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n 1e 32. eln 34.17 35. eln x2 30. ln 25 e 33. ln exy 36. eln 2x3 In Exercises 37–40, find the domain of the given function. 37. f 39. h x x 1 1 2 2 ln 1 log x 1 x 1 2 2 38. g x 1 2 40. k x 1 2 ln 1 log x 2 2 2 x 1 2 362 Chapter 5 Exponential and Logarithmic Functions 41. Compare the graphs of 2 log x. log x2 How are they alike? How are they and x x f 2 1 g different? 1 2 42. Compare the graphs of 3 log x. log x3 How are they alike? How are they and h x x 2 1 k different? 2 1 In Exercises 43–48, describe the transformation from g(x) ln x to the given function. Give the domain and range of the given function. 43. f 45. 47 ln x ln ln 1 1 x 4 x 3 2 2 4 44. 46. 48 ln x 7 ln ln 1 1 x 2 x 2 2 2 2 In Exercises 49–52, sketch the graph of the function. 49. 51. x f 1 2 h x 2 1 log x 3 1 2 log x 2 50. g 52. f x x 1 1 2 2 2 ln x 3 ln x 1 2 3 In Exercises 53–58, find a viewing window (or windows) that shows a complete graph of the function. 53. f x 1 2 55. h 57. f x x 1 1 2 2 x ln x ln x 2 x 10 log x x 54. g x 2 1 ln x x 56. k x 1 2 58. f x 1 2 2 ln x e log x x In Exercises 59–62, find the average rate of change of the function. (See Section 3.7.) 59. 60. 61. 62 ln x 2 2 1 x ln x,, as x goes from 3 to 5 as x goes from 0.5 to 1 log 1 x2 x 1 2, as x goes from 5 to 3 x log x 0 0, as x goes from 1 to 4 f 63. a. What is the average rate of change of 3 h? as x goes from 3 to b. What is the value of |
h when the average rate of 3 h, as x goes from 3 to ln x, ln x, x x f 1 2 change of is 0.25? 1 2 64. a. Find the average rate of change of f as x goes from 0.5 to 2. ln x2, x 1 2 b. Find the average rate of change of 2, as x goes from 3.5 to 5. x 3 ln g x 1 2 1 2 c. What is the relationship between your answers in parts a and b? Explain why this is so. 65. a. Use the doubling function D from Example 7 to find the time it takes to double your money at each of these interest rates: 4%, 6%, 8%, 12%, 18%, 24%, and 36%. b. Round the answers in part a to the nearest year.,,,,,, and 72 8 72 6 72 36 72 12 72 18 Use this and compare them with these numbers: 72 72 24 4 evidence to state a “rule of thumb” for determining approximate doubling time, without using the function D. This rule of thumb, which has long been used by bankers, is called the Rule of 72. 66. The height h above sea level (in meters) is related to air temperature t (in degrees Celsius), the atmospheric pressure p (in centimeters of mercury at height h), and the atmospheric pressure c at sea level by: h 1 30t 8000 c pb a ln 2 If the pressure at the top of Mount Rainier is 44 centimeters on a day when sea level pressure is 75.126 centimeters and the temperature is what is the height of Mount Rainier? 7°, 67. A class is tested at the end of the semester and weekly thereafter on the same material. The average score on the exam taken after t weeks is given by the following “forgetting function”. g t 2 1 77 10 ln t 1 1 2 a. What was the average score on the original exam? b. What was the average score after 2 weeks? after 5 weeks? 68. Students in a precalculus class were given a final exam. Each month thereafter, they took an equivalent exam. The class average on the exam taken after t months is given by the following function. F t 2 1 82 8 ln t 1 2 1 a. What was the average score on the original exam? b. What was the average score after 6 months? after 10 months? |
