = 7x β 2 15. β8 β
10 p + 7 β 7 = β24 18. β5e 9x β 8 β 8 = β62 21. e 2x β e x β 132 = 0 24. 4e 3x + 3 β 7 = 53 27. e 2x β e x β 6 = 0 13. e r + 10 β 10 = β42 16. 7e 3n β 5 + 5 = β89 19. β6e 9x + 8 + 2 = β74 22. 7e8x + 8 β 5 = β95 25. 8eβ5x β 2 β 4 = β90 28. 3e 3 β 3x + 6 = β31 For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. 29. log ξ’ 1 _ 30. log324(18) = 2 ξͺ = β2 1 _ 100 For the following exercises, use the definition of a logarithm to solve the equation. 31. 5log7(n) = 10 34. 2log(8n + 4) + 6 = 10 32. β8log9(x) = 16 35. 10 β 4ln(9 β 8x) = 6 33. 4 + log2(9k) = 2 For the following exercises, use the one-to-one property of logarithms to solve. 36. ln(10 β 3x) = ln(β4x) 39. ln(β3x) = ln(x2 β 6x) 42. log9(2n2 β 14n)= log9(β45 + n2) 37. log13(5n β 2) = log13(8 β 5n) 40. log4(6 β m) = log43(m) 43. ln(x2 β 10) + ln(9) = ln(10) 38. log(x + 3) β log(x) = log(74) 41. ln(x β 2) β ln(x) = ln(54) For the following exercises, solve each equation for x. 44. log(x + 12) = log(x) + log(12) 47. ln(7) |
+ ln(2 β 4x2) = ln(14) 50. log3(3x) β log3(6) = log3(77) 45. ln(x) + ln(x β 3) = ln(7x) 48. log8(x + 6) β log8(x) = log8(58) 46. log2(7x + 6) = 3 49. ln(3) β ln(3 β 3x) = ln(4) GRAPHICAl For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. 51. log9(x) β 5 = β4 54. ln(x β 5) = 1 52. log3(x) + 3 = 2 55. log(4) + log(β5x) = 2 53. ln(3x) = 2 56. β7 + log3 (4 β x) = β6 59. log11(β2x2 β 7x) = log11(x β 2) 62. log(x2 + 13) = log(7x + 3) 57. ln(4x β 10) β 6 = β5 60. ln(2x + 9) = ln(β5x) 63. 3 _ log2(10) β log(x β 9) = log(44) 58. log(4 β 2x) = log(β4x) 61. log9(3 β x) = log9(4x β 8) 64. ln(x) β ln(x + 3) = ln(6) 40 0 CHAPTER 4 exponential and logarithmic Functions For the following exercises, solve for the indicated value, and graph the situation showing the solution point. 65. An account with an initial deposit of $6,500 earns 7.25% annual interest, compounded continuously. How much will the account be worth after 20 years? 66. The formula for measuring sound intensity in I ξͺ, decibels D is defined by the equation D = 10 log ξ’ __ I0 where I is the intensity of the sound in watts per square meter and I0 = 10β12 is the lowest level of sound that the average person can hear |
. How many decibels are emitted from a jet plane with a sound intensity of 8.3 β
102 watts per square meter? 67. The population of a small town is modeled by the equation P = 1650e0.5t where t is measured in years. In approximately how many years will the townβs population reach 20,000? TeCHnOlOGY For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. 68. 1000(1.03)t = 5000 using the common log. 70. 3(1.04)3t = 8 using the common log 72. 50eβ0.12t = 10 using the natural log 69. e5x = 17 using the natural log 71. 34x β 5 = 38 using the common log For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. 73. 7e3x β 5 + 7.9 = 47 74. ln(3) + ln(4.4x + 6.8) = 2 75. log(β0.7x β 9) = 1 + 5log(5) 76. Atmospheric pressure P in pounds per square inch is represented by the formula P = 14.7eβ0.21x, where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5,280 feet in a mile) 77. The magnitude M of an earthquake is represented by E the equation M = 2 ξͺ where E is the amount log ξ’ __ _ E0 3 of energy released by the earthquake in joules and E0 = 104.4 is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing 1.4 Β· 1013 joules of energy? exTenSIOnS 78. Use the definition of a logarithm along with the one x = x. to-one property of logarithms to prove that blog b r ξͺ kt. _ 80. Recall the compound interest formula A = a ξ’ 1 + k Use the definition of a logarithm along with properties of |
logarithms to solve the formula for time t. 79. Recall the formula for continually compounding interest, y = Ae kt. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. 81. Newtonβs Law of Cooling states that the temperature T of an object at any time t can be described by the equation T = Ts + (T0 β Ts)eβkt, where Ts is the temperature of the surrounding environment, T0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. SECTION 4.7 exponential and logarithmic models 401 leARnInG OBjeCTIVeS In this section, you will: β’ β’ β’ β’ β’ Model exponential growth and decay. Use Newtonβs Law of Cooling. Use logistic-growth models. Choose an appropriate model for data. Express an exponential model in base e. 4.7 exPOnenTIAl AnD lOGARITHMIC MODelS Figure 1 A nuclear research reactor inside the neely nuclear Research Center on the Georgia Institute of Technology campus. (credit: Georgia Tech Research Institute) We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newtonβs Law of Cooling. Modeling exponential Growth and Decay In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function: y = A0e kt where A0 is equal to the value at time zero, e is Eulerβs constant, and k is a positive constant that determines the rate (percentage) of growth. We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation |
, the logistic model sometimes fits the data better than the exponential model. On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form y = A0e kt where A0 is the starting value, and e is Eulerβs constant. Now k is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes. 40 2 CHAPTER 4 exponential and logarithmic Functions In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in Figure 2 and Figure 3. It is important to remember that, although parts of each of the two graphs seem to lie on the x-axis, they are really a tiny distance above the x-axis. y 6 5 4 3 2 1 y = 2e3x,1 3 2e (0, 2) β 1 3, 2 e β5 β4 β3 β2 β1β1 β2 21 3 4 y = 0 5 x y = 3eβ2x y,β 1 2 3e y = 0 β5 β4 β3 β2 10 8 6 4 2 β1 β2 β4 β6 β8 β10 (0, 3) 1 2, 3 e 1 2 3 45 x Figure 2 A graph showing exponential growth. The equation is y = 2e 3x. Figure 3 A graph showing exponential decay. The equation is y = 3e β2x. Exponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use orders of magnitude. The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal. For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is 4.01134972 Γ 1013. So, we could describe this number as having order of magnitude 1013. characteristics of the |
exponential function, y = A0e kt An exponential function with the form y = A0e kt has the following characteristics: β’ one-to-one function β’ horizontal asymptote: y = 0 β’ domain: ( ββ, β) β’ range: (0, β) β’ x-intercept: none β’ y-intercept: (0, A0) β’ increasing if k > 0 (see Figure 4) β’ decreasing if k < 0 (see Figure 4) ( ) 1 k, A0e y = A0ekt k > 0 ( ), β (0, A0) A0 e 1 k y ( ) 1 β k, A0e y = A0ekt k < 0 (0, A0) y = 0 t y = 0 y ( ), A0 e 1 k t Figure 4 An exponential function models exponential growth when k > 0 and exponential decay when k < 0. Example 1 Graphing Exponential Growth A population of bacteria doubles every hour. If the culture started with 10 bacteria, graph the population as a function of time. Solution When an amount grows at a fixed percent per unit time, the growth is exponential. To find A0 we use the fact that A0 is the amount at time zero, so A0 = 10. To find k, use the fact that after one hour (t = 1) the population doubles from 10 to 20. The formula is derived as follows 20 = 10e k β
1 2 = e k ln2 = k Divide by 10 Take the natural logarithm so k = ln(2). Thus the equation we want to graph is y = 10e(ln2)t = 10(eln2)t = 10 Β· 2t. The graph is shown in Figure 5. SECTION 4.7 exponential and logarithmic models 403 y = 10e(ln 2)t y 320 280 240 200 160 120 80 40 1 2 3 4 5 6 t Figure 5 The graph of y = 10e (ln2)t. Analysis The population of bacteria after ten hours is 10,240. We could describe this amount is being of the order of magnitude 104. The population of bacteria after twenty hours is 10,485,760 which is of the order of magnitude 10 7, so we could say that the population has increased by three orders of magnitude in ten hours. Half-Life We now turn to exponential decay. One of |
the common terms associated with exponential decay, as stated above, is half-life, the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay. To find the half-life of a function describing exponential decay, solve the following equation: We find that the half-life depends only on the constant k and not on the starting quantity A0. The formula is derived as follows 1 __ A0 = A0e kt 2 1 __ A0 = A0e kt 2 1 __ = e kt 2 1 ξͺ = kt ln ξ’ __ 2 Divide by A0. Take the natural log. β ln(2) = kt Apply laws of logarithms. β ln(2) ____ k = t Divide by k. Since t, the time, is positive, k must, as expected, be negative. This gives us the half-life formula t = β ln(2) ____ k How Toβ¦ Given the half-life, find the decay rate. 1. Write A = A0 ekt. 1 _ 2. Replace A by A0 and replace t by the given half-life. 2 3. Solve to find k. Express k as an exact value (do not round). ln(2) _. t Note: It is also possible to find the decay rate using k = β 40 4 CHAPTER 4 exponential and logarithmic Functions Example 2 Finding the Function that Describes Radioactive Decay The half-life of carbon-14 is 5,730 years. Express the amount of carbon-14 remaining as a function of time, t. Solution This formula is derived as follows. A = A0e kt 0.5A0 = A0e k β
5730 0.5 = e5730k ln(0.5) = 5730k ln(0.5) ______ 5730 A = A0 e ξ’ ln(0.5) ______ 5730 k = ξͺ t The continuous growth formula. Substitute the half-life for t and 0.5A0 for f(t). Divide by A0. Take the natural log of both sides. Divide by the coefficient of k. Substitute for k in the continuous growth formula. ξͺ t. We |
observe that the coefficient of t, ln(0.5) ______ 5730 The function that describes this continuous decay is f(t) = A0 e ξ’ ln(0.5) _ 5730 β β1.2097 is negative, as expected in the case of exponential decay. Try It #14 The half-life of plutonium-244 is 80,000,000 years. Find function gives the amount of carbon-14 remaining as a function of time, measured in years. Radiocarbon Dating The formula for radioactive decay is important in radiocarbon dating, which is used to calculate the approximate date a plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libb y, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It is believed to be accurate to within about 1% error for plants or animals that died within the last 60,000 years. Carbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon dioxide in the air we breathe. Most of the carbon on earth is carbon-12, which has an atomic weight of 12 and is not radioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree rings and other organic samples of known datesβalthough the ratio has changed slightly over the centuries. As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the atmosphere. When it dies, the carbon-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to carbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated. Since the half-life of carbon-14 is 5,730 years, the formula for the amount of carbon-14 remaining after t years is ξͺ t ln(0.5) ______ 5730 A β A0 e ξ’ where β’ A is the amount of carbon-14 remaining β’ A0 is the amount of carbon-14 when the plant or animal began decaying. This formula is derived as follows: A = A0e kt 0.5A0 = |
A0e k β
5730 0.5 = e5730k ln(0.5) = 5730k ln(0.5) ______ 5730 A = A0 e ξ’ ln(0.5) ______ 5730 k = ξͺ t To find the age of an object, we solve this equation for t: A ξͺ ln ξ’ _ A0 _ β0.000121 t = The continuous growth formula. Substitute the half-life for t and 0.5A0 for f (t). Divide by A0. Take the natural log of both sides. Divide by the coefficient of k. Substitute for r in the continuous growth formula. SECTION 4.7 exponential and logarithmic models 405 Out of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon-14 dating, and we only look at the basic formula. The ratio of carbon-14 to carbon-12 in the atmosphere is approximately 0.0000000001%. Let r be the ratio of carbon-14 to carbon-12 in the organic artifact or fossil to be dated, determined by a method called liquid scintillation. From the equation A β A0eβ0.000121t we know the ratio of the percentage of β eβ0.000121t. We solve carbon-14 in the object we are dating to the percentage of carbon-14 in the atmosphere is r = this equation for t, to get A __ A0 t = ln(r) _ β0.000121 How Toβ¦ Given the percentage of carbon-14 in an object, determine its age. 1. Express the given percentage of carbon-14 as an equivalent decimal, k. 2. Substitute for k in the equation t = and solve for the age, t. ln(r) _________ β0.000121 Example 3 Finding the Age of a Bone A bone fragment is found that contains 20% of its original carbon-14. To the nearest year, how old is the bone? Solution We substitute 20% = 0.20 for k in the equation and solve for t : t = = ln(r) _ β0.000121 ln(0.20) _ β0.000121 Use the general form of the equation. Substitute for r. β 13301 Round to the nearest year. The bone fragment is about 13,301 years old. |
Analysis The instruments that measure the percentage of carbon-14 are extremely sensitive and, as we mention above, a scientist will need to do much more work than we did in order to be satisfied. Even so, carbon dating is only accurate to about 1%, so this age should be given as 13,301 years Β± 1% or 13,301 years Β± 133 years. Try It #15 Cesium-137 has a half-life of about 30 years. If we begin with 200 mg of cesium-137, will it take more or less than 230 years until only 1 milligram remains? Calculating Doubling Time For decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time. Given the basic exponential growth equation A = A0e kt, doubling time can be found by solving for when the original quantity has doubled, that is, by solving 2A0 = A0e kt. The formula is derived as follows: Thus the doubling time is 2A0 = A0e kt 2 = e kt ln(2) = kt t = ln(2) _ k t = ln(2) _ k Divide by A0. Take the natural logarithm. Divide by the coefficient of t. 40 6 CHAPTER 4 exponential and logarithmic Functions Example 4 Finding a Function That Describes Exponential Growth According to Mooreβs Law, the doubling time for the number of transistors that can be put on a computer chip is approximately two years. Give a function that describes this behavior. Solution The formula is derived as follows: t = 2 = ln(2) ___ k ln(2) ___ k ln(2) ___ 2 A = A0 e k = ln(2) _____ 2 t The doubling time formula. Use a doubling time of two years. Multiply by k and divide by 2. Substitute k into the continuous growth formula. The function is A0 e ln(2) t. _____ 2 Try It #16 Recent data suggests that, as of 2013, the rate of growth predicted by Mooreβs Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account |
. Using newtonβs law of Cooling Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the objectβs temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift of the generic exponential decay function. This translation leads to Newtonβs Law of Cooling, the scientific formula for temperature as a function of time as an objectβs temperature is equalized with the ambient temperature This formula is derived as follows: T(t) = Ae kt + Ts T(t) = Ab ct + Ts T(t) = Ae ln(bct) + Ts T(t) = Ae ctln(b) + Ts T(t) = Ae kt + Ts Laws of logarithms. Laws of logarithms. Rename the constant cln(b), calling it k. Newtonβs law of cooling The temperature of an object, T, in surrounding air with temperature Ts will behave according to the formula T(t) = Ae kt + Ts where β’ t is time β’ A is the difference between the initial temperature of the object and the surroundings β’ k is a constant, the continuous rate of cooling of the object How Toβ¦ Given a set of conditions, apply Newtonβs Law of Cooling. 1. Set Ts equal to the y-coordinate of the horizontal asymptote (usually the ambient temperature). 2. Substitute the given values into the continuous growth formula T(t) = Ae kt + Ts to find the parameters A and k. 3. Substitute in the desired time to find the temperature or the desired temperature to find the time. SECTION 4.7 exponential and logarithmic models 407 Example 5 Using Newtonβs Law of Cooling A cheesecake is taken out of the oven with an ideal internal temperature of 165Β°F, and is placed into a 35Β°F refrigerator. After 10 minutes, the cheesecake has cooled to 150Β°F. If we must wait until the cheesecake has cooled to 70Β°F before we eat it, how long will we have to wait? Solution Because the surrounding air temperature in the refrigerator is 35 degrees, the cheesecakeβ |
s temperature will decay exponentially toward 35, following the equation We know the initial temperature was 165, so T(0) = 165. T(t) = Ae kt + 35 165 = Ae k0 + 35 Substitute (0, 165). A = 130 Solve for A. We were given another data point, T(10) = 150, which we can use to solve for k. 150 = 130e k10 + 35 Substitute (10, 150). 115 = 130e k10 115 ___ 130 = e 10k Subtract 35. Divide by 130. 115 ___ 130 ln ξ’ ξͺ = 10k 115 ___ ln ξ’ ξͺ 130 _ 10 This gives us the equation for the cooling of the cheesecake: T(t) = 130e β0.0123t + 35. β β0.0123 Divide by the coefficient of k. k = Take the natural log of both sides. Now we can solve for the time it will take for the temperature to cool to 70 degrees. 70 = 130eβ0.0123t + 35 35 = 130eβ0.0123t Substitute in 70 for T(t). Subtract 35. 35 ___ 130 = eβ0.0123t Divide by 130. ln ξ’ 35 ___ 130 Take the natural log of both sides ξͺ = β0.0123t 35 ___ ξͺ ln ξ’ 130 _ β 106.68 Divide by the coefficient of t. β0.0123 t = It will take about 107 minutes, or one hour and 47 minutes, for the cheesecake to cool to 70Β°F. Try It #17 A pitcher of water at 40 degrees Fahrenheit is placed into a 70 degree room. One hour later, the temperature has risen to 45 degrees. How long will it take for the temperature to rise to 60 degrees? Using logistic Growth Models Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-half-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin |
to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value. 40 8 CHAPTER 4 exponential and logarithmic Functions The logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the modelβs upper bound, called the carrying capacity. For constants a, b, and c, the logistic growth of a population over time x is represented by the model f (x) = c _______ 1 + aeβb x The graph in Figure 6 shows how the growth rate changes over time. The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases. f (x+a logistic growth The logistic growth model is where c _____ 1 + a β’ is the initial value Carrying capacity f (x) = c 1 + aeβbx ln(a) b ( )c, 2 Point of maximum growth Initial value of population x Figure 6 f (x) = c _ 1 + aeβb x β’ c is the carrying capacity, or limiting value β’ b is a constant determined by the rate of growth. Example 6 Using the Logistic-Growth Model An influenza epidemic spreads through a population rapidly, at a rate that depends on two factors: The more people who have the flu, the more rapidly it spreads, and also the more uninfected people there are, the more rapidly it spreads. These two factors make the logistic model a good one to study the spread of communicable diseases. And, clearly, there is a maximum value for the number of people infected: the entire population. For example, at time t = 0 there is one person in a community of 1,000 people who has the flu. So, in that community, at most 1,000 people can have the flu. Researchers find that for this particular strain of the flu, the logistic growth constant is b = 0.6030. Estimate the number of people in this community who will have had this flu after ten days. Predict how many people in this community will have had this flu after a long period of time has passed. Solution We substitute the given data into the logistic growth model |
f (x) = c _______ 1 + aeβb x This model predicts that, after ten days, the number of people who have had the flu is f (x) = Because at most 1,000 people, the entire population of the community, can get the flu, we know the limiting value is c = 1000. To find a, we use the formula that the number of cases at time t = 0 is = 1, from which it follows that a = 999. 1000 ____________ 1 + 999eβ0.6030x β 293.8. Because the actual number must be a whole number (a person has either had the flu or not) we round to 294. In the long term, the number of people who will contract the flu is the limiting value, c = 1000. c _ 1 + a Analysis Remember that, because we are dealing with a virus, we cannot predict with certainty the number of people infected. The model only approximates the number of people infected and will not give us exact or actual values. The graph in Figure 7 gives a good picture of how this model fits the data. SECTION 4.7 exponential and logarithmic models 409 y = 1000 1,000 cases on day 21 1,100 1,000 900 800 700 s e s a C 600 500 400 300 200 100 294 cases on day 10 1 case on day 0 20 cases on day 5 2 4 6 8 10 12 14 Days 16 18 20 22 24 26 Figure 7 The graph of f (x) = 1000 __ 1 + 999e β0.6030x Try It #18 Using the model in Example 6, estimate the number of cases of flu on day 15. Choosing an Appropriate Model for Data Now that we have discussed various mathematical models, we need to learn how to choose the appropriate model for the raw data we have. Many factors influence the choice of a mathematical model, among which are experience, scientific laws, and patterns in the data itself. Not all data can be described by elementary functions. Sometimes, a function is chosen that approximates the data over a given interval. For instance, suppose data were gathered on the number of homes bought in the United States from the years 1960 to 2013. After plotting these data in a scatter plot, we notice that the shape of the data from the years 2000 to 2013 follow a logarithmic curve. We could restrict the interval from 2000 to 2010, apply regression analysis using a logarithmic model, and use |
it to predict the number of home buyers for the year 2015. Three kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and logarithmic functions. If the data lies on a straight line, or seems to lie approximately along a straight line, a linear model may be best. If the data is non-linear, we often consider an exponential or logarithmic model, though other models, such as quadratic models, may also be considered. In choosing between an exponential model and a logarithmic model, we look at the way the data curves. This is called the concavity. If we draw a line between two data points, and all (or most) of the data between those two points lies above that line, we say the curve is concave down. We can think of it as a bowl that bends downward and therefore cannot hold water. If all (or most) of the data between those two points lies below the line, we say the curve is concave up. In this case, we can think of a bowl that bends upward and can therefore hold water. An exponential curve, whether rising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote. A logarithmic curve is always concave away from its vertical asymptote. In the case of positive data, which is the most common case, an exponential curve is always concave up, and a logarithmic curve always concave down. A logistic curve changes concavity. It starts out concave up and then changes to concave down beyond a certain point, called a point of inflection. After using the graph to help us choose a type of function to use as a model, we substitute points, and solve to find the parameters. We reduce round-off error by choosing points as far apart as possible. 41 0 CHAPTER 4 exponential and logarithmic Functions Example 7 Choosing a Mathematical Model Does a linear, exponential, logarithmic, or logistic model best fit the values listed in Table 1? Find the model, and use a graph to check your choice.386 2.197 2.773 3.219 3.584 3.892 4.159 4.394 Table 1 Solution First, plot the data on a graph as in Figure 8. For the purpose of graphing, round the data to two significant digits. y 5.5 5 4.5 |
4 3.5 3 2.5 2 1.5 1 0. 10 Figure 8 x Clearly, the points do not lie on a straight line, so we reject a linear model. If we draw a line between any two of the points, most or all of the points between those two points lie above the line, so the graph is concave down, suggesting a logarithmic model. We can try y = aln(b x). Plugging in the first point, (1,0), gives 0 = alnb. We reject the case that a = 0 (if it were, all outputs would be 0), so we know ln(b) = 0. Thus b = 1 and y = aln(x). Next we can use the point (9,4.394) to solve for a: y = aln(x) 4.394 = aln(9) a = 4.394 _____ ln(9) Because a = β 2, an appropriate model for the data is y = 2ln(x). 4.394 _ ln(9) To check the accuracy of the model, we graph the function together with the given points as in Figure 9. y = 2 ln(x) x = 0 y 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0. 10 Figure 9 The graph of y = 2lnx. x We can conclude that the model is a good fit to the data. Compare Figure 9 to the graph of y = ln(x2) shown in Figure 10. SECTION 4.7 exponential and logarithmic models 411 y x = 0 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 y = ln(x2 10 Figure 10 The graph of y = ln(x 2) x The graphs appear to be identical when x > 0. A quick check confirms this conclusion: y = ln(x 2) = 2ln(x) for x > 0. However, if x < 0, the graph of y = ln(x 2) includes a βextraβ branch, as shown in Figure 11. This occurs because, while y = 2ln(x) cannot have negative values in the domain (as such values would force the argument to be negative), the function y = ln(x 2) can have negative domain values. y y |
= ln(x2) 642 8 10 x β10 β8 β6 β4 10 8 6 4 2 β2 β2 β4 β6 β8 β10 Figure 11 Try It #19 Does a linear, exponential, or logarithmic model best fit the data in Table 2? Find the model.297 5.437 8.963 14.778 24.365 40.172 66.231 109.196 180.034 Table 2 expressing an exponential Model in Base e While powers and logarithms of any base can be used in modeling, the two most common bases are 10 and e. In science and mathematics, the base e is often preferred. We can use laws of exponents and laws of logarithms to change any base to base e. How Toβ¦ Given a model with the form y = ab x, change it to the form y = A0e kx. 1. Rewrite y = ab x as y = aeln(b x). 2. Use the power rule of logarithms to rewrite y as y = ae xln(b) = aeln(b)x. 3. Note that a = A0 and k = ln(b) in the equation y = A0e kx. 41 2 CHAPTER 4 exponential and logarithmic Functions Example 8 Changing to base e Change the function y = 2.5(3.1)x so that this same function is written in the form y = A0e kx. Solution The formula is derived as follows y = 2.5(3.1)x = 2.5e ln(3.1x ) = 2.5e xln3.1 Insert exponential and its inverse. Laws of logs. = 2.5e (ln3.1)x Commutative law of multiplication Try It #20 Change the function y = 3(0.5)x to one having e as the base. Access these online resources for additional instruction and practice with exponential and logarithmic models. β’ logarithm Application β pH (http://openstaxcollege.org/l/logph) β’ exponential Model β Age Using Half-life (http://openstaxcollege.org/l/expmodelhalf) β’ newtonβs law of Cooling (http://openstaxcollege.org/l/newtoncooling) β’ exponential Growth Given Doubling Time ( |
http://openstaxcollege.org/l/expgrowthdbl) β’ exponential Growth β Find Initial Amount Given Doubling Time (http://openstaxcollege.org/l/initialdouble) SECTION 4.7 section exercises 413 4.7 SeCTIOn exeRCISeS VeRBAl 1. With what kind of exponential model would half-life be associated? What role does half-life play in these models? 2. What is carbon dating? Why does it work? Give an example in which carbon dating would be useful. 3. With what kind of exponential model would doubling time be associated? What role does doubling time play in these models? 4. Define Newtonβs Law of Cooling. Then name at least three real-world situations where Newtonβs Law of Cooling would be applied. 5. What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain. nUMeRIC 6. The temperature of an object in degrees Fahrenheit after t minutes is represented by the equation T(t) = 68eβ0.0174t + 72. To the nearest degree, what is the temperature of the object after one and a half hours? For the following exercises, use the logistic growth model f (x) = 150 _ 1 + 8eβ2x. 7. Find and interpret f (0). Round to the nearest tenth. 9. Find the carrying capacity. 11. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data. 12. Rewrite f (x) = 1.68(0.65)x as an exponential equation with base e to five significant digits. 8. Find and interpret f (4). Round to the nearest tenth. 10. Graph the model. 0 β2 β1 x 4 1 2 3 5 f (x) 0.694 0.833 1 1.2 1.44 1.728 2.074 2.488 TeCHnOlOGY For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. x f (x) x 1 2 1 13. 14. 15. 16. 2 3 4 5 6 7 8 9 10 4 |
.079 5.296 6.159 6.828 7.375 7.838 8.238 8.592 8.908 2 3 4 5 6 7 f (x) 2.4 2.88 3.456 4.147 4.977 5.972 7.166 x 4 5 6 7 8 9 10 8 8.6 11 9 10 10.32 12.383 12 13 f (x) 9.429 9.972 10.415 10.79 11.115 11.401 11.657 11.889 12.101 12.295 x f (x) 1.25 5.75 2.25 8.75 3.56 12.68 4.2 14.6 5.65 6.75 7.25 18.95 22.25 23.75 8.6 27.8 9.25 29.75 10.5 33.5 For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in t years is modeled by the equation P(t) = 1000 _ 1 + 9eβ0.6t. 17. Graph the function. 19. To the nearest tenth, what is the doubling time for 18. What is the initial population of fish? 20. To the nearest whole number, what will the fish the fish population? population be after 2 years? 21. To the nearest tenth, how long will it take for the 22. What is the carrying capacity for the fish population? population to reach 900? Justify your answer using the graph of P. 41 4 CHAPTER 4 exponential and logarithmic Functions exTenSIOnS 23. A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay? 25. Recall the formula for calculating the magnitude of S an earthquake, M = 2 ξͺ. Show each step for log ξ’ __ _ S0 3 solving this equation algebraically for the seismic moment S. 27. Prove that b x = e xln(b) for positive b β 1. 24. The formula for an increasing population is given by P(t) = P0e rt where P0 is the initial population and r > 0. Derive a general formula for the time t it takes for the population to |
increase by a factor of M. 26. What is the y-intercept of the logistic growth model c ________ 1 + aeβrx? Show the steps for calculation. What y = does this point tell us about the population? ReAl-WORlD APPlICATIOnS For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. 28. To the nearest hour, what is the half-life of the drug? 29. Write an exponential model representing the amount of the drug remaining in the patientβs system after t hours. Then use the formula to find the amount of the drug that would remain in the patientβs system after 3 hours. Round to the nearest milligram. 30. Using the model found in the previous exercise, find f (10) and interpret the result. Round to the nearest hundredth. For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day. 31. To the nearest day, how long will it take for half of 32. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after t days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram. 34. The half-life of Radium-226 is 1590 years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits. 36. A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is 5730 years.) the Iodine-125 to decay? 33. A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance? 35. The half-life of Erbium-165 is 10.4 hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits. 37. A research student is working with |
a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours? SECTION 4.7 section exercises 415 For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1,000 bacteria present after 20 minutes. 38. To the nearest whole number, what was the initial 39. Rounding to six significant digits, write an population in the culture? exponential equation representing this situation. To the nearest minute, how long did it take the population to double? For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of 100Β° Fahrenheit was taken off the stove to cool in a 69Β° F room. After fifteen minutes, the internal temperature of the soup was 95Β° F. 40. Use Newtonβs Law of Cooling to write a formula that 41. To the nearest minute, how long will it take the soup models this situation. to cool to 80Β° F? 42. To the nearest degree, what will the temperature be after 2 and a half hours? For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of 165Β° Fahrenheit and is allowed to cool in a 75Β° F room. After half an hour, the internal temperature of the turkey is 145Β° F. 43. Write a formula that models this situation. 44. To the nearest degree, what will the temperature be after 50 minutes? 45. To the nearest minute, how long will it take the turkey to cool to 110Β° F? For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth. 46. log (x) 47. log (x) β5 β4 β3 β2 β1 0 1 2 34 5 β5 β4 β3 β2 β1 0 1 2 34 5 48. Plot each set of approximate values of intensity of W ___ sounds on a logarithmic scale: Whisper: 10β10 m2, Vacuum: 10β4 W ___ m2, Jet: 102 W ___ m2 49. Recall the formula for calculating the magnitude of an earthquake, M = 2 S ξͺ. One earthquake log ξ’ __ __ S0 3 has magnitude 3.9 on the |
MMS scale. If a second earthquake has 750 times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth. For the following exercises, use this scenario: The equation N(t) = who have heard a rumor after t days. 500 _ 1 + 49eβ0.7t models the number of people in a town 50. How many people started the rumor? 51. To the nearest whole number, how many people will have heard the rumor after 3 days? 52. As t increases without bound, what value does N(t) approach? Interpret your answer. For the following exercise, choose the correct answer choice. 53. A doctor and injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patientβs system. Which is an appropriate model for this situation? a. f (t) = 13(0.0805)t c. f (t) = 13e(β0.0839t) b. f (t) = 13e0.9195t d. f (t) = 4.75 __________ 1 + 13eβ0.83925t 41 6 CHAPTER 4 exponential and logarithmic Functions leARnInG OBjeCTIVeS In this section, you will: β’ β’ β’ Build an exponential model from data. Build a logarithmic model from data. Build a logistic model from data. 4.8 FITTInG exPOnenTIAl MODelS TO DATA In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from real-world observations. With regression analysis, we donβt expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events. Do not be confused by the word model. In mathematics, we often use the terms function, equation, and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a |
real-world situation. We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work weβve done so far, and then explore the ways regression is used to model real-world phenomena. Building an exponential Model from Data As weβve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But thatβs not the whole story. Itβs the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so letβs review exponential growth and decay. Recall that exponential functions have the form y = ab x or y = A0e kx. When performing regression analysis, we use the form most commonly used on graphing utilities, y = ab x. Take a moment to reflect on the characteristics weβve already learned about the exponential function y = ab x (assume a > 0): β’ b must be greater than zero and not equal to one. β’ The initial value of the model is y = a. β’ If b > 1, the function models exponential growth. As x increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound. β’ If 0 < b < 1, the function models exponential decay. As x increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the x-axis. In other words, the outputs never become equal to or less than zero. As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable r, or r 2. (You may have to change the calculatorβs settings for these to be shown.) The values are an indication of the βgoodness of |