69. One person with a flu virus visited the campus. The number T of days it took for the virus to infect x people is given by T. T 0.93 ln 7000 x 6999x b a Section 5.5 Properties and Laws of Logarithms 363 a. How many days did it take for 6000 people to become infected? b. After 2 weeks, how many people were infected? 70. Critical Thinking For each positive integer n, let fn be the polynomial function whose rule is x fn1 2 x x 2 2 x 3 3 x4 4 x 5 5 p – x n n if n is odd if n is even. In the viewing window with where the sign of the last term is and 1 x 1 and 1 x g 2 For what values of x does approximation of g? 4 y 1, f4 1 and ln x x 1 2 2 1 graph on the same screen. f4 appear to be a good 71. Critical Thinking Using the viewing window in Exercise 70, find a value of n for which the graph of the function (as defined in Exercise 70) fn x ln 1 x appears to coincide with the graph of g from graph to graph to see how good this approximation actually is. Use the trace feature to move. 1 2 2 1 72. A bicycle store finds that N the number of bikes sold, is related to d, the number of dollars spent on advertising. N 51 100 ln d 100 a 2 b a. How many bikes will be sold if nothing is is spent? if $1000 spent on advertising? if $10,000 is spent? b. If the average profit is worthwhile to spend $10,000? What about $25 $1000 per bike, is it on advertising? c. What are the answers in part b if the average profit per bike is $35? 5.5 Properties and Laws of Logarithms Objectives • Use properties and laws of logarithms to simplify and evaluate expressions The definitions of common and natural logarithms differ only in their bases. Therefore, common and natural logarithms share the same basic properties and laws. Basic Properties of Logarithms Logarithms are only defined for positive real numbers. That is, NOTE Any number raised to the zero power, except zero, is 1. x0 1, where x 0 log v and ln v are defined only when v 77 0. y log x y ln x and both contain the point |
(1, 0) because The graphs of 100 1 and e0 1. The values of log nential statements. 104 log 1 0 and ln 1 0 and ln e9 can be found by writing equivalent expo- In general, If log 104 x, then 10x 104. So x 4. If ln e9 x, then ex e9. So x 9. log 10k k, for every real number k. ln ek k, for every real number k. 364 Chapter 5 Exponential and Logarithmic Functions By definition, log 678 is the exponent to which 10 must be raised to produce 678. 10log 678 678 Similarly, ln 54 is the exponent to which e must be raised to produce 54. eln 54 54 In general, 10logv v and eln v v, for every v 77 0. The facts presented above are summarized in the table below. Common logarithms Natural logarithms 1. log v is defined only when and 2. 3. log log 10 1 for every real log 1 0 10k k v 77 0 1. ln v is defined only when ln e 1 and 2. for every real 3. ln v 77 0 number k 10log v v 4. for every v 77 0 4. for every v 77 0 ln 1 0 ek k number k eln v v Properties 3 and 4 are restatements of the fact that the composition of inverse functions produces the identity function. Basic Properties of Logarithms That is, if f x 1 2 10x f g g f 21 21 1 1 g and log x 10x 2 log x, then 10log x x for all x 7 0 2 log 10x x for all x ex and x x g f ln x. Analogous statements are true for 1 The properties of logarithms can be used to simplify expressions and solve k 2x2 7x 9 equations. For example, applying Property 3 with allows you to rewrite the expression 2x2 7x 9. ln e2x 27x9 as 1 2 2 Example 1 Solving Equations by Using Properties of Logarithms Use the basic properties of logarithms to solve the equation ln x 1 1 2 2. 2 3 2 Solution Because f x 1 2 8 ex x1 is a function, if 2 e2 e ln x 1 e2 1 x e2 1 x 6.3891 |