fitβ of the regression equation to the data. We more commonly use the value of r 2 instead of r, but the closer either value is to 1, the better the regression equation approximates the data. SECTION 4.8 Fitting exponential models to data 417 exponential regression Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command βExpRegβ on a graphing utility to fit an exponential function to a set of data points. This returns an equation of the form, y = ab x Note that: β’ b must be non-negative. β’ when b > 1, we have an exponential growth model. β’ when 0 < b < 1, we have an exponential decay model. How Toβ¦ Given a set of data, perform exponential regression using a graphing utility. 1. Use the STAT then EDIT menu to enter given data. a. Clear any existing data from the lists. b. List the input values in the L1 column. c. List the output values in the L2 column. 2. Graph and observe a scatter plot of the data using the STATPLOT feature. a. Use ZOOM [9] to adjust axes to fit the data. b. Verify the data follow an exponential pattern. 3. Find the equation that models the data. a. Select βExpRegβ from the STAT then CALC menu. b. Use the values returned for a and b to record the model, y = ab x. 4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 1 Using Exponential Regression to Fit a Model to Data In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871 crashes were used to measure the association of a personβs blood alcohol level (BAC) with the risk of being in an accident. Table 1 shows results from the study[24]. The relative risk is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of 0.09 is 3.54 times as likely to crash as a person who has not been drinking alcohol. BAC Relative Risk of Crashing BAC Relative Risk of Crashing 0 1 0.11 6.41 0.01 1.03 0.13 |
12.6 Table 1 0.03 1.06 0.15 22.1 0.05 1.38 0.17 0.07 2.09 0.19 0.09 3.54 0.21 39.05 65.32 99.78 a. Let x represent the BAC level, and let y represent the corresponding relative risk. Use exponential regression to fit a model to these data. b. After 6 drinks, a person weighing 160 pounds will have a BAC of about 0.16. How many times more likely is a person with this weight to crash if they drive after having a 6-pack of beer? Round to the nearest hundredth. 24 Source: Indiana University Center for Studies of Law in Action, 2007 41 8 CHAPTER 4 exponential and logarithmic Functions Solution a. Using the STAT then EDIT menu on a graphing utility, list the BAC values in L1 and the relative risk values in L2. Then use the STATPLOT feature to verify that the scatterplot follows the exponential pattern shown in Figure 1: y 110 100 90 80 70 60 50 40 30 20 10.02.04.06.08.10.12.14.16.18.20.22 x Figure 1 Use the βExpRegβ command from the STAT then CALC menu to obtain the exponential model, y = 0.58304829(2.20720213E10)x Converting from scientific notation, we have: y = 0.58304829(22,072,021,300)x Notice that r 2 β 0.97 which indicates the model is a good fit to the data. To see this, graph the model in the same window as the scatterplot to verify it is a good fit as shown in Figure 2: y 110 100 90 80 70 60 50 40 30 20 10.02.04.06.08.10.12.14.16.18.20.22 x Figure 2 b. Use the model to estimate the risk associated with a BAC of 0.16. Substitute 0.16 for x in the model and solve for y. y = 0.58304829(22,072,021,300)x Use the regression model found in part (a). = 0.58304829(22,072,021,300)0.16 Substitute 0.16 for x. β 26.35 Round to the nearest hundredth. |
If a 160-pound person drives after having 6 drinks, he or she is about 26.35 times more likely to crash than if driving while sober. SECTION 4.8 Fitting exponential models to data 419 Try It #1 Table 2 shows a recent graduateβs credit card balance each month after graduation. Month 1 2 3 4 5 6 7 8 Debt ($) 620.00 761.88 899.80 1039.93 1270.63 1589.04 1851.31 2154.92 Table 2 a. Use exponential regression to fit a model to these data. b. If spending continues at this rate, what will the graduateβs credit card debt be one year after graduating? Q & Aβ¦ Is it reasonable to assume that an exponential regression model will represent a situation indefinitely? No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation). Building a logarithmic Model from Data Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best. Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics weβve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis, we use the form of the logarithmic function most commonly used on graphing utilities, y = a + bln(x). For this function β’ All input values, x, must be greater than zero. β’ The point (1, a) is on the graph of the model. β’ If b > 0, the model is increasing. Growth increases rapidly at first and then steadily slows over time. β’ If b < 0, the model is decreasing. Decay occurs rapidly at first and then steadily slows over time. logarithmic |
regression Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command βLnRegβ on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form, y = a + bln(x) Note that: β’ all input values, x, must be non-negative. β’ when b > 0, the model is increasing. β’ when b < 0, the model is decreasing. How Toβ¦ Given a set of data, perform logarithmic regression using a graphing utility. 1. Use the STAT then EDIT menu to enter given data. a. Clear any existing data from the lists. b. List the input values in the L1 column. 42 0 CHAPTER 4 exponential and logarithmic Functions c. List the output values in the L2 column. 2. Graph and observe a scatter plot of the data using the STATPLOT feature. a. Use ZOOM [9] to adjust axes to fit the data. b. Verify the data follow a logarithmic pattern. 3. Find the equation that models the data. a. Select βLnRegβ from the STAT then CALC menu. b. Use the values returned for a and b to record the model, y = a + bln(x). 4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 2 Using Logarithmic Regression to Fit a Model to Data Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century. Table 3 shows the average life expectancies, in years, of Americans from 1900β2010[25]. Year Life Expectancy (Years) 1900 47.3 1910 50.0 1920 54.1 1930 59.7 1940 62.9 1950 68.2 Year Life Expectancy (Years) 1960 69.7 1970 70.8 1980 73.7 1990 75.4 2000 76.8 2010 78.7 Table 3 a. Let x represent time in decades starting with x = 1 for the year 1900, x = 2 for the year 1910, and so on. Let y represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data. b. Use the model to predict the average American life expectancy for the |
year 2030. Solution a. Using the STAT then EDIT menu on a graphing utility, list the years using values 1β12 in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in Figure 3: y 85 80 75 70 65 60 55 50 45 40 1 2 3 4 5 6 7 8 9 10 11 12 13 x Figure 3 Use the βLnRegβ command from the STAT then CALC menu to obtain the logarithmic model, y = 42.52722583 + 13.85752327ln(x) Next, graph the model in the same window as the scatterplot to verify it is a good fit as shown in Figure 4: 25 Source: Center for Disease Control and Prevention, 2013 SECTION 4.8 Fitting exponential models to data 421 y 85 80 75 70 65 60 55 50 45 40 1 2 3 4 5 6 7 8 9 10 11 12 13 x Figure 4 b. To predict the life expectancy of an American in the year 2030, substitute x = 14 for the in the model and solve for y: y = 42.52722583 + 13.85752327ln(x) Use the regression model found in part ( a). = 42.52722583 + 13.85752327ln(14) Substitute 14 for x. β 79.1 Round to the nearest tenth If life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the year 2030. Try It #2 Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 4 shows the number of games sold, in thousands, from the years 2000β2010. Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Number Sold (Thousands) 142 149 154 155 159 161 163 164 164 166 167 a. Let x represent time in years starting with x = 1 for the year 2000. Let y represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data. b. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand. Table 4 42 2 CHAPTER 4 exponential and logarithmic Functions Building a logistic Model from Data Like exponential and logarithmic growth, logistic growth increases over time. |
One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients. It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities: y = c _______ 1 + aeβb x Recall that: c _____ 1 + a is the initial value of the model. β’ β’ when b > 0, the model increases rapidly at first until it reaches its point of maximum growth rate, ξ’ that point, growth steadily slows and the function becomes asymptotic to the upper bound y = c. ln(a) c ξͺ. At _ _, 2 b β’ c is the limiting value, sometimes called the carrying capacity, of the model. logistic regression Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command βLogisticβ on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form Note that y = c _______ 1 + aeβb x β’ The initial value of the model is c _. 1 + a β’ Output values for the model grow closer and closer to y = c as time increases. How Toβ¦ Given a set of data, perform logistic regression using a graphing utility. 1. Use the STAT then EDIT menu to enter given data. a. Clear any existing data from the lists. b. List the input values in the L1 column. c. List the output values in the L2 column. 2. Graph and observe a scatter plot of the data using the STATPLOT feature. a. Use ZOOM [9] to adjust axes to fit the data. b. Verify the data follow a logistic pattern. 3. Find the equation that models the data. a. Select βLogisticβ from the STAT then CALC menu. b. Use the values returned for a, b, and c to record the model, y = c _ 1 + a |
eβb x. 4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data. Example 3 Using Logistic Regression to Fit a Model to Data Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. Table 5 shows the percentage of Americans with cellular service between the years 1995 and 2012[26]. 26 Source: The World Bankn, 2013 SECTION 4.8 Fitting exponential models to data 423 Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 Americans with Cellular Service (%) 12.69 16.35 20.29 25.08 30.81 38.75 45.00 49.16 55.15 Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 Americans with Cellular Service (%) 62.852 68.63 76.64 82.47 85.68 89.14 91.86 95.28 98.17 Table 5 a. Let x represent time in years starting with x = 0 for the year 1995. Let y represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data. b. Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent. c. Discuss the value returned for the upper limit c. What does this tell you about the model? What would the limiting value be if the model were exact? Solution a. Using the STAT then EDIT menu on a graphing utility, list the years using values 0β15 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in Figure 5: y 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 7 89 1 0 11 12 13 14 15 16 17 18 19 20 x Figure 5 Use the βLogisticβ command from the STAT then CALC menu to obtain the logistic model, 105.7379526 ___________________ 1 + 6.88328979eβ0.2595440013x Next, graph the model in the same window as shown in Figure 6 the scatterplot to verify it is a good fit: y = y 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x Figure 6 42 4 CHAPTER 4 exponential and logarithmic Functions b |
. To approximate the percentage of Americans with cellular service in the year 2013, substitute x = 18 for the in the model and solve for y: y = = 105.7379526 ____________________ 1 + 6.88328979e β0.2595440013x 105.7379526 _____________________ 1 + 6.88328979e β0.2595440013(18) Substitute 18 for x. Use the regression model found in part ( a). β 99.3% Round to the nearest tenth According to the model, about 99.3% of Americans had cellular service in 2013. c. The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be c = 100 and the modelβs outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service! Try It #3 Table 6 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012. Year 1997 1998 1999 2000 2001 2002 2003 2004 Seal Population (Thousands) 3.493 5.282 6.357 9.201 11.224 12.964 16.226 18.137 Year 2005 2006 2007 2008 2009 2010 2011 2012 Seal Population (Thousands) 19.590 21.955 22.862 23.869 24.243 24.344 24.919 25.108 Table 6 a. Let x represent time in years starting with x = 0 for the year 1997. Let y represent the number of seals in thousands. Use logistic regression to fit a model to these data. b. Use the model to predict the seal population for the year 2020. c. To the nearest whole number, what is the limiting value of this model? Access this online resource for additional instruction and practice with exponential function models. β’ exponential Regression on a Calculator (http://openstaxcollege.org/l/pregresscalc) SECTION 4.8 section exercises 425 4.8 SeCTIOn exeRCISeS VeRBAl 1. What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit. 2. What is a carrying capacity? What kind of model has |
a carrying capacity built into its formula? Why does this make sense? 3. What is regression analysis? Describe the process of performing regression analysis on a graphing utility. 4. What might a scatterplot of data points look like if it were best described by a logarithmic model? 5. What does the y-intercept on the graph of a logistic equation correspond to for a population modeled by that equation? GRAPHICAl For the following exercises, match the given function of best fit with the appropriate scatterplot in Figure 7 through Figure 11. Answer using the letter beneath the matching graph. y y 16 14 12 10 8 6 4 2 16 14 12 10 8 6 4 2 y 16 14 12 10 8 6 4 2 y 16 14 12 10 a) 6 7 8 9 10 1 2 3 4 x 6 7 8 9 10 1 2 3 4 x 5 (b) 6 7 8 9 10 x 5 (c) Figure 7 Figure 8 Figure 9 y 16 14 12 10 d) 6 7 8 9 10 10 5 (e) Figure 10 Figure 11 6. y = 10.209eβ0.294x 7. y = 5.598 β 1.912ln(x) 8. y = 2.104(1.479)x 9. y = 4.607 + 2.733ln(x) 10. y = 14.005 __________ 1 + 2.79eβ0.812x 42 6 CHAPTER 4 exponential and logarithmic Functions nUMeRIC 11. To the nearest whole number, what is the initial value of a population modeled by the logistic equation P(t) = 175 ____________ 1 + 6.995eβ0.68t? What is the carrying capacity? 12. Rewrite the exponential model A(t) = 1550(1.085)x as an equivalent model with base e. Express the exponent to four significant digits. 13. A logarithmic model is given by the equation h(p) = 67.682 β 5.792ln(p). To the nearest hundredth, for what value of p does h(p) = 62? 15. What is the y-intercept on the graph of the logistic model given in the previous exercise? TeCHnOlOGY 14. A logistic model is given by the equation 90 P(t) = ________ 1 + 5eβ0.42t. To the nearest hundredth |
, for what value of t does P(t) = 45? For the following exercises, use this scenario: The population P of a koi pond over x months is modeled by the function P(x) = 68 __ 1 + 16eβ0.28x. 16. Graph the population model to show the population 17. What was the initial population of koi? over a span of 3 years. 18. How many koi will the pond have after one and a 19. How many months will it take before there are 20 half years? koi in the pond? 20. Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity. For the following exercises, use this scenario: The population P of an endangered species habitat for wolves 558 __ is modeled by the function P(x) = 1 + 54.8eβ0.462x, where x is given in years. 21. Graph the population model to show the population 22. What was the initial population of wolves over a span of 10 years. transported to the habitat? 23. How many wolves will the habitat have after 3 years? 24. How many years will it take before there are 100 wolves in the habitat? 25. Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity. For the following exercises, refer to Table 7. x f (x) 1 1125 2 1495 3 2310 Table 7 4 3294 5 4650 6 6361 26. Use a graphing calculator to create a scatter diagram 27. Use the regression feature to find an exponential of the data. function that best fits the data in the table. 28. Write the exponential function as an exponential 29. Graph the exponential equation on the scatter equation with base e. diagram. 30. Use the intersect feature to find the value of x for which f (x) = 4000. SECTION 4.8 section exercises 427 For the following exercises, refer to Table 8. x f (x) 1 555 2 383 3 307 Table 8 4 210 5 158 6 122 31. Use a graphing calculator to create a scatter diagram 32. Use the regression feature to find an exponential of the data. function that best fits the data in the table. 33. Write the exponential function as an exponential 34. Graph the exponential equation on the scatter equation with base e. diagram. 35. Use the intersect feature to find the value of x |
for which f (x) = 250. For the following exercises, refer to Table 9. x f (x) 1 5.1 2 6.3 3 7.3 Table 9 4 7.7 5 8.1 6 8.6 36. Use a graphing calculator to create a scatter diagram 37. Use the LOGarithm option of the REGression of the data. feature to find a logarithmic function of the form y = a + bln(x) that best fits the data in the table. 38. Use the logarithmic function to find the value of the 39. Graph the logarithmic equation on the scatter function when x = 10. diagram. 40. Use the intersect feature to find the value of x for which f (x) = 7. For the following exercises, refer to Table 10. x f (x) 1 7.5 2 6 3 5.2 4 4.3 5 3.9 6 3.4 7 3.1 8 2.9 Table 10 41. Use a graphing calculator to create a scatter diagram 42. Use the LOGarithm option of the REGression of the data. feature to find a logarithmic function of the form y = a + bln(x) that best fits the data in the table. 43. Use the logarithmic function to find the value of the 44. Graph the logarithmic equation on the scatter function when x = 10. diagram. 45. Use the intersect feature to find the value of x for which f (x) = 8. For the following exercises, refer to Table 11. x f (x) 1 8.7 2 12.3 3 15.4 4 18.5 5 20.7 6 22.5 7 23.3 8 24 9 24.6 10 24.8 Table 11 46. Use a graphing calculator to create a scatter diagram of the data. 47. Use the LOGISTIC regression option to find a logistic growth model of the form y = best fits the data in the table. c _ 1 + aeβb x that 42 8 CHAPTER 4 exponential and logarithmic Functions 48. Graph the logistic equation on the scatter diagram. 49. To the nearest whole number, what is the predicted carrying capacity of the model? 50. Use the intersect feature to find the value of x for which the model reaches half its carrying capacity. For the following exercises, refer to Table 12. |
x f (x) 0 12 2 28.6 4 52.8 5 70.3 7 8 10 11 15 17 99.9 112.5 125.8 127.9 135.1 135.9 Table 12 51. Use a graphing calculator to create a scatter diagram of the data. 52. Use the LOGISTIC regression option to find a logistic growth model of the form y = best fits the data in the table. c ________ 1 + aeβb x that 53. Graph the logistic equation on the scatter diagram. 54. To the nearest whole number, what is the predicted carrying capacity of the model? 55. Use the intersect feature to find the value of x for which the model reaches half its carrying capacity. exTenSIOnS 56. Recall that the general form of a logistic equation for a population is given by P(t) = c _ 1 + aeβbt, such that the initial population at time t = 0 is P(0) = P0. Show algebraically that c β P0 _ c β P(t) _ P0 P(t) eβbt. = 57. Use a graphing utility to find an exponential regression formula f (x) and a logarithmic regression formula g(x) for the points (1.5, 1.5) and (8.5, 8.5). Round all numbers to 6 decimal places. Graph the points and both formulas along with the line y = x on the same axis. Make a conjecture about the relationship of the regression formulas. 58. Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary. 59. Find the inverse function f β1 (x) for the logistic c _ 1 + aeβb x. Show all steps. function f (x) = the logistic model P(t) = 60. Use the result from the previous exercise to graph ________ 1 + 4eβ0.5t along with its inverse on the same axis. What are the intercepts and asymptotes of each function? 20 CHAPTER 4 review 429 CHAPTeR 4 ReVIeW Key Terms annual percentage rate (APR) the yearly interest rate earned by an investment account, also called nominal rate carrying capacity in a logistic model, the limiting value of the output change-of-base formula a formula for converting a logarithm with any base to a quotient of log |
arithms with any other base. common logarithm the exponent to which 10 must be raised to get x; log10(x) is written simply as log(x). compound interest interest earned on the total balance, not just the principal doubling time the time it takes for a quantity to double exponential growth a model that grows by a rate proportional to the amount present extraneous solution a solution introduced while solving an equation that does not satisfy the conditions of the original equation half-life the length of time it takes for a substance to exponentially decay to half of its original quantity logarithm the exponent to which b must be raised to get x; written y = logb(x) c _ logistic growth model a function of the form f (x) = 1 + a c ________ 1+ aeβb x where limiting value, and b is a constant determined by the rate of growth is the initial value, c is the carrying capacity, or natural logarithm the exponent to which the number e must be raised to get x; loge(x) is written as ln(x). Newtonβs Law of Cooling the scientific formula for temperature as a function of time as an objectβs temperature is equalized with the ambient temperature nominal rate the yearly interest rate earned by an investment account, also called annual percentage rate order of magnitude the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal power rule for logarithms a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base product rule for logarithms a rule of logarithms that states that the log of a product is equal to a sum of logarithms quotient rule for logarithms a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms Key equations definition of the exponential function f (x) = b x, where b > 0, b β 1 definition of exponential growth f (x) = ab x, where a > 0, b > 0, b β 1 compound interest formula continuous growth formula General Form for the Translation of the Parent Function f (x) = b x nt, where r _ n ξͺ A(t) = P ξ’ 1 + A(t) is the account value at time |
t t is the number of years P is the initial investment, often called the principal r is the annual percentage rate (APR), or nominal rate n is the number of compounding periods in one year A(t) = ae rt, where t is the number of unit time periods of growth a is the starting amount (in the continuous compounding formula a is replaced with P, the principal) e is the mathematical constant, e β 2.718282 f (x) = ab x + c + d Definition of the logarithmic function Definition of the common logarithm For x > 0, b > 0, b β 1, y = logb(x) if and only if b y = x. For x > 0, y = log(x) if and only if 10y = x. 43 0 CHAPTER 4 exponential and logarithmic Functions Definition of the natural logarithm General Form for the Translation of the Parent Logarithmic Function f (x) = logb(x) The Product Rule for Logarithms The Quotient Rule for Logarithms The Power Rule for Logarithms The Change-of-Base Formula For x > 0, y = ln(x) if and only if ey = x. f (x) = alogb(x + c) + d logb(MN) = logb(M) + logb(N) M ξͺ = logbM β logbN logb ξ’ __ N logb(Mn) = nlogbM logbM = lognM _ lognb n > 0, n β 1, b β 1 One-to-one property for exponential functions Definition of a logarithm One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number b, where bS = bT if and only if S = T. For any algebraic expression S and positive real numbers b and c, where b β 1, logb(S) = c if and only if bc = S. For any algebraic expressions S and T and any positive real number b, where b β 1, logbS = logbT if and only if S = T. If A = A0e kt, k < 0, the half-life is t = β. t = ln(2 |
) _ k A ln ξ’ ξͺ _ A0 _ β0.000121 is the amount of carbon-14 when the plant or animal died, A0 A is the amount of carbon-14 remaining today, t is the age of the fossil in years If A = A0e kt, k > 0, the doubling time is t = T(t) = Ae kt + Ts, where Ts is the ambient temperature, A = T(0) β Ts, and k is the continuous rate of cooling. ln(2) ___ k Half-life formula Carbon-14 dating Doubling time formula Newtonβs Law of Cooling Key Concepts 4.1 Exponential Functions β’ An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent. See Example 1. β’ A function is evaluated by solving at a specific value. See Example 2 and Example 3. β’ An exponential model can be found when the growth rate and initial value are known. See Example 4. β’ An exponential model can be found when the two data points from the model are known. See Example 5. β’ An exponential model can be found using two data points from the graph of the model. See Example 6. β’ An exponential model can be found using two data points from the graph and a calculator. See Example 7. β’ The value of an account at any time t can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See Example 8. β’ The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See Example 9. β’ The number e is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is e β 2.718282. β’ Scientific and graphing calculators have the key [e x] or [e xp(x)] for calculating powers of e. See Example 10. β’ Continuous growth or decay models are exponential models that use e as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See Example 11 and Example 12. CHAPTER 4 review 431 4.2 Graphs of Exponential Functions β’ The graph of the function f (x) = b x has a y-intercept at (0, 1), domain (ββ |
, β), range (0, β), and horizontal asymptote y = 0. See Example 1. β’ If b > 1, the function is increasing. The left tail of the graph will approach the asymptote y = 0, and the right tail will increase without bound. β’ If 0 < b < 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote y = 0. β’ The equation f (x) = b x + d represents a vertical shift of the parent function f (x) = b x. β’ The equation f (x) = b x + c represents a horizontal shift of the parent function f (x) = b x. See Example 2. β’ Approximate solutions of the equation f (x) = b x + c + d can be found using a graphing calculator. See Example 3. β’ The equation f (x) = ab x, where a > 0, represents a vertical stretch if β£ a β£ > 1 or compression if 0 < β£ a β£ < 1 of the parent function f (x) = b x. See Example 4. β’ When the parent function f (x) = b x is multiplied by β1, the result, f (x) = βb x, is a reflection about the x-axis. When the input is multiplied by β1, the result, f (x) = bβx, is a reflection about the y-axis. See Example 5. β’ All translations of the exponential function can be summarized by the general equation f (x) = ab x + c + d. See Table 3. β’ Using the general equation f (x) = ab x + c + d, we can write the equation of a function given its description. See Example 6. 4.3 Logarithmic Functions β’ The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function. β’ Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See Example 1. β’ Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See Example 2. β’ Logarithmic functions with base b can be evaluated mentally using previous knowledge of powers of b. See Example 3 and Example 4. |
β’ Common logarithms can be evaluated mentally using previous knowledge of powers of 10. See Example 5. β’ When common logarithms cannot be evaluated mentally, a calculator can be used. See Example 6. β’ Real-world exponential problems with base 10 can be rewritten as a common logarithm and then evaluated using a calculator. See Example 7. β’ Natural logarithms can be evaluated using a calculator Example 8. 4.4 Graphs of Logarithmic Functions β’ To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for x. See Example 1 and Example 2 β’ The graph of the parent function f (x) = logb(x) has an x-intercept at (1, 0), domain (0, β), range (ββ, β), vertical asymptote x = 0, and β’ if b > 1, the function is increasing. β’ if 0 < b < 1, the function is decreasing. See Example 3. β’ The equation f (x) = logb(x + c) shifts the parent function y = logb(x) horizontally β’ left c units if c > 0. β’ right c units if c < 0. See Example 4. β’ The equation f (x) = logb(x) + d shifts the parent function y = logb(x) vertically β’ up d units if d > 0. β’ down d units if d < 0. See Example 5. 43 2 CHAPTER 4 exponential and logarithmic Functions β’ For any constant a > 0, the equation f (x) = alogb(x) β’ stretches the parent function y = logb(x) vertically by a factor of a if β£ a β£ > 1. β’ compresses the parent function y = logb(x) vertically by a factor of a if β£ a β£ < 1. See Example 6 and Example 7. β’ When the parent function y = logb(x) is multiplied by β1, the result is a reflection about the x-axis. When the input is multiplied by β1, the result is a reflection about the y-axis. β’ The equation f (x) = βlogb(x) represents a reflection of the parent function about the x-axis. β’ The equation f (x) = logb(βx) represents a reflection of the parent function about the y- |
axis. See Example 8. β’ A graphing calculator may be used to approximate solutions to some logarithmic equations See Example 9. β’ All translations of the logarithmic function can be summarized by the general equation f (x) = alogb(x + c) + d. See Table 4. β’ Given an equation with the general form f (x) = alogb(x + c) + d, we can identify the vertical asymptote x = βc for the transformation. See Example 10. β’ Using the general equation f (x) = alogb(x + c) + d, we can write the equation of a logarithmic function given its graph. See Example 11. 4.5 Logarithmic Properties β’ We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See Example 1. β’ We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See Example 2. β’ We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See Example 3, Example 4, and Example 5. β’ We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See Example 6, Example 7, and Example 8. β’ The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See Example 9, Example 10, Example 11, and Example 12. β’ We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of- base formula. See Example 13. β’ The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient of natural or common logs. That way a calculator can be used to evaluate. See Example 14. 4.6 Exponential and Logarithmic Equations β’ We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the |
unknown. β’ When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See Example 1. β’ When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See Example 2, Example 3, and Example 4. β’ When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See Example 5. β’ We can solve exponential equations with base e, by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See Example 6 and Example 7. β’ After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See Example 8. CHAPTER 4 review 433 β’ When given an equation of the form logb(S) = c, where S is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation bc = S, and solve for the unknown. See Example 9 and Example 10. β’ We can also use graphing to solve equations with the form logb(S) = c. We graph both equations y = logb(S) and y = c on the same coordinate plane and identify the solution as the x-value of the intersecting point. See Example 11. β’ When given an equation of the form logbS = logbT, where S and T are algebraic expressions, we can use the one- to-one property of logarithms to solve the equation S = T for the unknown. See Example 12. β’ Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See Example 13. 4.7 Exponential and Logarithmic Models β’ The basic exponential function is f (x) = ab x. If b > 1, we have exponential growth; if 0 < b < 1, we have exponential decay. β’ We can also write this formula in terms of continuous growth as A = A0e kx, where A0 is the starting value. If A0 is positive, then we have exponential growth when k > |
0 and exponential decay when k < 0. See Example 1. β’ In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See Example 2. β’ We can find the age, t, of an organic artifact by measuring the amount, k, of carbon-14 remaining in the artifact and using the formula t = to solve for t. See Example 3. ln(k) _ β0.000121 β’ Given a substanceβs doubling time or half-life we can find a function that represents its exponential growth or decay. See Example 4. β’ We can use Newtonβs Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See Example 5. β’ We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See Example 6. β’ We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See Example 7. β’ Any exponential function with the form y = ab x can be rewritten as an equivalent exponential function with the form y = A0e kx where k = lnb. See Example 8. 4.8 Fitting Exponential Models to Data β’ Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. β’ We use the command βExpRegβ on a graphing utility to fit function of the form y = ab x to a set of data points. See Example 1. β’ Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. β’ We use the command βLnRegβ on a graphing utility to fit a function of the form y = a + bln(x) to a set of data points. See Example 2. β’ Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit. β’ We use the command βLogistic |
β on a graphing utility to fit a function of the form y = points. See Example 3. c _________ 1 + aeβb x to a set of data 43 4 CHAPTER 4 exponential and logarithmic Functions CHAPTeR 4 ReVIeW exeRCISeS exPOnenTIAl FUnCTIOnS 1. Determine whether the function y = 156(0.825)t represents exponential growth, exponential decay, or neither. Explain 2. The population of a herd of deer is represented by the function A(t) = 205(1.13)t, where t is given in years. To the nearest whole number, what will the herd population be after 6 years? 3. Find an exponential equation that passes through the points (2, 2.25) and (5, 60.75). 4. Determine whether Table 1 could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. x f (x) 1 3 2 0.9 Table 1 3 0.27 4 0.081 5. A retirement account is opened with an initial deposit of $8,500 and earns 8.12% interest compounded monthly. What will the account be worth in 20 years? 7. Does the equation y = 2.294eβ0.654t represent continuous growth, continuous decay, or neither? Explain. 6. Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with 7.5% APR, compounded daily, in order to reach her goal in 3 years? 8. Suppose an investment account is opened with an initial deposit of $10,500 earning 6.25% interest, compounded continuously. How much will the account be worth after 25 years? GRAPHS OF exPOnenTIAl FUnCTIOnS 9. Graph the function f (x) = 3.5(2)x. State the domain and range and give the y-intercept. 1 ξͺ 10. Graph the function f (x) = 4 ξ’ __ 8 x and its reflection about the y-axis on the same axes, and give the y-intercept. 11. The graph of f (x) = 6.5x is reflected about the y-axis and stretched vertically |