ln 1 x 1 2 2, then eln 1 x1 2 e2. Apply Property 4 with v x 1 Figure 5.5-1 The intersection of the graphs of Figure 5.5-1, confirms the solution. Y1 ln 1 x 1 2 and Y2 2, shown in ■ Section 5.5 Properties and Laws of Logarithms 365 Laws of Logarithms bm The Product Law of Exponents states that rithms are exponents, the following law holds. ˛bn bmn. Because loga- Product Law of Logarithms For all v, w 77 0, log (vw) log v log w ln (vw) ln v ln w. Proof According to Property 4 of logarithms, 10log w w. Then, by the Product Law of Exponents: vw 10log v 10log w 10log vlog w 10log v v and Again by Property 4 of logarithms: 10log vw vw Therefore, one-to-one, natural logarithms. 10log vw 10log vlog w log vw log v log w ; and because exponential functions are. A similar argument can be made for Example 2 Using the Product Law of Logarithms Use the Product Law of Logarithms to evaluate each logarithm. log 3 0.4771 ln 7 1.9459 and log 11 1.0414, ln 9 2.1972, and find ln 63. find log 33. a. Given that b. Given that Solution a. b. log 33 log ln 63 ln 1 7 9 1 2 2 3 11 log 3 log 11 0.4771 1.0414 1.5185 ln 7 ln 9 1.9459 2.1972 4.1431 ■ CAUTION Graphing Exploration A common error in applying the Product Law of Logarithms is to write the false statement ln 7 ln 9 7 9 ln 16 ln 1 2 instead of the correct statement ln 7 ln 9 7 9 ln 63. ln 1 2 10 x 10 Using the viewing window with graph both functions below on the same screen. and 8 y 8, f x 1 2 ln x ln 9 g x 1 2 ln 1 x 9 2 Explain how the graph illustrates the caution in the margin. The Quotient Law of Exponents states that b |
m bn bmn. When the expo- nents are logarithms, the Quotient Law is still valid. 366 Chapter 5 Exponential and Logarithmic Functions Quotient Law of Logarithms For all v, w 77 0, log a ln a v wb v wb log v log w ln v ln w. The proof of the Quotient Law of Logarithms is similar to the proof of the Product Law of Logarithms. Example 3 Using the Quotient Law of Logarithms Use the Quotient Law of Logarithms to evaluate each logarithm. Given that log 28 1.4472 and log 7 0.8451, find log 4. Given that ln 18 2.8904 and ln 6 1.7918, find ln 3. b. a. Solution a. log 4 log 28 7 b a log 28 log 7 1.4472 0.8451 0.6021 b. ln 3 ln 18 6 b a ln 18 ln 6 2.8904 1.7918 1.0986 CAUTION Do not confuse ln 7 9b a 0.2513 with the quotient ln 7 0.8856. ln 9 They are different numbers. Graphing Exploration Using the viewing window with both functions below on the same screen. 0 x 8 and 4 y 2, graph f x 2 1 ln x 9b a g ln x ln 9 x 2 1 Explain how the graph illustrates the caution in the margin. ■ The Power Law of Exponents, which states that translated into a logarithmic statement. 1 bm 2 k bmk, can also be Power Law of Logarithms For all k and v 77 0, logvk klog v, ln vk k ln v. Section 5.5 Properties and Laws of Logarithms 367 Proof According to Property 4 of logarithms, Law of Exponents: 10log v v. Then, by the Power v k 10log v 1 2 k 10k log v Again by Property 4 of logarithms: 10log v k v k 10log v k 10k log v So, and therefore, can be made for natural logarithms. log v k k log v. A similar argument Example 4 Using the Power Law of Logarithms Use the Power Law |
of Logarithms to evaluate each logarithm. a. b. Given that log 6 0.7782, find log 26. Given that ln 50 3.9120, find ln 23 50. Solution a. log26 log 6 b. ln23 50 ln 50 1 2 1 2 3 1 3 1 log 6 1 2 1 ln 50 1 3 1 0.7782 3.9120 2 2 0.3891 1.3040 The laws of logarithms can be used to simplify various expressions. Example 5 Simplifying Expressions Write ln 3x 4 ln x ln 3xy as a single logarithm. Solution ln 3x 4 ln x ln 3xy ln 3x ln x4 ln 3xy ln 3xy 2 ln 1 ln ln 3x x4 3x5 3xyb x4 y b a a Power Law Product Law Quotient Law Example 6 Simplifying Expressions Simplify ln 2x x b a ln A 24 ex2. B ■ ■ 368 Chapter 5 Exponential and Logarithmic Functions Solution 2x x b ln a ln A 24 ex2 B 1 2 1 4 2 1 4 ex2 ln ln 1 ln 2 ex2 1 ex2 x x b a 1 ln ln x 2 2 1 ln x 1 1 2 4 ln x 1 1 4 1 2 ln x 1 1 4 1 2 ln e 1 ln x 1 1 2 2 4 1 2 ln e ln x2 ln e 2 ln x ln x ln e 1 4 1 4 Power Law Product Law 2 Power Law 2 ln e 1 ■ NOTE The zero earthquake has ground motion amplitude of less than 1 micron on