by a factor of 7. What is the equation of the new function, g (x)? State its y-intercept, domain, and range. 12. The graph here shows transformations of the graph of f (x) = 2x. What is the equation for the transformation1β1 β2 β3 β6 β5 β4 β3 β2 21 3 4 5 6 x lOGARITHMIC FUnCTIOnS 13. Rewrite log17(4913) = x as an equivalent exponential 14. Rewrite ln(s) = t as an equivalent exponential equation. equation. Figure 1 β 2 __ = b as an equivalent logarithmic 5 15. Rewrite a equation. 1 ξͺ to exponential form. 17. Solve for xlog64(x) = ξ’ _ 3 19. Evaluate log(0.000001) without using a calculator. 16. Rewrite eβ3.5 = h as an equivalent logarithmic equation. 18. Evaluate log5 ξ’ 20. Evaluate log(4.005) using a calculator. Round to the ξͺ without using a calculator. 1 _ 125 nearest thousandth. CHAPTER 4 review 435 21. Evaluate ln(eβ0.8648) without using a calculator. 22. Evaluate ln ξ’ 3 β β 18 ξͺ using a calculator. Round to the GRAPHS OF lOGARITHMIC FUnCTIOnS 23. Graph the function g(x) = log(7x + 21) β 4. 25. State the domain, vertical asymptote, and end behavior of the function g (x) = ln(4x + 20) β 17. lOGARITHMIC PROPeRTIeS nearest thousandth. 24. Graph the function h(x) = 2ln(9 β 3x) + 1. 26. Rewrite ln(7r Β· 11st) in expanded form. 27. Rewrite log8(x) + log8(5) + log8(y) + log8(13) in ξͺ in expanded form. 67 28. Rewrite logm ξ’ ___ 83 1 30. Rewrite ln ξ’ x5 ξͺ as a product. __ 32. Use properties of logarithms to expand log ξ’ r 2s11 t14 |
ξͺ. _ compact form. 29. Rewrite ln(z) β ln(x) β ln(y) in compact form. 1 ξͺ as a single logarithm. 31. Rewrite βlogy ξ’ __ 12 33. Use properties of logarithms to expand ln ξ’ 2b β ______ b + 1 ξͺ. _____ b β 1 to a single logarithm. 37. Rewrite 512x β 17 = 125 as a logarithm. Then apply the change of base formula to solve for x using the common log. Round to the nearest thousandth. 125 __ βx β 3 = 53 by rewriting each side with a 39. Solve 1 ξ’ ξͺ _ 625 common base. 34. Condense the expression 5ln(b) + ln(c) + 35. Condense the expression 3log7v + 6log7w β ln(4 β a) _ 2 log 7 u _ 3 to a single logarithm. 36. Rewrite log3(12.75) to base e. exPOnenTIAl AnD lOGARITHMIC eQUATIOnS 38. Solve 2163x Β· 216x = 363x + 2 by rewriting each side with a common base. 40. Use logarithms to find the exact solution for 41. Use logarithms to find the exact solution for 7 Β· 17β9x β 7 = 49. If there is no solution, write no solution. 3e6n β 2 + 1 = β60. If there is no solution, write no solution. 42. Find the exact solution for 5e3x β 4 = 6. If there is 43. Find the exact solution for 2e5x β 2 β 9 = β56. no solution, write no solution. If there is no solution, write no solution. 44. Find the exact solution for 52x β 3 = 7x + 1. If there is 45. Find the exact solution for e 2x β e x β 110 = 0. If no solution, write no solution. there is no solution, write no solution. 46. Use the definition of a logarithm to solve. 47. Use the definition of a logarithm to find the exact β5log7(10n) = 5. solution for 9 + 6ln(a + 3) |
= 33. 48. Use the one-to-one property of logarithms to find an exact solution for log8(7) + log8(β4x) = log8(5). If there is no solution, write no solution. 49. Use the one-to-one property of logarithms to find an exact solution for ln(5) + ln(5x2 β 5) = ln(56). If there is no solution, write no solution. 50. The formula for measuring sound intensity in decibels D is defined by the equation D = 10log I _ ξͺ, where I is the intensity of the sound in watts ξ’ I0 per square meter and I0 = 10β12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of 6.3 Β· 10β3 watts per square meter? 52. Find the inverse function f β1 for the exponential function f (x. 51. The population of a city is modeled by the equation P(t) = 256, 114e0.25t where t is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million? 53. Find the inverse function f β1 for the logarithmic function f (x) = 0.25 Β· log2(x3 + 1). 43 6 CHAPTER 4 exponential and logarithmic Functions exPOnenTIAl AnD lOGARITHMIC MODelS For the following exercises, use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour. 54. To the nearest minute, what is the half-life of the drug? 55. Write an exponential model representing the amount of the drug remaining in the patientβs system after t hours. Then use the formula to find the amount of the drug that would remain in the patientβs system after 24 hours. Round to the nearest hundredth of a gram. For the following exercises, use this scenario: A soup with an internal temperature of 350Β° Fahrenheit was taken off the stove to cool in a 71Β°F room. After fifteen minutes, the internal temperature of the soup was 175Β°F. 56. Use Newtonβs Law of Cooling to write a formula that 57. How many minutes will it |
take the soup to cool models this situation. to 85Β°F? For the following exercises, use this scenario: The equation N(t) = school who have heard a rumor after t days. 1200 __ 1 + 199eβ0.625t models the number of people in a 58. How many people started the rumor? 59. To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity? 60. What is the carrying capacity? For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic. 61. 62. x f (x) x f (x) 1 3.05 0.5 18.05 2 4.42 1 17 3 6.4 4 9.28 5 13.46 6 19.52 7 28.3 8 41.04 9 59.5 10 86.28 3 15.33 5 14.55 7 14.04 10 13.5 12 13.22 13 13.1 15 12.88 17 12.69 20 12.45 63. Find a formula for an exponential equation that goes through the points (β2, 100) and (0, 4). Then express the formula as an equivalent equation with base e. FITTInG exPOnenTIAl MODelS TO DATA 64. What is the carrying capacity for a population modeled by the logistic equation P(t) = initial population for the model? 65. The population of a culture of bacteria is modeled by the logistic equation P(t) = 250, 000 ___________ 1 + 499eβ0.45t? What is the 14, 250 __ 1 + 29eβ0.62t, where t is in days. To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity? For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. 66. 67. 68. x f (x) x f (x) x f (x) 1 409.4 2 260 |
.7 3 170.4 4 110.6 5 74 6 44.7 7 32.4 8 19.5 9 12.7 0.15 36.21 0.25 28.88 0.5 24.39 0.75 18.28 1 16.5 1.5 12.99 2 9.91 2.25 8.57 2.75 7.23 3 5.99 0 9 2 22.6 4 44.2 5 62.1 7 96.9 8 113.4 10 133.4 11 137.6 15 148.4 10 8.1 3.5 4.81 17 149.3 CHAPTER 4 practice test 437 CHAPTeR 4 PRACTICe TeST 1. The population of a pod of bottlenose dolphins is 2. Find an exponential equation that passes through modeled by the function A(t) = 8(1.17)t, where t is given in years. To the nearest whole number, what will the pod population be after 3 years? the points (0, 4) and (2, 9). 3. Drew wants to save $2,500 to go to the next 4. An investment account was opened with an World Cup. To the nearest dollar, how much will he need to invest in an account now with 6.25% APR, compounding daily, in order to reach his goal in 4 years? 5. Graph the function f (x) = 5(0.5)βx and its reflection across the y-axis on the same axes, and give the y-intercept. initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? 6. The graph below shows transformations of the 1 ξͺ graph of f (x) = ξ’ __ 2 transformation? x. What is the equation for the β4 β3 β2 y 4 3 2 1 0 β1β1 β2 β3 β4 β5 β6 β7 β8 21 3 4 5 6 7 8 x 7. Rewrite log8.5(614.125) = a as an equivalent exponential equation. 1 __ 8. Rewrite e = m as an equivalent logarithmic 2 equation. 9. Solve for x by converting the logarithmic equation 10. Evaluate log(10,000,000) without using a calculator. log 1 _ 7 (x) = 2 to exponential |
form. 11. Evaluate ln(0.716) using a calculator. Round to the 12. Graph the function g (x) = log(12 β 6x) + 3. nearest thousandth. 13. State the domain, vertical asymptote, and end 14. Rewrite log(17a Β· 2b) as a sum. behavior of the function f (x) = log5(39 β 13x) + 7. 15. Rewrite logt(96) β logt(8) in compact form. 17. Use properties of logarithm to expand ln (y 3z 2 Β· 3 β β x β 4 ). 19. Rewrite 163x β 5 = 1000 as a logarithm. Then apply the change of base formula to solve for x using the natural log. Round to the nearest thousandth. 21. Use logarithms to find the exact solution for β9e10a β 8 β5 = β41. If there is no solution, write no solution. 16. Rewrite log8 ξ’ a 1 __ b ξͺ as a product. 18. Condense the expression 4ln(c) + ln(d) + logarithm. ln(a) _ 3 + ln(b + 3) _ 3 to a single 20. Solve ξ’ 1 ξͺ = ξ’ _ 9 side with a common base. 1 _ 243 1 ξͺ _ 81 x Β· β3x β 1 by rewriting each 22. Find the exact solution for 10e 4x + 2 + 5 = 56. If there is no solution, write no solution. 23. Find the exact solution for β5eβ4x β 1 β 4 = 64. If 24. Find the exact solution for 2x β 3 = 62x β 1. If there is there is no solution, write no solution. no solution, write no solution. 25. Find the exact solution for e2x β e x β 72 = 0. If there is no solution, write no solution. 26. Use the definition of a logarithm to find the exact solution for 4log(2n) β 7 = β11. 43 8 CHAPTER 4 exponential and logarithmic Functions 27. Use the one-to-one property of logarithms to find an exact solution for log(4x2 β 10) + log(3) = |
log(51) If there is no solution, write no solution. 28. The formula for measuring sound intensity in decibels D is defined by the equation I ξͺ D = 10log ξ’ __ I0 where I is the intensity of the sound in watts per square meter and I0 = 10β12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of 4.7 Β· 10β1 watts per square meter? 30. Write the formula found in the previous exercise as an equivalent equation with base e. Express the exponent to five significant digits. 32. The population of a wildlife habitat is modeled __ 1 + 6.2eβ0.35t, where t is by the equation P(t) = 360 given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity? 29. A radiation safety officer is working with 112 grams of a radioactive substance. After 17 days, the sample has decayed to 80 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance? 31. A bottle of soda with a temperature of 71Β° Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of 35Β° F. After ten minutes, the internal temperature of the soda was 63Β° F. Use Newtonβs Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour? 33. Enter the data from Table 2 into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic. x f (x) 1 3 2 8.55 3 11.79 4 14.09 5 15.88 Table 2 6 17.33 7 18.57 8 19.64 9 20.58 10 21.42 34. The population of a lake of fish is modeled by the logistic equation P(t) = __ 1 + 25eβ0.75t, where t is time in years. To the nearest hundredth, how many years will it take the lake to reach 80% of its carrying capacity? 16, 120 For the following exercises, use a graphing utility to create a scatter diagram of the data given |
in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. 35. 36. 37. x f (x) x f (x) x f (x) 1 20 2 21.6 3 29.2 4 36.4 5 46.6 6 55.7 7 72.6 8 87.1 9 107.2 10 138.1 3 13.98 4 17.84 5 20.01 6 22.7 7 24.1 8 26.15 9 27.37 10 28.38 11 29.97 12 31.07 13 31.43 0 2.2 0.5 2.9 1 3.9 1.5 4.8 2 6.4 3 9.3 4 12.3 5 15 6 16.2 7 17.3 8 17.9 Trigonometric Functions 5 Figure 1 The tide rises and falls at regular, predictable intervals. (credit: Andrea Schaffer, Flickr) CHAPTeR OUTlIne 5.1 Angles 5.2 Unit Circle: Sine and Cosine Functions 5.3 The Other Trigonometric Functions 5.4 Right Triangle Trigonometry Introduction Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. Similarly, the progression from day to night occurs as a result of Earthβs rotation, and the pattern of the seasons repeats in response to Earthβs revolution around the sun. Outside of nature, many stocks that mirror a companyβs profits are influenced by changes in the economic business cycle. In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the functionβs outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions. 439 44 0 CHAPTER 5 trigonometric Functions leARnInG |
OBjeCTIVeS In this section, you will: β’ β’ β’ β’ β’ Draw angles in standard position. Convert between degrees and radians. Find coterminal angles. Find the length of a circular arc. Use linear and angular speed to describe motion on a circular path. 5.1 AnGleS A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles. Drawing Angles in Standard Position Properly defining an angle first requires that we define a ray. A ray consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific β ray by stating its endpoint and any other point on it. The ray in Figure 1 can be named as ray EF, or in symbol form EF. Ray EF F endpoint E Figure 1 An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. The angle in Figure 2 is formed from ED and EF. Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form β DEF. β β Angle DEF D F E Vertex Figure 2 Greek letters are often used as variables for the measure of an angle. Table 1 is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 3. ΞΈ theta Ο or Ο phi Ξ± alpha Table 1 Ξ² beta Ξ³ gamma ΞΈ Figure 3 Angle theta, shown as β ΞΈ SECTION 5.1 angles 441 Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex as in Figure 4 |
. Terminal side Vertex Initial side Figure 4 As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is of a circular 1 _ 360 rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit βdegreesβ after the number, or include the degree symbol Β°. For example, 90 degrees = 90Β°. To formalize our work, we will begin by drawing angles on an x-y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See Figure 5. Standard Position y Terminal side Initial side x Figure 5 If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle. Drawing an angle in standard position always starts the same wayβdraw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do 1 _ =. 4 So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive that by dividing the angle measure in degrees by 360Β°. For example, to draw a 90Β° angle, we calculate that 90Β° _ 360Β° x-axis. To draw a 360Β° angle, we calculate that = 1. So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x-axis. In this case, the initial side and the terminal side overlap. See Figure 6. 360Β° _ 360Β° Drawing a 90Β° angle y Drawing a 360Β° angle y Terminal side 90Β° x Initial side Figure 6 360Β° Initial side x Terminal side 44 2 CHAPTER 5 trigonometric Functions Since we define an angle in standard position by its initial side, we have a special type of angle whose terminal side lies on an axis, a quad |
rantal angle. This type of angle can have a measure of 0Β°, 90Β°, 180Β°, 270Β° or 360Β°. See Figure 7. II III I 0Β° IV II III I 90Β° IV II I 180Β° II I 270Β° III IV III IV Figure 7 Quadrantal angles have a terminal side that lies along an axis. examples are shown. quadrantal angles Quadrantal angles are angles in standard position whose terminal side lies on an axis, including 0Β°, 90Β°, 180Β°, 270Β°, or 360Β°. How Toβ¦ Given an angle measure in degrees, draw the angle in standard position. 1. Express the angle measure as a fraction of 360Β°. 2. Reduce the fraction to simplest form. 3. Draw an angle that contains that same fraction of the circle, beginning on the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles. Example 1 Drawing an Angle in Standard Position Measured in Degrees a. Sketch an angle of 30Β° in standard position. b. Sketch an angle of β135Β° in standard position. Solution a. Divide the angle measure by 360Β°. To rewrite the fraction in a more familiar fraction, we can recognize that 30Β° ____ 360Β° = 1 __ 12 ξ’ 1 __ ξͺ 4 One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30Β° as in Figure 8. = 1 __ 3 1 __ 12 y 30Β° x SECTION 5.1 angles 443 b. Divide the angle measure by 360Β°. In this case, we can recognize that Figure 8 β135Β° _____ 360Β° = β 3 __ 8 ξ’ 1 __ ξͺ 4 Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in Figure 9. = β 3 __ 2 β 3 __ 8 y Figure 9 x β135Β° Try It #1 Show an angle of 240Β° on a circle in standard position. Converting Between Degrees and Radians Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc |
. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle. The circumference of a circle is C = 2Οr. If we divide both sides of this equation by r, we create the ratio of the circumference to the radius, which is always 2Ο regardless of the length of the radius. So the circumference of any circle is 2Ο β 6.28 times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in Figure 10. 3 2 4 1 Fractional piece 5 6 Figure 10 44 4 CHAPTER 5 trigonometric Functions This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals 2Ο times the radius, a full circular rotation is 2Ο radians. So 2Ο radians = 360Β° Ο radians = 360Β° ____ 2 = 180Β° 1 radian = β 57.3Β° 180Β° ____ Ο See Figure 11. Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out. r t r Figure 11 The angle t sweeps out a measure of one radian. note that the length of the intercepted arc is the same as the length of the radius of the circle. Relating Arc Lengths to Radius An arc length s is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius. This ratio, called the radian measure, is the same regardless of the radius of the circleβit depends only on the angle. This property allows us to define a |
measure of any angle as the ratio of the arc length s to the radius r. See Figure 12. If s = r, then ΞΈ = r _ r = 1 radian. s = rΞΈ ΞΈ = s _ r 2r 2 radians 1 radian r s r 3 radians 1 radian (s = r) A full revolution 0, 2Ο 4 3 + Ο radians 1 + Ο radians 2 + Ο radians (a) (b) (c) Figure 12 (a) In an angle of 1 radian, the arc length s equals the radius r. (b) An angle of 2 radians has an arc length s = 2r. (c) A full revolution is 2Ο or about 6.28 radians. To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is C = 2Οr, where r is the radius. The smaller circle then has circumference 2Ο(2) = 4Ο and the larger has circumference 2Ο(3) = 6Ο. Now we draw a 45Β°angle on the two circles, as in Figure 13. SECTION 5.1 angles 445 Ο 45Β° = β radians 4 45Β° r = 3 r = 2 Figure 13 A 45Β° angle contains one-eighth of the circumference of a circle, regardless of the radius. Notice what happens if we find the ratio of the arc length divided by the radius of the circle. Smaller circle: Larger circle: 1 __ Ο 2 = 1 __ _ Ο 4 2 3 __ Ο 4 = 1 __ _ Ο 4 3 1 _ Since both ratios are Ο, the angle measures of both circles are the same, even though the arc length and radius differ. 4 radians One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution (360Β°) equals 2Ο radians. A half revolution (180Β°) is equivalent to Ο radians. The radian measure of an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle. In other words, if s is the length of an arc of a circle, and r is the radius of the circle, then the central angle s _ r radians. In a circle of |