a standard seismograph 100 kilometers from the epicenter. Applications A logarithmic scale is a scale that is determined by a logarithmic function. Because logarithmic growth is slow, measurements on a logarithmic scale can sometimes be deceptive. The Richter scale is an illustration of this. of an earthquake on the Richter scale is given by The magnitude log R i R i 1 2 2 1, where i is the amplitude of the ground motion of the earth- i i0b a i0 is the amplitude of the ground motion of the zero earthquake. quake and A moderate earthquake might have 1000 times the ground motion of the zero |
earthquake, or Its magnitude would be i 1000i0. log 1000i0 i0 a b log 1000 3 An earthquake with 10 times this ground motion, or have a magnitude of i 10,000i0, would log 10,000i0 i0 a b log 10,000 4 So a tenfold increase in ground motion produces only a 1-point change on the Richter scale. In general, increasing the ground motion by a factor of the Richter magnitude by k units. 10k increases Example 7 Richter Scale The 1989 World Series earthquake in San Francisco measured 7.0 on the Richter scale, and the great earthquake of 1906 measured 8.3. How much more intense was the ground motion of the 1906 earthquake than that of the 1989 earthquake? Section 5.5 Properties and Laws of Logarithms 369 Solution The difference in Richter magnitude is 101.3 20 earthquake was in terms of ground motion. Therefore, the 1906 times more intense than the 1989 earthquake 8.3 7.0 1.3. ■ Exercises 5.5 In Exercises 1 – 4, solve each equation by using the basic properties of logarithms. 1. log 3. ln. 4. 2x log 2 1 5 ln 1 3 x 1 2 8 In Exercises 5–10, use laws of logarithms and the values given below logarithmic expression. to evaluate each log 7 0.8451 log 5 0.6990 log 3 0.4771 log 2 0.3010 5. log 8 7. log 5 7b a 9. log 0.6 6. log 12 8. log 3 14b a 10. log 1.5 In Exercises 11–20, write the given expression as a single logarithm. 11. ln x2 3 ln y 12. ln 2x 2 ln x ln 3y 25. ln A 23 x2 1y B 26. ln a 2x 2y 23 y b 27. a. Graph y x y e ln x and in separate viewing windows. For what values of x are the graphs identical? b. Use the properties of logarithms to explain your answer in part a. 28. a. Graph and y x y ln e x in separate viewing windows. For what values of x are the graphs identical? b. Use the properties of logarithms to explain your answer in part a. In Ex |
ercises 29–34, use graphical or algebraic means to determine whether the statement is true or false. 29. ln x 0 0 ln x 0 0 31. log x5 5 log x 33. ln x3 ln x 1 3 2 30. ln 1 xb a 1 ln x 32. 34. x 7 0 ex ln x xx 1 2 log 2x 2log x In Exercises 35 and 36, find values of a and b for which the statement is false. 13. 14. log x2 9 1 log 3x 2 15. 2 ln x 3 1 2 2 y log x 3 2 1 log x log 3 ln x2 ln x 1 2 2 4 35. 36. 16. ln a e 2x b ln A 2ex B 17. 3 ln e2 e 2 1 3 37. If log a log b log a bb a log a b 2 1 ln b7 7, log a log b what is b? 18. 2 2 log 20 19. log 10x log 20y 1 20. ln e2 x 1 2 ln ey 1 2 3 u ln x In Exercises 21–26, let given expression in terms of u and v. For example, ln x3y ln x3 ln y 3 ln x ln y 3u v. v ln y. and Write the 21. ln x2y5 1 2 23. ln A 2x y2 B 22. ln 1 x3y2 2 2x y b 24. ln a 38. Suppose f x 2 constants. If and B? 1 A ln x B, 10 f and 1 1 2 where A and B are, what are A f 1 e 1 2 39. If f x A ln x B 1 2 find A and B. and f e 1 2 5 and f 8, e2 2 1 40. Show that g x 1 2 ln function of f x 1 2 a x 1 xb 1 1 e x. is the inverse (See Section 3.6.) 370 Chapter 5 Exponential and Logarithmic Functions In Exercises 41–44, state the magnitude on the Richter scale of an earthquake that satisfies the given condition. 41. 100 times stronger than the zero quake 42. 104.7 times stronger than the zero quake 43. 350 times stronger than the zero quake 44. 2500 |