radius 1, the radian measure corresponds to the length of containing that arc measures the arc. Q & Aβ¦ A measure of 1 radian looks to be about 60Β°. Is that correct? Yes. It is approximately 57.3Β°. Because 2Ο radians equals 360Β°, 1 radian equals 360Β° _ 2Ο β 57.3Β°. Using Radians Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in Figure 12, suppose the radius was 2 inches and the distance along the arc was also 2 inches. When we calculate the radian measure of the angle, the βinchesβ cancel, and we have a result without units. Therefore, it is not necessary to write the label βradiansβ after a radian measure, and if we see an angle that is not labeled with βdegreesβ or the degree symbol, we can assume that it is a radian measure. Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360Β°. We can also track one rotation around a circle by finding the circumference, C = 2Οr, and for the unit circle C = 2Ο. These two different ways to rotate around a circle give us a way to convert from degrees to radians. 1 rotation = 360Β° = 2Ο radians 1 __ rotation = 180Β° = Ο radians 2 Ο __ 1 __ rotation = 90Β° = radians 4 2 44 6 CHAPTER 5 trigonometric Functions Identifying Special Angles Measured in Radians In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in Figure 14. Memorizing these angles will be very useful as we study the properties associated with angles. 90Β° 120Β° 135Β° 150Β° 60Β° 45Β° 30Β° 5Ο 6 Ο 2 90Β° 2Ο 3 120Β° 3Ο 4 135Β° 150Β° Ο 3 60Β° Ο 4 45Β° 30Β° Ο 6 180Β° 0Β° Ο 180ΒΊ 0Β° 2Ο 210Β° 225Β° 240Β° 330Β° 315Β° 300Β° 270Β° Figure 14 Commonly encountered angles measured in degrees 210Β° 330Β° |
7Ο 6 225Β° 240Β° 5Ο 4 11Ο 6 315Β° 7Ο 4 300Β° 5Ο 3 270ΒΊ 3Ο 2 Figure 15 Commonly encountered angles measured in radians 4Ο 3 Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure 14, which are shown in Figure 15. Be sure you can verify each of these measures. Example 2 Finding a Radian Measure Find the radian measure of one-third of a full rotation. Solution For any circle, the arc length along such a rotation would be one-third of the circumference. We know that So, 1 rotation = 2Οr s = 1 __ 3 2Οr ___ 3 = (2Οr) The radian measure would be the arc length divided by the radius. radian measure = 2Οr _ 3 _ r 2Οr ___ 3r 2Ο ___ 3 = = Try It #2 Find the radian measure of three-fourths of a full rotation. Converting Between Radians and Degrees Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion. ΞΈ _ 180 = ΞΈ R _ Ο This proportion shows that the measure of angle ΞΈ in degrees divided by 180 equals the measure of angle ΞΈ in radians divided by Ο. Or, phrased another way, degrees is to 180 as radians is to Ο. Degrees _ 180 = Radians _ Ο SECTION 5.1 angles 447 converting between radians and degrees To convert between degrees and radians, use the proportion ΞΈ ___ 180 = ΞΈ R _ Ο Example 3 Converting Radians to Degrees Convert each radian measure to degrees. b. 3 Ο __ a. 6 Solution Because we are given radians and we want degrees, we should set up a proportion and solve it. a. We use the proportion, substituting the given information. ΞΈ _ 180 ΞΈ _ 180 = ΞΈ R _ Ο Ο __ 6 _ = Ο b. We use the proportion, substituting the given information. ΞΈ = 180 _ 6 ΞΈ = 30Β° ΞΈ _ 180 ΞΈ _ 180 = ΞΈ R _ Ο 3 _ Ο = 3(180) _ Ο ΞΈ β 172Β° ΞΈ = Try It #3 Convert β 3Ο _ radians to degrees. 4 Example 4 Conver |
ting Degrees to Radians Convert 15 degrees to radians. Solution In this example, we start with degrees and want radians, so we again set up a proportion and solve it, but we substitute the given information into a different part of the proportion. ΞΈ _ 180 = 15 _ 180 = ΞΈ R _ Ο ΞΈ R _ Ο 15Ο _ 180 = ΞΈR Analysis Another way to think about this problem is by remembering that 30Β° = Ο. Because 15Β° = 1 _ _ (30Β°), we can find that 2 6 Ο 1 ξ’ _ _ ξͺ is 2 6 Ο _. 12 Ο _ 12 = ΞΈR 44 8 CHAPTER 5 trigonometric Functions Try It #4 Convert 126Β° to radians. Finding Coterminal Angles Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 0Β° to 360Β°, or 0 to 2Ο. It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution. It is possible for more than one angle to have the same terminal side. Look at Figure 16. The angle of 140Β° is a positive angle, measured counterclockwise. The angle of β220Β° is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than 360Β° or less than 0Β° is coterminal with an angle between 0Β° and 360Β°, and it is often more convenient to find the coterminal angle within the range of 0Β° to 360Β° than to work with an angle that is outside that range. y 140Β° x β220Β° Figure 16 An angle of 140Β° and an angle of β220Β° are coterminal angles. Any angle has infinitely many coterminal angles because each time we add 360Β° to that angleβor subtract 360Β° from itβ the resulting value has a terminal side in the same location. For example, 100Β° and 460Β° are coterminal for this reason, as is β260Β°. Recognizing that any angle has infinitely many coterminal angles explains the repetitive shape in the graphs of trig |
onometric functions. An angleβs reference angle is the measure of the smallest, positive, acute angle t formed by the terminal side of the angle t and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See Figure 17 for examples of reference angles for angles in different quadrants. Quadrant I y Quadrant II y Quadrant III y Quadrant IV y II I t x II t' I II I II t x t x III IV III IV t' III IV III IV t I t' x t' = t t' = Ο β t = 180Β° β t t' = t β Ο = t β 180Β° t' = 2Ο β t = 360Β° β t Figure 17 coterminal and reference angles Coterminal angles are two angles in standard position that have the same terminal side. An angleβs reference angle is the size of the smallest acute angle, tβ², formed by the terminal side of the angle t and the horizontal axis. SECTION 5.1 angles 449 How Toβ¦ Given an angle greater than 360Β°, find a coterminal angle between 0Β° and 360Β°. 1. Subtract 360Β° from the given angle. 2. If the result is still greater than 360Β°, subtract 360Β° again till the result is between 0Β° and 360Β°. 3. The resulting angle is coterminal with the original angle. Example 5 Finding an Angle Coterminal with an Angle of Measure Greater Than 360Β° Find the least positive angle ΞΈ that is coterminal with an angle measuring 800Β°, where 0Β° β€ ΞΈ < 360Β°. Solution An angle with measure 800Β° is coterminal with an angle with measure 800 β 360 = 440Β°, but 440Β° is still greater than 360Β°, so we subtract 360Β° again to find another coterminal angle: 440 β 360 = 80Β°. y The angle ΞΈ = 80Β° is coterminal with 800Β°. To put it another way, 800Β° equals 80Β° plus two full rotations, as shown in Figure 18. 80Β° 800Β° x Try It #5 Find an angle Ξ± that is coterminal with an angle measuring 870Β°, where 0Β° β€ Ξ± < 360Β°. Figure 18 How Toβ¦ Given an angle with measure less than 0Β°, find a cotermin |
al angle having a measure between 0Β° and 360Β°. 1. Add 360Β° to the given angle. 2. If the result is still less than 0Β°, add 360Β° again until the result is between 0Β° and 360Β°. 3. The resulting angle is coterminal with the original angle. Example 6 Finding an Angle Coterminal with an Angle Measuring Less Than 0Β° Show the angle with measure β45Β° on a circle and find a positive coterminal angle Ξ± such that 0Β° β€ Ξ± < 360Β°. Solution Since 45Β° is half of 90Β°, we can start at the positive horizontal axis and measure clockwise half of a 90Β° angle. y Because we can find coterminal angles by adding or subtracting a full rotation of 360Β°, we can find a positive coterminal angle here by adding 360Β°: We can then show the angle on a circle, as in Figure 19. β45Β° + 360Β° = 315Β° x β45Β° 315Β° Figure 19 Try It #6 Find an angle Ξ² that is coterminal with an angle measuring β300Β° such that 0Β° β€ Ξ² < 360Β°. 45 0 CHAPTER 5 trigonometric Functions Finding Coterminal Angles Measured in Radians We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations. How Toβ¦ Given an angle greater than 2Ο, find a coterminal angle between 0 and 2Ο. 1. Subtract 2Ο from the given angle. 2. If the result is still greater than 2Ο, subtract 2Ο again until the result is between 0 and 2Ο. 3. The resulting angle is coterminal with the original angle. Example 7 Finding Coterminal Angles Using Radians Find an angle Ξ² that is coterminal with, where 0 β€ Ξ² < 2Ο. 19Ο _ 4 Solution When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of 2Ο radians: 19Ο ___ 4 β 2Ο = β 8Ο ___ 4 19Ο ___ 4 11Ο ___ 4 = The angle 11Ο _ 4 is coterminal |
, but not less than 2Ο, so we subtract another rotation: 11Ο ___ 4 β 2Ο = 8Ο ___ 4 β 11Ο ___ 4 3Ο ___ 4 = The angle is coterminal with, as shown in Figure 20. 3Ο _ 4 19Ο _ 4 19Ο 4 y 3Ο 4 Figure 20 x Try It #7 Find an angle of measure ΞΈ that is coterminal with an angle of measure β 17Ο _ 6 where 0 β€ ΞΈ < 2Ο. Determining the length of an Arc Recall that the radian measure ΞΈ of an angle was defined as the ratio of the arc length s of a circular arc to the radius s _ r. From this relationship, we can find arc length along a circle, given an angle. r of the circle, ΞΈ = SECTION 5.1 angles 451 arc length on a circle In a circle of radius r, the length of an arc s subtended by an angle with measure ΞΈ in radians, shown in Figure 21, is s = rΞΈ s ΞΈ r Figure 21 How Toβ¦ Given a circle of radius r, calculate the length s of the arc subtended by a given angle of measure ΞΈ. 1. If necessary, convert ΞΈ to radians. 2. Multiply the radius r by the radian measure of ΞΈ : s = r ΞΈ. Example 8 Finding the Length of an Arc Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun. a. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day? b. Use your answer from part (a) to determine the radian measure for Mercuryβs movement in one Earth day. Solution a. Letβs begin by finding the circumference of Mercuryβs orbit. C = 2Οr = 2Ο(36 million miles) β 226 million miles Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled: (0.0114)226 million miles = 2.58 million miles b. Now, we convert to radians: radian = arclength _ radius = 2.58 million miles __ 36 million miles = 0.0717 Try It #8 Find the arc length along a circle of radius 10 units subtended by an angle of 215Β°. Finding the Area of |
a Sector of a Circle In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius r can be found using the formula A = Οr 2. If the two radii form an angle of ΞΈ, measured in radians, then is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction multiplied by the entire area. (Always remember that this formula only applies if ΞΈ is in radians.) ΞΈ _ 2Ο ΞΈ _ 2Ο Area of sector = ξ’ ΞΈ __ 2Ο ξͺ Οr2 = ΞΈΟr 2 ____ 2Ο 1 __ = ΞΈr 2 2 45 2 CHAPTER 5 trigonometric Functions area of a sector The area of a sector of a circle with radius r subtended by an angle ΞΈ, measured in radians, is See Figure 22. A = 1 __ Figure 22 The area of the sector equals half the square of the radius times the central angle measured in radians. How Toβ¦ Given a circle of radius r, find the area of a sector defined by a given angle ΞΈ. 1. If necessary, convert ΞΈ to radians. 2. Multiply half the radian measure of ΞΈ by the square of the radius r : A = 1 _ ΞΈ r2. 2 Example 9 Finding the Area of a Sector An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 23. What is the area of the sector of grass the sprinkler waters? Solution First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert: 20 ft 30Β° Figure 23 The sprinkler sprays 20 ft within an arc of 30Β°. The area of the sector is then So the area is about 104.72 ft2. 30 degrees = 30 Β· Ο ___ 180 Ο __ = radians 6 Ο __ Area = 1 __ ξͺ (20)2 ξ’ 2 6 β 104.72 Try It #9 In |
central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places. Use linear and Angular Speed to Describe Motion on a Circular Path In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 10Ο inches, every second. So the linear speed of the point is 10Ο in./s. The equation for linear speed is as follows where v is linear speed, s is displacement, and t is time. v = s _ t SECTION 5.1 angles 453 Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes 360 degrees __ 4 seconds a full rotation every 4 seconds, we can calculate its angular speed as = 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where Ο (read as omega) is angular speed, ΞΈ is the angle traversed, and t is time. Ο = ΞΈ __ t Combining the definition of angular speed with the arc length equation, s = rΞΈ, we can find a relationship between angular and linear speeds. The angular speed equation can be solved for ΞΈ, giving ΞΈ = Οt. Substituting this into the arc length equation gives: Substituting this into the linear speed equation gives: s = rΞΈ = rΟt v = s _ t = rΟt ___ t = rΟ angular and linear speed As a point moves along a circle of radius r, its angular speed, Ο, is the angular rotation ΞΈ per unit time, t. Ο = ΞΈ __ t The linear speed, |
v, of the point can be found as the distance traveled, arc length s, per unit time, t. v = s _ t When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation v = rΟ This equation states that the angular speed in radians, Ο, representing the amount of rotation occurring in a unit of time, can be multiplied by the radius r to calculate the total arc length traveled in a unit of time, which is the definition of linear speed. How Toβ¦ Given the amount of angle rotation and the time elapsed, calculate the angular speed. 1. If necessary, convert the angle measure to radians. 2. Divide the angle in radians by the number of time units elapsed: Ο = ΞΈ _. t 3. The resulting speed will be in radians per time unit. Example 10 Finding Angular Speed A water wheel, shown in Figure 24, completes 1 rotation every 5 seconds. Find the angular speed in radians per second. Solution The wheel completes 1 rotation, or passes through an angle of 2Ο radians in 5 seconds, so the angular speed would be Ο = β 1.257 radians per second. 2Ο _ 5 Figure 24 45 4 CHAPTER 5 trigonometric Functions Try It #10 An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second. How Toβ¦ Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed. 1. Convert the total rotation to radians if necessary. 2. Divide the total rotation in radians by the elapsed time to find the angular speed: apply Ο = ΞΈ _ t. 3. Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply v = rΟ. Example 11 Finding a Linear Speed A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road. Solution Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road. We begin by converting from rotations per minute to |
radians per minute. It can be helpful to utilize the units to make this conversion: 180 rotations _______ Β· minute 2Ο radians ________ rotation = 360Ο radians ______ minute Using the formula from above along with the radius of the wheels, we can find the linear speed: v = (14 inches) ξ’ 360Ο = 5040Ο inches ______ minute Remember that radians are a unitless measure, so it is not necessary to include them. radians ξͺ ______ minute Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour. 5040Ο inches ______ minute Β· 1 feet _______ 12 inches Β· 1 mile _______ 5280 feet Β· 60 minutes _________ 1 hour β 14.99 miles per hour (mph) Try It #11 A satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour. Access these online resources for additional instruction and practice with angles, arc length, and areas of sectors. β’ Angles in Standard Position (http://openstaxcollege.org/l/standardpos) β’ Angle of Rotation (http://openstaxcollege.org/l/angleofrotation) β’ Coterminal Angles (http://openstaxcollege.org/l/coterminal) β’ Determining Coterminal Angles (http://openstaxcollege.org/l/detcoterm) β’ Positive and negative Coterminal Angles (http://openstaxcollege.org/l/posnegcoterm) β’ Radian Measure (http://openstaxcollege.org/l/radianmeas) β’ Coterminal Angles in Radians (http://openstaxcollege.org/l/cotermrad) β’ Arc length and Area of a Sector (http://openstaxcollege.org/l/arclength) SECTION 5.1 section exercises 455 5.1 SeCTIOn exeRCISeS VeRBAl 1. Draw an angle in standard position. Label the vertex, 2. Explain why there are an infinite number of angles initial side, and terminal side. that are coterminal to a certain angle. 3. State what a positive or negative angle signifies, and explain how to draw each. 4. How |
does radian measure of an angle compare to the degree measure? Include an explanation of 1 radian in your paragraph. 5. Explain the differences between linear speed and angular speed when describing motion along a circular path. For the following exercises, draw an angle in standard position with the given measure. 10. β150Β° 8. β80Β° 7. 300Β° 9. 135Β° 6. 30Β° 12. 7Ο ___ 4 18. β315Β° 13. 19. 5Ο ___ 6 22Ο ___ 3 14. Ο __ 2 20. β Ο __ 6 15. β Ο __ 10 21. β 4Ο ___ 3 16. 415Β° 11. 2Ο ___ 3 17. β120Β° For the following exercises, refer to Figure 25. Round to two decimal places. For the following exercises, refer to Figure 26. Round to two decimal places. r = 3 in 140Β° Figure 25 22. Find the arc length. 23. Find the area of the sector. 2Ο 5 r = 4.5 cm Figure 26 24. Find the arc length. 25. Find the area of the sector. 23. 24. AlGeBRAIC For the following exercises, convert angles in radians to degrees. 27. Ο __ 9 31. β 5Ο ___ 12 3Ο ___ radians 4 30. β 7Ο ___ 3 radians radians radians 32. 26. 28. β 5Ο ___ radians 4 11Ο ___ 6 radians For the following exercises, convert angles in degrees to radians. 33. 90Β° 34. 100Β° 38. β315Β° 35. β540Β° 39. 150Β° 37. 180Β° 29. Ο __ 3 radians 36. β120Β° For the following exercises, use to given information to find the length of a circular arc. Round to two decimal places. 40. Find the length of the arc of a circle of radius Ο _ 12 inches subtended by a central angle of radians. 4 42. Find the length of the arc of a circle of diameter 5Ο _. 14 meters subtended by the central angle of 6 44. Find the length of the arc of a circle of radius 5 inches subtended by the central angle of 220Β°. 41. Find the length of the arc of a circle of radius Ο _ 5.02 miles subtended by the central angle of. 3 43. Find the length of the arc of a circle of radius 10 centimeters subtended by the central angle of 50 |