times stronger than the zero quake Exercises 45–48 deal with the energy intensity i of a sound, which is related to the loudness of the sound by the function L(i) 10 log i i0b, a where i0 is the minimum intensity detectable by the human ear and L(i) is measured in decibels. Find the decibel measure of the sound. 45. ticking watch (intensity is 100 times ) i0 46. soft music (intensity is 10,000 times ) i0 47. loud conversation (intensity is 4 million times ) i0 48. Victoria Falls in Africa (intensity is 10 billion times )i0 49. How much louder is the sound in Exercise 46 than the sound in Exercise 45? 50. The perceived loudness L of a sound of intensity I L k ln I, where k is a certain is given by constant. By how much must the intensity be increased to double the loudness? (That is, what must be done to I to produce 2L?) 51. Compute each of the following pairs of numbers: a. log 18 and ln 18 ln 10 b. log 8950 and ln 8950 ln 10 c. What do the results in parts a and b suggest? 52. Find each of the following logarithms. c. b. e. log 8.753 log 8753 log 87.53 log 87,530 a. d. f. How are the numbers 8.753, 87.53, 875.3, 8753, and 87,530 related to one another? How are their logarithms related? State a general conclusion that this evidence suggests. log 875.3 5.5.A Excursion: Logarithmic Functions to Other Bases Objectives • Evaluate logarithms to any base with and without a calculator • Solve exponential and logarithmic equations to any base by using an equivalent equation • Identify transformations of logarithmic functions to any base • Use properties and laws of logarithms to simplify and evaluate logarithmic expressions to any base Common and natural logarithms were defined by considering the inverse functions of the exponential functions In this section, you will see that a similar procedure can be carried out with any positive number b in place of 10 and e. 10x ex. and x x f f 2 1 1 2 NOTE b 7 1. valid for In the discussion below, b is a fixed |
positive number with The discussion on exponents and logarithms to base b is also 0 6 b 6 1, but in that case the graphs have a different shape. Defining Logarithmic Functions to Other Bases bx 2 1 f x Because is an increasing function, it is a one-to-one function and therefore has an inverse function. (See Section 3.6) Recall that the graphs y x. of inverse functions are reflections of one another across the line and its inverse function are graphed in An exponential function Figure 5.5.A-1. bx x f 2 1 y y = x f(x) 1 x 1 g(x) Figure 5.5.A-1 Section 5.5.A Excursion: Logarithmic Functions to Other Bases 371 This inverse function g is called the logarithmic function to the base b. and is called the logThe value of g(x) at the number x is denoted arithm to the base b of the number x. logb x Because the functions f x 2 logb v u 1 bx logb x and g x 1 if and only if 2 bu v. are inverse functions, Because all logarithms are exponents, every statement about logarithms is equivalent to a statement about exponents. Logarithmic statement logb v u log3 81 4 log4 64 3 log125 5 1 3 log8a 1 4b 2 3 Equivalent exponential statement bu v 34 81 43 64 125 1 3 5 8 3 1 2 4 Example 1 Evaluating Logarithms to Other Bases Without using a calculator, find each value. 25 log2 16 b. a. c. 9 log1 3 log51 2 Solution a. If b. If c. If log2 16 x, 9 x, log 1 3 then then log51 25 x, 2 Because Because 2x 16. x 1 a 3b then 9. 5x 25. 24 16, log2 16 4. 9, log 2 1 3b a 1 3 9 2 exponent of 5 that produces a negative number, defined. log51 2 Because there is no real number is not 25 ■ Example 2 Solving Logarithmic Equations Solve each equation for x. a. log5 x 3 b. log6 1 x c. log1 6 1 3 2 x d. log6 6 x 372 Chapter 5 Exponential and Logarithmic Functions Solution a |
. If b. If c. If log5 x 3, log6 1 x, 3 log1 6 2 1 then then x, then 53 x. 6x 1. 1 6b 3 log1 1 6 6x 6. a Therefore, Therefore, x x 125. x 0. 3. x 2 Therefore, x 1. d. If log6 6 x, then negative number, has no real solution. Because no real power of 1 6 is a ■ Basic Properties of Logarithms to Other Bases Logarithms are only defined for positive real numbers. That is, logb v is defined only when v 77 0. The graph of b 7 0. That is, y logb x contains the point (1, 0) because b0 1 for any logb 1 0 The value of statement. log5 54 can be found by writing an equivalent exponential In general, If log5 54 x, then 5x 54. So x 4. logb bk k for every real number k. By definition, duce 104. Therefore, log3 104 is the exponent to which 3 must be raised to pro- In general, 3log3 104 104. blogb v v for every v 77 0. The facts presented above are summarized in the table below. Basic Properties of Logarithms For b 77 0 and b 1, 1. 2. 3. 4. logb v logb 1 0 logb bk k blogb v v is defined only when v 77 0 and logb b 1 for every real number k for every v 77 0 Properties 3 and 4 are restatements of the fact that the composition of inverse functions produces the identity function. Section 5.5.A Excursion: Logarithmic Functions to Other Bases 373 If f x 2 1 bx and 21 2 1 logb x, f 1 g 1 x x 2 2 logb x bx 2 2 then blogb x x for all x 7 0 logb bx x for all x Equations that involve both logarithmic and constant terms may be solved by using basic properties of logarithms. blogb v v for b 7 0 and b 1 Example 3 Solving Logarithmic Equations Solve the equation log31 x 1 2 4. Solution log31 3log3 4 x 1 2 x1 2 34 1 x 1 34 x 82 exponentiate both sides b log b v v ■ Laws of Logarithms to Other Bases |
Because all logarithms are a form of exponents, the laws of exponents translate to the corresponding laws of logarithms to any base. Laws of Logarithms For all b, v, w, and k, with b, v, and w positive and b 1: Product Law: logb(vw) logb v logb w Quotient Law: logba v wb logb v logb w Power Law: logb(vk) k logb v Example 4 Using the Laws of Logarithms Use the Laws of Logarithms to evaluate each expression, given that log7 2 0.3562, log7 3 0.5646, a. log7 5 0.8271. and b. c. log7 2.5 log7 48 log7 10 Solution a. Use the Product Law. 2 5 log7 10 log71 2 log7 2 log7 5 0.3562 0.8271 1.1833 374 Chapter 5 Exponential and Logarithmic Functions b. Use the Quotient Law. log7 2.5 log7 a 5 2b log7 5 log7 2 0.8271 0.3562 0.4709 c. Use the Product and Power Laws. log7 48 log71 2 3 24 log7 3 log7 24 log7 3 4 log7 2 0.5646 4 1.9894 1 0.3562 2 Example 5 Using the Laws of Logarithms Simplify and write each expression as a single logarithm. 2 125x log3 y log31 2 a. b. x 2 log31 3 log51 Solution x2 4 a. log31 x 2 log3 y log31 2 x2 4 2 2 log31 x2 4 2 4 log33 1 log3S 1 log3S log3 x 2 y 2 x 2 y 2 x2 21 1 y 2 x 2 2 T NOTE log5 x be expressed as log 5˛a 1 xb. can also 1 or log5 x b. 3 log51 125x 2 log5 125 log5 x 3 1 3 3 log5 x log5 x 2 Change-of-Base Formula Scientific and graphing calculators have a LOG key and a LN key for calculating logarithms. No calculators have a key for logarithms to other bases. One way to evaluate logarithms to other bases is to use the formula below. |
Change-of-Base Formula For any positive number v, logb v log v log b and logb v ln v ln b ■ ■ Section 5.5.A Excursion: Logarithmic Functions to Other Bases 375 Proof By Property 4 of the Basic Properties of Logarithms blog b v v. ln 1 logb v blog b v v ln v blog b v 2 ln v ln b 2 1 logb v ln v ln b take logarithms of both sides apply the Power Law A similar argument can be made by taking common logarithms of both sides. Example 6 Evaluating Logarithms to Other Bases Evaluate log8 9. Solution Use the change-of-base formula and a calculator. log8 9 log 9 log 