Β°. 45. Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63Β°. 45 6 CHAPTER 5 trigonometric Functions For the following exercises, use the given information to find the area of the sector. Round to four decimal places. 47. A sector of a circle has a central angle of 30Β° and a 46. A sector of a circle has a central angle of 45Β° and a radius 6 cm. radius of 20 cm. 48. A sector of a circle with diameter 10 feet and an Ο _ angle of radians. 2 49. A sector of a circle with radius of 0.7 inches and an angle of Ο radians. For the following exercises, find the angle between 0Β° and 360Β° that is coterminal to the given angle. 50. β40Β° 51. β110Β° 53. 1400Β° 52. 700Β° For the following exercises, find the angle between 0 and 2Ο in radians that is coterminal to the given angle. 54. β Ο __ 9 10Ο ___ 3 13Ο ___ 6 44Ο ___ 9 56. 57. 55. ReAl-WORlD APPlICATIOnS 58. A truck with 32-inch diameter wheels is traveling at 60 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make? 59. A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/ min. How many revolutions per minute do the wheels make? 60. A wheel of radius 8 inches is rotating 15Β°/s. What is 61. A wheel of radius 14 inches is rotating 0.5 rad/s. What the linear speed v, the angular speed in RPM, and the angular speed in rad/s? is the linear speed v, the angular speed in RPM, and the angular speed in deg/s? 62. A CD has diameter of 120 millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 200 RPM (revolutions per minute). Find the linear speed. 64. A person is standing on the equator of Earth (radius 3960 miles). What are his linear and angular speeds? 66. Find the distance along an arc on the surface of Earth that subt |
ends a central angle of 7 minutes 1 ξ’ 1 minute = degree ξͺ. The radius of Earth is _ 60 3,960 miles. exTenSIOnS 63. When being burned in a writable CD-R drive, the angular speed of a CD varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4,800 RPM (revolutions per minute). Find the linear speed if the CD has diameter of 120 millimeters. 65. Find the distance along an arc on the surface of Earth 1 _ 60 that subtends a central angle of 5 minutes ξ’ 1 minute = degree ξͺ. The radius of Earth is 3,960 mi. 67. Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in 20 minutes? 68. Two cities have the same longitude. The latitude of city A is 9.00 degrees north and the latitude of city B is 30.00 degree north. Assume the radius of the earth is 3960 miles. Find the distance between the two cities. 69. A city is located at 40 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city. 70. A city is located at 75 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city. 71. Find the linear speed of the moon if the average distance between the earth and moon is 239,000 miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour. 72. A bicycle has wheels 28 inches in diameter. A 73. A car travels 3 miles. Its tires make 2640 revolutions. tachometer determines that the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is travelling down the road. 74. A wheel on a tractor has a 24-inch diameter. How many revolutions does the wheel make if the tractor travels 4 miles? What is the radius of a tire in inches? SECTION 5.2 unit circle: sine and cosine Functions 457 leARnInG OBjeCTIVeS In this section, you will: β’ β’ β’ ξͺ |
and 60Β° or ξ’ Ο ξͺ, 45Β° or ξ’ Ο Find function values for the sine and cosine of 30Β° or ξ’ Ο _ _ _ ξͺ. 3 4 6 Identify the domain and range of sine and cosine functions. Use reference angles to evaluate trigonometric functions. 5.2 UnIT CIRCle: SIne AnD COSIne FUnCTIOnS Figure 1 The Singapore Flyer is the worldβs tallest Ferris wheel. (credit: βVibin jKβ/Flickr) Looking for a thrill? Then consider a ride on the Singapore Flyer, the worldβs tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feetβa little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs. Finding Function Values for the Sine and Cosine To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in Figure 2. The angle (in radians) that t intercepts forms an arc of length s. Using the formula s = rt, and knowing that r = 1, we see that for a unit circle, s = t. Recall that the x- and y-axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV. For any angle t, we can label the intersection of the terminal side and the unit circle as by its coordinates, (x, y). The coordinates x and y will be the outputs of the trigonometric functions f (t) = cos t and f (t) = sin t, respectively. This means x = cos t and y = sin t. y II I (x, y) s x 1 t sin t cos t III IV Figure 2 Unit circle where the central angle is t radians 45 8 CHAPTER 5 trigon |
ometric Functions unit circle A unit circle has a center at (0, 0) and radius 1. In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle t. Let (x, y) be the endpoint on the unit circle of an arc of arc length s. The (x, y) coordinates of this point can be described as functions of the angle. Defining Sine and Cosine Functions Now that we have our unit circle labeled, we can learn how the (x, y) coordinates relate to the arc length and angle. The sine function relates a real number t to the y-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle t equals the y-value of the endpoint on the unit circle of an arc of length t. In Figure 2, the sine is equal to y. Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y-coordinate of the corresponding point on the unit circle. The cosine function of an angle t equals the x-value of the endpoint on the unit circle of an arc of length t. In Figure 3, the cosine is equal to x. y (x, y) = (cos t, sin t) 1 sin t t t cos t Figure 3 x Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: sin t is the same as sin(t) and cos t is the same as cos(t). Likewise, cos2 t is a commonly used shorthand notation for (cos(t))2. Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer. sine and cosine functions If t is a real number and a point (x, y) on the unit circle corresponds to an angle of t, then cos t = x sin t = y How To⦠Given a point P (x, y) on the unit circle corresponding to an angle of t, find the sine and cosine. 1. The sine of t is equal to the y-coordinate of point P: sin t = y. 2. The cosine of t is equal to the x-coordinate of point P: cos t = x. Example 1 Finding |
Function Values for Sine and Cosine Point P is a point on the unit circle corresponding to an angle of t, as shown in Figure 4. Find cos(t) and sin(t). SECTION 5.2 unit circle: sine and cosine Functions 459 y 1 t cos t sin t x = (cos t, sin t) Figure 4 Solution We know that cos t is the x-coordinate of the corresponding point on the unit circle and sin t is the y-coordinate of the corresponding point on the unit circle. So: 1 __ x = cos t = 2 3 β ___ 2 y = sin t = β Try It #1 A certain angle t corresponds to a point on the unit circle at ξ’ β β β 2 2 β β ξͺ as shown in Figure 5. Find cos t and sin t Figure 5 Finding Sines and Cosines of Angles on an Axis For quadrantral angles, the corresponding point on the unit circle falls on the x- or y-axis. In that case, we can easily calculate cosine and sine from the values of x and y. Example 2 Calculating Sines and Cosines along an Axis Find cos(90Β°) and sin(90Β°). Solution Moving 90Β° counterclockwise around the unit circle from the positive x-axis brings us to the top of the circle, where the (x, y) coordinates are (0, 1), as shown in Figure 6. Using our definitions of cosine and sine, x = cos t = cos(90Β°) = 0 y = sin t = sin(90Β°) = 1 The cosine of 90Β° is 0; the sine of 90Β° is 1. y (0, 1) t Figure 6 x 46 0 CHAPTER 5 trigonometric Functions Try It #2 Find cosine and sine of the angle Ο. The Pythagorean Identity Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is x2 + y2 = 1. Because x = cos t and y = sin t, we can substitute for x and y to get cos2 t + sin2 t = 1. This equation, cos2 t + sin2 t = 1, is known as the Pythagorean Identity. See Figure 7. 1 = x 2 + y 2 = cos2 t + sin2 t 1 t |
x y Figure 7 We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution. Pythagorean Identity The Pythagorean Identity states that, for any real number t, cos2 t + sin2 t = 1 How Toβ¦ Given the sine of some angle t and its quadrant location, find the cosine of t. 1. Substitute the known value of sin(t) into the Pythagorean Identity. 2. Solve for cos(t). 3. Choose the solution with the appropriate sign for the x-values in the quadrant where t is located. Example 3 Finding a Cosine from a Sine or a Sine from a Cosine 3 _ If sin(t) = and t is in the second quadrant, find cos(t). 7 Solution If we drop a vertical line from the point on the unit circle corresponding to t, we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See Figure 8. Substituting the known value for sine into the Pythagorean Identity, cos2(t) + sin2(t) = 1 cos2(t) + 9 __ 49 cos2(t) = 40 __ 49 = 1 cos(t) = Β± β ___ 40 __ 49 = Β± = Β± β 40 β ____ 7 β 10 2 β _ 7 y 3 7 1 x, x x Figure 8 Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative. So cos(t) = β β 10 2 β _ 7 SECTION 5.2 unit circle: sine and cosine Functions 461 Try It #3 If cos(t) = 24 ___ 25 and t is in the fourth quadrant, find sin(t). Finding Sines and Cosines of Special Angles We have already learned some properties of the special angles, such as the conversion from radians to degrees. We can also calculate sines and cosines of the special angles using the Pythagorean Identity and our knowledge of triangles. Finding Sines and Cosines of 45Β° Angles |
Ο _, as shown in Figure 9. A 45Β° β 45Β° β 90Β° triangle is an isosceles triangle, so the First, we will look at angles of 45Β° or 4 x- and y-coordinates of the corresponding point on the circle are the same. Because the x- and y-values are the same, the sine and cosine values will also be equal. (x, y) = (x, x) 1 45Β° x y Figure 9 Ο __ At t =, which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies 4 along the line y = x. A unit circle has a radius equal to 1. So, the right triangle formed below the line y = x has sides x and y (y = x), and a radius = 1. See Figure 10. y (0, 11, 0) (β1, 0) From the Pythagorean Theorem we get Substituting y = x, we get Combining like terms we get And solving for x, we get (0, β1) Figure 10 x2 + y2 = 1 x2 + x2 = 1 2x2 = 1 x2 = 1 __ 2 x = Β± 1 _ 2 β β In quadrant I, x = 1 _. 2 β β 46 2 CHAPTER 5 trigonometric Functions Ο _ At t = or 45 degreesx, y) = (x, x β β, sin t = 1 cos t = 1 _ _ 2 2 β β β β β β If we then rationalize the denominators, we get β β β β β = cos t = 1 _ Β· 2 β 2 β ____ 2 sin t = 1 _ Β· 2 β 2 β ____ 2 = β β Therefore, the (x, y) coordinates of a point on a circle of radius 1 at an angle of 45Β° are ξ’ Finding Sines and Cosines of 30Β° and 60Β° Angles β β Next, we will find the cosine and sine at an angle of 30Β°, or. First, we will draw a triangle inside a circle with one 6 side at an angle of 30Β°, and another at an angle of β30Β°, as shown in Figure 11. If the resulting two right triangles are combined into one large triangle, notice that all three angles of this larger |
triangle will be 60Β°, as shown in Figure 12. 30Β° 30Β° r r 60Β° y y 60Β° Figure 12 (x, y) r 30Β° Figure 11 Because all the angles are equal, the sides are also equal. The vertical line has length 2y, and since the sides are all equal, we can also conclude that r = 2y or y = 1 _ r. Since sin t = y, 2 ξͺ = 1 __ sin ξ’ Ο __ r 2 6 And since r = 1 in our unit circle, sin ξ’ Ο __ 6 (1) ξͺ = 1 __ 2 = 1 __ 2 SECTION 5.2 unit circle: sine and cosine Functions 463 Using the Pythagorean Identity, we can find the cosine value. cos2 Ο __ 6 cos2 ξ’ Ο __ 6 2 + sin2 ξ’ Ο __ ξͺ = 1 6 ξͺ + ξ’ 1 __ ξͺ 2 ξͺ = 3 __ cos2 ξ’ Ο __ 4 6 = 1 Use the square root property. cos ξ’ Ο __ ξͺ = Since y is positive, choose the positive root. The (x, y) coordinates for the point on a circle of radius 1 at an angle of 30Β° are ξ’ β 3 β Ο 1 ξͺ. At t = _ _ _ (60Β°), the radius of, 3 2 2 the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle, BAD, as shown in Figure 13. Angle A has measure 60Β°. At point B, we draw an angle ABC with measure of 60Β°. We know the angles in a triangle sum to 180Β°, so the measure of angle C is also 60Β°. Now we have an equilateral triangle. Because each side of the equilateral triangle ABC is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1. y (0, 1) B 30Β°, Ο 6 1 (β1, 01, 0) y x 60Β°, Ο 3 (0, β1) Figure 13 The measure of angle ABD is 30Β°. So, if double, angle ABC is 60Β°. BD is the perpendicular bisector of AC, so it cuts 1 1 _ _ AC in half. This means that AD is |
. Notice that AD is the x-coordinate of point B, which is at the the radius, or 2 2 intersection of the 60Β° angle and the unit circle. This gives us a triangle BAD with hypotenuse of 1 and side x of 1 _ length. 2 From the Pythagorean Theorem, we get Substituting x = 1 _, we get 2 Solving for y, we get __ ξͺ 2 1 __ + __ 4 y 2 = 3 __ 4 β 3 y = Β± β ____ 2 Ο _ Since t = has the terminal side in quadrant I where the y-coordinate is positive, we choose y = 3 value. β 3 β _ 2, the positive 46 4 CHAPTER 5 trigonometric Functions (60Β°), the (x, y) coordinates for the point on a circle of radius 1 at an angle of 60Β° are ξ’ Ο 1 _ _ At t =, 2 3 find the sine and cosine. (x, y) = ξ’ 1 __, 2 x = 1 __, y = 2 cos t = 1 __, sin β ξͺ, so we can _ 2 We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. Table 1 summarizes these values. Angle Cosine Sine 0 1 0 Ο __, or 30Β° 6 Ο __, or 45Β° 4 Ο __, or 60Β° 3 Ο __, or 90Β° 2 β 3 β _ 2 1 __ 2 Table __ 2 β 3 β _ 2 0 1 Figure 14 shows the common angles in the first quadrant of the unit circle. 90Β°, Ο 2 (0, 1) 1 2 60Β°, 45Β°, 30Β°, 0Β° = 360Β° (1, 0) 1 2 Figure 14 Using a Calculator to Find Sine and Cosine To find the cosine and sine of angles other than the special angles, we turn to a computer or calculator. Be aware: Most calculators can be set into βdegreeβ or βradianβ mode, which tells the calculator the units for the input value. When we evaluate cos(30) on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode. How To |
β¦ Given an angle in radians, use a graphing calculator to find the cosine. 1. If the calculator has degree mode and radian mode, set it to radian mode. 2. Press the COS key. 3. Enter the radian value of the angle and press the close-parentheses key β)β. 4. Press ENTER. SECTION 5.2 unit circle: sine and cosine Functions 465 Using a Graphing Calculator to Find Sine and Cosine Example 4 5Ο Evaluate cos ξ’ ξͺ using a graphing calculator or computer. _ 3 Solution Enter the following keystrokes: COS(5 Γ Ο Γ· 3) ENTER cos ξ’ 5Ο ___ ξͺ = 0.5 3 Analysis We can find the cosine or sine of an angle in degrees directly on a calculator with degree mode. For calculators or software that use only radian mode, we can find the sign of 20Β°, for example, by including the conversion factor to radians as part of the input: SIN(20 Γ Ο Γ· 180) ENTER Try It #4 Ο Evaluate sin ξ’ _ ξͺ. 3 Identifying the Domain and Range of Sine and Cosine Functions Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than 0 and angles larger than 2Ο can still be graphed on the unit circle and have real values of x, y, and r, there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number. What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in Figure 15. The bounds of the x-coordinate are [β1, 1]. The bounds of the y-coordinate are also [β1, 1]. Therefore, the range of both the sine and cosine functions is [β1, 1]. y (0, 1) (β1, 0) x (1 |
, 0) (0, β1) Figure 15 Finding Reference Angles We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y-coordinate on the unit circle, the other angle with the same sine will share the same y-value, but have the opposite x-value. Therefore, its cosine value will be the opposite of the first angleβs cosine value. Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the same cosine will share the same x-value but will have the opposite y-value. Therefore, its sine value will be the opposite of the original angleβs sine value. As shown in Figure 16, angle Ξ± has the same sine value as angle t; the cosine values are opposites. Angle Ξ² has the same cosine value as angle t; the sine values are opposites. sin(t) = sin(Ξ±) and cos(t) = βcos(Ξ±) sin(t) = βsin(Ξ²) and cos(t) = cos(Ξ²) 46 6 CHAPTER 5 trigonometric Functions II r Ξ± r (x, y) I t II Ξ² (x, y) I t r r III IV III IV Figure 16 Recall that an angleβs reference angle is the acute angle, t, formed by the terminal side of the angle t and the horizontal Ο _ axis. A reference angle is always an angle between 0 and 90Β°, or 0 and radians. As we can see from Figure 17, for 2 any angle in quadrants II, III, or IV, there is a reference angle in quadrant I. Quadrant I y Quadrant II y II I t, t' x II t' I t III IV III IV t' = t Quadrant III y t' = Ο β t = 180Β° β t Quadrant IV y t II t' III I II I t x IV III t' IV t' = t β Ο = t β 180Β° t' = 2Ο β t = 360Β° β t Figure 17 x x How Toβ¦ Given an angle between 0 and 2Ο, find its reference angle |