8 1.0566 or log8 9 ln 9 ln 8 1.0566 ■ Figure 5.5A-2 Graphing Logarithmic Functions to Other Bases The graph of a logarithmic function to any base b shares characteristics with the graphs of natural logarithms and common logarithms. The following table compares the graphs of exponential and logarithmic functions for base b, where b is any real number, b 7 0 b 1, and. y y = x (1, b) Exponential function f(x) bx Logarithmic function g(x) logb x (0, 1) y = bx (b, 1) x (1, 0) Domain all real numbers all positive real numbers Range all positive real numbers all real numbers y = logb x Figure 5.5A-3 x f 1 2 increases as x increses g x 1 2 increases as x increases approaches the x-axis x f 1 as x decreases 2 approaches the y-axis as x g x approaches 0 1 2 Reference points 1, a 1 b b, 0, 1 1, 0, b 1 b, 1, 1 2 2 Example 7 Transforming Logarithmic Functions Describe the transformation from Give the domain and range of h. g x 1 2 log2 x to h x 2 1 log21 x 1 2 3. 376 Chapter 5 Exponential and Logarithmic Functions Solution g x 1 3, h x Because after a horizontal translation of 1 unit to the left and a vertical translation of 3 units down. its graph is the graph of g x 2 1 2 1 2 1 |
log2 x Domain of h: The domain of g x log2 x is all positive real 1 2 numbers. The horizontal translation of 1 unit to the left changes the domain to all real numbers greater than 1. Range of h: The range of log2 x vertical translation has no effect on the range. is all real numbers, so the x g 1 2 The points 12 the points, 0, 2 0, 3, 1 b and (2, 1) on the graph of g are translated to, and 1 2 1, 2 2 on the graph of h. To graph these Y1 ln x ln 2 for g x 2 1 log2 x and functions with a calculator, graph ln Y2 1 x 1 ln 2 2 3 for h x 1 2 log21 x 1 2 3. The graphs of g and h are shown in Figure 5.5A-4. ■ 2 4 8 Figure 5.5A-4 Exercises 5.5.A Note: Unless stated otherwise, all letters represent positive numbers and b 1. In Exercises 1–10, translate the given exponential statement into an equivalent logarithmic statement. 1. 10 2 0.01 2. 103 1000 3. 23 10 10 1 3 4. 100.4771 3 5. 107k r 6. 101 ab 2 c 7. 78 5,764,801 8. 3 1 2 8 9. 2 1 3 9 10. b14 3379 In Exercises 11 – 20, translate the given logarithmic statement into an equivalent exponential statement. 11. log 10,000 4 12. log 0.001 3 13. log 750 2.8751 14. log 0.8 0.0969 15. log5 125 3 16. log8 a 1 4b 2 3 17. log2 a 1 4b 2 18. log2 22 1 2 x2 2y z w 19. log 20. log 1 1 a c 2 2 d In Exercises 21–28, evaluate the given expression without using a calculator. 21. log 10243 24. log3.51 3.51 x21 2 2 27. log23 1 27 2 22. 25. 1717 log17 1 log16 4 2 28. log23 a 1 9b 23. log 102x2y2 26. log2 64 In Exercises 29–36, find the missing entries in each table. 29. x log4? Section 5.5.A |
Excursion: Logarithmic Functions to Other Bases 377 30. x log5 x g x 2 1 31. x log6 x h x 1 2 32. x x k 1 2 log3˛ 1 x 3 2 33. x 2 log7 x f x 1 2 34. x 3 log x g x 1 2 35. x 1 25?? 2 10.75 1 h x 2 1 3 log2 1 x 3 2?? 36. x 2 ln x x k 1 2 1 e? 1? 25 25? 1? 6? 27?? 216? 12? 49? 100 1000? 1? e?? 29? e2? In Exercises 37–40, a graph or a table of values is given for the function f (x) logb x. Find b. 37. y 3 2 1 −1 −2 38. y 5 4 3 2 1 −5 −2 0.05 1.05 1 25 4 x 5 10 15 20 25 1 0 1 0 400 2 5 2 225 1 2 125 6 39. x f x 2 1 40. x f x 1 2 In Exercises 41–46, solve each equation for x. 41. log3 243 x 42. log81 27 x 43. log27 x 1 3 45. logx 64 3 44. log5 x 4 46. logx ˛a 1 9b 2 3 In Exercises 47–60, write the given expression as the logarithm of a single quantity. (See Example 5.) 47. 2 log x 3 log y 6 log z 48. 49. 50. 51. x 3 5 log8 x 3 log8 y 2 log8 z log x log˛1 y 2 log3 ˛1 1 2 ˛ log2˛ 1 2 log3 ˛1 log˛1 y 3 25c2 52. 2 2 2 x2 9 2 log3 y 1 3 ˛ log2 ˛1 27b6 2 x 53. 2 log4 7c 1 2 54. 1 3 ˛ log5 10 11 55. 2 ln x 1 1 z 3 2 ln 2 2 ln x 2 1 z 3 1 2 2 56. ln 1 378 Chapter 5 Exponential and Logarithmic Functions 1 57. log2 59. 2 ln 1 2x 2 1 e2 e 2 2 58. 60. 25z 2 2 |