. 1. An angle in the first quadrant is its own reference angle. 2. For an angle in the second or third quadrant, the reference angle is β£Ο β tβ£ or β£180Β° β tβ£. 3. For an angle in the fourth quadrant, the reference angle is 2Ο β t or 360Β° β t. 4. If an angle is less than 0 or greater than 2Ο, add or subtract 2Ο as many times as needed to find an equivalent angle between 0 and 2Ο. SECTION 5.2 unit circle: sine and cosine Functions 467 Example 5 Finding a Reference Angle Find the reference angle of 225Β° as shown in Figure 18. y 225Β° II III I IV x Solution Because 225Β° is in the third quadrant, the reference angle is Figure 18 β£(180Β° β 225Β°)β£ = β£β45Β°β£ = 45Β° Try It #5 Find the reference angle of 5Ο _. 3 Using Reference Angles Now letβs take a moment to reconsider the Ferris wheel introduced at the beginning of this section. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. What is the riderβs new elevation? To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle. Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find (x, y) coordinates for those angles. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies. Using Reference Angles to Evaluate Trigonometric Functions We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x-values in that quadrant. The sine will be positive or negative depending on the sign of the y-values in that quadrant. using reference angles to find cosine and sine Angles have cosines and sines with the same absolute value as cos |
ines and sines of their reference angles. The sign (positive or negative) can be determined from the quadrant of the angle. How Toβ¦ Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle. 1. Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle. 2. Determine the values of the cosine and sine of the reference angle. 3. Give the cosine the same sign as the x-values in the quadrant of the original angle. 4. Give the sine the same sign as the y-values in the quadrant of the original angle. 46 8 CHAPTER 5 trigonometric Functions Example 6 Using Reference Angles to Find Sine and Cosine a. Using a reference angle, find the exact value of cos(150Β°) and sin(150Β°). b. Using the reference angle, find cos and sin 5Ο _ 4 5Ο _. 4 Solution a. 150Β° is located in the second quadrant. The angle it makes with the x-axis is 180Β° β 150Β° = 30Β°, so the reference angle is 30Β°. This tells us that 150Β° has the same sine and cosine values as 30Β°, except for the sign. We know that 3 β and sin(30Β°) = 1 __ ___. 2 2 cos(30Β°) = β Since 150Β° is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. The y-coordinate is positive, so the sine value is positive. 3 and sin(150Β°) = 1 β __ ___ 2 2 β Ο = Ο is in the third quadrant. Its reference angle is 5Ο b. 5Ο. The cosine and sine of Ο __ _ _ _ are both 4 4 4 4 quadrant, both x and y are negative, so: cos(150Β°) = β β 2 β _ 2 β. In the third cos = β and sin = β 5Ο ___ 4 β 2 β ____ 2 5Ο ___ 4 β 2 β ___ 2 Try It #6 a. Use the reference angle of 315Β° to find cos(315Β°) and sin(315Β°). b. Use the reference angle of β Ο ξͺ and sin ξ’ β Ο to |
find cos ξ’ β Ο _ _ _ ξͺ. 6 6 6 Using Reference Angles to Find Coordinates Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle. They are shown in Figure 19. Take time to learn the (x, y) coordinates of all of the major angles in the first quadrant. 120Β°, 2Ο 3 135Β°, 3Ο 4 150Β°, 5Ο 6 180Β°, Ο (β1, 0) 210Β°, 7Ο 6 225Β°, 5Ο 4 240Β°, 4Ο 3 90Β°, Ο 2 (0, 1) 60Β°, Ο 3 45Β°, Ο 4 30Β°, Ο 6 0Β°, 0 (1, 0) 360Β°, 2Ο (1, 0) 330Β°, 11Ο 6 315Β°, 7Ο 4 300Β°, 5Ο 3 270Β°, 3Ο 2 Figure 19 Special angles and coordinates of corresponding points on the unit circle (0, β1) In addition to learning the values for special angles, we can use reference angles to find (x, y) coordinates of any point on the unit circle, using what we know of reference angles along with the identities x = cos t y = sin t SECTION 5.2 unit circle: sine and cosine Functions 469 First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y- and x-values of the quadrant. How Toβ¦ Given the angle of a point on a circle and the radius of the circle, find the (x, y) coordinates of the point. 1. Find the reference angle by measuring the smallest angle to the x-axis. 2. Find the cosine and sine of the reference angle. 3. Determine the appropriate signs for x and y in the given quadrant. Example 7 Using the Unit Circle to Find Coordinates Find the coordinates of the point on the unit circle at an angle of Solution We know that the angle is in the third quadrant. 7Ο _ 6 7Ο _. 6 First, letβs find the reference angle by measuring the angle to the x-axis. To find the reference angle of an angle whose |
terminal side is in quadrant III, we find the difference of the angle and Ο. Next, we will find the cosine and sine of the reference angle: 7Ο ___ 6 Ο __ β Ο = 6 β Ο __ ξͺ = cos ξ’ 6 3 β ____ 2 ξͺ = 1 Ο __ __ and sin ξ’ 2 6 We must determine the appropriate signs for x and y in the given quadrant. Because our original angle is in the third quadrant, where both x and y are negative, both cosine and sine are negative. cos ξ’ sin ξ’ β 3 ξͺ = β β 7Ο ___ ____ 6 2 ξͺ = β 1 7Ο ___ __ 6 2 Now we can calculate the (x, y) coordinates using the identities x = cos ΞΈ and y = sin ΞΈ. 3 β The coordinates of the point are ξ’ β, β 1 ξͺ on the unit circle. _ _ 2 2 β Try It #7 Find the coordinates of the point on the unit circle at an angle of 5Ο _. 3 Access these online resources for additional instruction and practice with sine and cosine functions. β’ Trigonometric Functions Using the Unit Circle (http://openstaxcollege.org/l/trigunitcir) β’ Sine and Cosine from the Unit Circle (http://openstaxcollege.org/l/sincosuc) β’ Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Six (http://openstaxcollege.org/l/sincosmult) β’ Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Four (http://openstaxcollege.org/l/sincosmult4) β’ Trigonometric Functions Using Reference Angles (http://openstaxcollege.org/l/trigrefang) 47 0 CHAPTER 5 trigonometric Functions 5.2 SeCTIOn exeRCISeS VeRBAl 1. Describe the unit circle. 2. What do the x- and y-coordinates of the points on 3. Discuss the difference between a coterminal angle and a reference angle. 5. Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle. the unit circle represent? 4 |
. Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle. AlGeBRAIC For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by t lies. 6. sin(t) < 0 and cos(t) < 0 8. sin(t) > 0 and cos(t) < 0 7. sin(t) > 0 and cos(t) > 0 9. sin(t) < 0 and cos(t) > 0 For the following exercises, find the exact value of each trigonometric function. Ο __ 11. sin 3 Ο __ 15. cos 4 Ο __ 12. cos 2 Ο __ 16. sin 6 19. cos Ο 20. cos 0 Ο __ 10. sin 2 Ο __ 14. sin 4 3Ο ___ 2 22. sin 0 18. sin nUMeRIC For the following exercises, state the reference angle for the given angle. 24. β170Β° 23. 240Β° 25. 100Β° 27. 135Β° 31. β11Ο _ 3 28. 32. 5Ο ___ 4 β7Ο _ 4 29. 33. 2Ο ___ 3 βΟ ___ 8 Ο __ 13. cos 3 17. sin Ο Ο __ 21. cos 6 26. β315Β° 30. 5Ο ___ 6 For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places. 34. 225Β° 38. 210Β° 35. 300Β° 39. 120Β° 36. 320Β° 40. 250Β° 37. 135Β° 41. 150Β° 42. 46. 5Ο ___ 4 4Ο ___ 3 43. 47. 7Ο ___ 6 2Ο ___ 3 44. 48. 5Ο ___ 3 5Ο ___ 6 45. 49. 3Ο ___ 4 7Ο ___ 4 For the following exercises, find the requested value. 50. If cos(t) = 1 _ and t is in the 4th quadrant, find sin(t). 7 52. If sin(t) = 3 _ and t is in the 2nd quadrant, find cos(t). 8 51. If cos(t) = 2 _ and t is in the 1 |
st quadrant, find sin(t). 9 53. If sin(t) = β 1 _ and t is in the 3rd quadrant, find cos(t). 4 54. Find the coordinates of the point on a circle with radius 15 corresponding to an angle of 220Β°. 55. Find the coordinates of the point on a circle with radius 20 corresponding to an angle of 120Β°. SECTION 5.2 section exercises 471 56. Find the coordinates of the point on a circle with 57. Find the coordinates of the point on a circle with radius 8 corresponding to an angle of 7Ο _. 4 radius 16 corresponding to an angle of 5Ο _ 9. 58. State the domain of the sine and cosine functions. 59. State the range of the sine and cosine functions. GRAPHICAl For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of t. 60. 62. 61. t 63. 64. t 65. t t 66. 69. ( 1, 0) β 72. t t t t t t (1, 0) t 0.803 β0.596 (0.803, β0.596) 67. 68. t 70. (0.111, 0.994) 71. 0.994 t 0.111 73. 74. t t 47 2 75. CHAPTER 5 trigonometric Functions 76. 77. (β 0.649, 0.761) 0.761 t β 0.649 t 78. 79. β0.948 t (β0.948, β0.317) β0.317 t (0, β1) (0, 1) t TeCHnOlOGY For the following exercises, use a graphing calculator to evaluate. 80. sin 84. sin 5Ο ___ 9 3Ο ___ 4 81. cos 85. cos 5Ο ___ 9 3Ο ___ 4 82. sin Ο __ 10 86. sin 98Β° 83. cos Ο __ 10 87. cos 98Β° 88. cos 310Β° 89. sin 310Β° exTenSIOnS 11Ο ___ 90. sin ξ’ 3 ξͺ cos ξ’ β5Ο ____ ξͺ 6 91. sin ξ’ 3Ο ___ ξͺ cos ξ’ 4 5Ο ___ ξͺ 3 Ο 92. sin ξ’ β 4Ο __ ___ ξͺ |
cos ξ’ ξͺ 3 2 93. sin ξ’ β9Ο ____ 4 ξͺ cos ξ’ βΟ ___ ξͺ 6 Ο __ ξͺ cos ξ’ 94. sin ξ’ 6 βΟ ___ ξͺ 3 95. sin ξ’ 7Ο ___ ξͺ cos ξ’ 4 β2Ο ____ ξͺ 3 96. cos ξ’ 5Ο ___ ξͺ cos ξ’ 6 2Ο ___ ξͺ 3 97. cos ξ’ βΟ ___ 3 Ο __ ξͺ ξͺ cos ξ’ 4 98. sin ξ’ β5Ο ____ 4 ξͺ sin ξ’ 11Ο ___ ξͺ 6 Ο __ ξͺ 99. sin(Ο)sin ξ’ 6 ReAl-WORlD APPlICATIOnS For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point (0, 1), that is, on the due north position. Assume the carousel revolves counter clockwise. 100. What are the coordinates of the child after 101. What are the coordinates of the child after 45 seconds? 90 seconds? 102. What is the coordinates of the child after 125 seconds? 104. When will the child have coordinates (β0.866, β0.5) if the ride last 6 minutes? 103. When will the child have coordinates (0.707, β0.707) if the ride lasts 6 minutes? (There are multiple answers.) SECTION 5.3 the other trigonometric Functions 473 leARnInG OBjeCTIVeS In this section, you will: β’ β’ β’ β’ β’ Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of Ο _, and Ο _. 6 3 Use reference angles to evaluate the trigonometric functions secant, cosecant, tangent, and cotangent., Ο _ 4 Use properties of even and odd trigonometric functions. Recognize and use fundamental identities. Evaluate trigonometric functions with a calculator. 5.3 THe OTHeR TRIGOnOMeTRIC FUnCTIOnS A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with |
the or less, regardless of its length. A tangent represents a ratio, so this means that for every ground whose tangent is 1 _ 12 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions. Finding exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent To define the remaining functions, we will once again draw a unit circle with a point (x, y) corresponding to an angle of t, as shown in Figure 1. As with the sine and cosine, we can use the (x, y) coordinates to find the other functions. y (x, y) 1 t x Figure 1 The first function we will define is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the y _ x, x β 0. Because the y-value corresponding point on the unit circle. In Figure 1, the tangent of angle t is equal to sin t _ is equal to the sine of t, and the x-value is equal to the cosine of t, the tangent of angle t can also be defined as, cos t cos t β 0. The tangent function is abbreviated as tan. The remaining three functions can all be expressed as reciprocals of functions we have already defined. β’ The secant function is the reciprocal of the cosine function. In Figure 1, the secant of angle t is equal to 1 _ cos t 1 _ x, x β 0. The secant function is abbreviated as sec. = β’ The cotangent function is the reciprocal of the tangent function. In Figure 1, the cotangent of angle t is equal to cos t _ sin t x _ y, y β 0. The cotangent function is abbreviated as cot. = β’ The cosecant function is the reciprocal of the sine function. In Figure 1, the cosecant of angle t is equal to 1 _ sin t 1 _ y, y |
β 0. The cosecant function is abbre viated as csc. = tangent, secant, cosecant, and cotangent functions If t is a real number and (x, y) is a point where the terminal side of an angle of t radians intercepts the unit circle, then y _ tan t = x, x β 0 csc t = 1 __, y β 0 y, x β 0 sec t = 1 __ x cot t = x __, y β 0 y 47 4 CHAPTER 5 trigonometric Functions Example 1 Finding Trigonometric Functions from a Point on the Unit Circle β 3 β 1 ξͺ is on the unit circle, as shown in Figure 2. Find sin t, cos t, tan t, sec t, csc t, and cot t. The point ξ’ β _ _, 2 2 1 t β1 1 Figure 2 Solution Because we know the (x, y) coordinates of the point on the unit circle indicated by angle t, we can use those coordinates to find the six functions: 1 _ sin t = y = 2 β 3 cos = tan = sec = csc = cot Try It #1 The point ξ’ β 2 β _ 2 β, β 2 β ξͺ is on the unit circle, as shown in Figure 3. Find sin t, cos t, tan t, sec t, csc t, and cot t. _ 2 1 β1 t 1 β1 Figure 3 SECTION 5.3 the other trigonometric Functions 475 Example 2 Finding the Trigonometric Functions of an Angle Ο _ Find sin t, cos t, tan t, sec t, csc t, and cot t when t =. 6 Ο = 1 __ __ Solution We have previously used the properties of equilateral triangles to demonstrate that sin 2 6 β and Ο _ = cos 6 3 β _. 2 We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values. Ο __ sin 6 _ Ο __ cos 6 Ο _ = tan __ = sec 6 Ο _ = csc 6 Ο _ = cot 6 = 1 __ cos __ sin 6 Ο _ cos 6 _ Ο _ sin Try It #2 Ο _ Find sin t |
, cos t, tan t, sec t, csc t, and cot t when t =. 3 Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x equal to the cosine and y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in Table 1. Angle Cosine Sine Tangent Secant 0 1 0 0 1 Ο __, or 30Β° 6 β 3 β ___ 2 1 __ 2 3 β ___ 3 3 2 β ____ 3 β β Cosecant Undefined 2 Cotangent Undefined β 3 β Table 1 Ο __, or 45Β° 4 β 2 β ___ 2 β 2 β ___ __, or 60Β° 3 Ο __, or 90Β° 2 1 __ 2 3 β ___ 2 β β 3 β 0 1 Undefined 2 Undefined β 3 2 β ____ 3 β 3 β ___ 3 1 0 47 6 CHAPTER 5 trigonometric Functions Using Reference Angles to evaluate Tangent, Secant, Cosecant, and Cotangent We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant. Figure 4 shows which functions are positive in which quadrant. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase βA Smart Trig Class.β Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is βA,β all of the six trigonometric functions are positive. In quadrant II, βSmart,β only sine and its reciprocal function, cosecant, are positive. In quadrant III, βTrig,β only tangent and its reciprocal function, c |
otangent, are positive. Finally, in quadrant IV, βClass,β only cosine and its reciprocal function, secant, are positive. y II sin t csc t I sin t cos t tan t sec t csc t cot t x III tan t cot t IV cos t sec t Figure 4 How Toβ¦ Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions. 1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle. 2. Evaluate the function at the reference angle. 3. Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative. Using Reference Angles to Find Trigonometric Functions Example 3 Use reference angles to find all six trigonometric functions of β 5Ο _. 6 Ο _ Solution The angle between this angleβs terminal side and the x-axis is 6 β 5Ο _ is in the third quadrant, where both x and y are negative, cosine, sine, secant, and cosecant will be negative, while 6, so that is the reference angle. Since tangent and cotangent will be positive. cos ξ’ β 5Ο ___ ξͺ = β 6 3 β ____, sin ξ’ β 2 1 5Ο ___ __, tan ξ’ β ξͺ = β 6 2 5Ο ___ ξͺ = 6 β 3 β ____ 3 β sec ξ’ β 5Ο ___ ξͺ = β 6 3 2 β ____, csc ξ’ β 3 5Ο ___ ξͺ = β 2, cot ξ’ β 6 5Ο ___ ξͺ = β 6 β 3 β Try It #3 Use reference angles to find all six trigonometric functions of β 7Ο _. 4 SECTION 5.3 the other trigonometric Functions 477 Using even and Odd Trigonometric Functions To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard. Consider the function f (x) = x 2, shown in Figure 5. The graph of the function is symmetrical about the y-axis |
. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation: (4)2 = (β4)2, (β5)2 = (5)2, and so on. So f (x) = x 2 is an even function, a function such that two inputs that are opposites have the same output. That means f (βx) = f (x). y 30 25 20 15 10 5 β1 β5 β10 (2, 4) 4 321 5 6 x (β2, 4) β3 β4 β2 β6 β5 Figure 5 The function f (x) = x2 is an even function. Now consider the function f (x) = x3, shown in Figure 6. The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So f (x) = x3 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f (βx) = βf (x). y 10 8 6 4 2 β1 β2 β4 β6 β8 β10 β2 β5 β3 β4 (β1, β1) (1, 1) 321 4 5 x Figure 6 The function f (x) = x3 is an odd function. We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure 7. The sine of the positive angle is y. The sine of the negative angle is βy. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table 2. y Figure 7 (x, y) t βt (x, βy) x 47 8 CHAPTER 5 trigonometric Functions sin t = y sin(βt) = βy cos t = x cos(βt) = x y _ x tan(t) = y _ x tan(βt) = β sin t β sin(βt) cos t = cos(βt) tan t β tan(βt) 1 _ x sec t = 1 _ x sec(βt) = sec t = sec(βt) 1 _ y csc t = 1 _ βy csc |
(βt) = x _ y cot t = x _ βy cot(βt) = csc t β csc(βt) cot t β cot(βt) Table 2 even and odd trigonometric functions An even function is one in which f (βx) = f (x). An odd function is one in which f (βx) = βf (x). Cosine and secant are even: Sine, tangent, cosecant, and cotangent are odd: cos(βt) = cos t sec(βt) = sec t sin(βt) = βsin t tan(βt) = βtan t csc(βt) = βcsc t cot(βt) = βcot t Example 4 Using Even and Odd Properties of Trigonometric Functions If the secant of angle t is 2, what is the secant of βt? Solution Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is 2, the secant of βt is also 2. Try It #4 If the cotangent of angle t is β β 3, what is the cotangent of βt? Recognizing and Using Fundamental Identities We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine. fundamental identities We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships: tan t = sin t _ cos t csc t = 1 _ sin t sec t = 1 _ cos t cot t = 1 _ tan t = cos t _ sin t SECTION 5.3 the other trigonometric Functions 479 Example 5 Using Identities to Evaluate Trigonometric Functions a. Given sin (45Β°) = β β 2 β _, cos (45Β°) = 2, 5Ο 1, cos ξ’ |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.