log5 1 4 4 log5 20 1 2 80. If logb 9.21 7.4 and logb 359.62 19.61, then what is logb 359.62 logb 9.21? In Exercises 61–68, use a calculator and the change-ofbase formula to evaluate the logarithm. In Exercises 81–84, assume that a and b are positive, with b 1. a 1 and 61. log2 10 62. log2 22 63. log7 5 81. Express logb ˛u in terms of logarithms to the base a. 64. log5 7 65. log500 1000 66. log500 250 67. log12 56 68. log12 725 In Exercises 69–72, describe the transformation from f to g, and give the domain and range of g. 82. Show that logb a 1 loga b. 83. How are log u and log100 u related? 84. Show that alog b blog a. 69. 70. 71. 72 log5 x and g log7 x and g log2 x and g log4 x and 3x 4 log5 ˛1 2 log7˛ 1 1 3 log 2˛1 x 1 7 2 3 log4 ˛1 2x 2 85. If logb x 1 2 logb v 3, show that x b3 2 ˛2v. 1 86. Graph the functions log ˛1 x 7 2 f x log x log 7 on the same screen. For what and 1 2 1 2 x g g values of x is it true that x What do 2 1 log 6 log 7 you conclude about the statement log In Exercises 73–78, answer true or false. Explain your answer. 87. Graph the functions f x 1 2 log ˛a x 4b and 73. logb ˛a r 5b logb r logb 5 74. logb a logb c logb ˛a a c b 75. logb r t logb ˛1 1 t r 2 76. logb 77. log5 78. logb 1 1 1 logb c logb d cd 2 log5 x 2 5x ab 5 1 2 t t˛1 2 t logb a 2 397398 log 398 2.5999 or 79. Which is larger: 2.5988 and increasing function |
. 398397? and Hint: x f log 397 10x is an 1 2 log x log 4. g x 1 2 say about a statement such as Are they the same? What does this log 48 log 4 48 4 b log ˛a? In Exercises 88 – 90, sketch a complete graph of the function, labeling any holes, asymptotes, or local extrema. 88. f x 1 2 89. h 90. g x x 2 2 1 1 log5 x 2 x log x2 log20 x2 Section 5.6 Solving Exponential and Logarithmic Equations 379 5.6 Solving Exponential and Logarithmic Equations Objectives • Solve exponential and logarithmic equations • Solve a variety of application problems by using exponential and logarithmic equations Exponential and logarithmic equations have been solved in this chapter so far by using the graphing method or by writing equivalent statements that can be easily solved. Most of them could also have been solved algebraically by using the techniques presented in this section, which depend primarily on the properties and laws of logarithms. By definition of a function, if This results in two statements. u v and f is a function, then u f 1 2 f v. 2 1 If u v, then bu bv for all real numbers b 77 0. If u v, then log b u log bv for all real numbers b 77 0. Because exponential and logarithmic functions are one-to-one functions, the converse is also true. If b u b v, then u v. If log b u log b v, then u v. Exponential Equations The easiest exponential equations to solve are those in which both sides are powers of the same base. Example 1 Powers of the Same Base Solve the equation 8x 2x1. Confirm your solution with a graph. Solution Write the equation so that each side is a power of the same base. 1 8x 2x1 x 2x1 23 2 23x 2x1 3x x 1 2x 1 x 1 2 If bu bv, then u v. Y1 8x Y2 2x1, To find a window for the graphs of consider the basic shapes of the graphs and any transformations. Because both bases of these exponential functions are greater than 1, the graphs are increasing. Because there is no vertical shift on either function, both graphs |
are asymptotic to the x-axis. The intersection of the graphs of and Y2 shown at the left, confirms the solution. 2x1, 8x and Y1 ■ 4 4 4 2 Figure 5.6-1 380 Chapter 5 Exponential and Logarithmic Functions CAUTION Example 2 Powers of Different Bases ln 2 ln 5 ln 2 ln 5 and ln 2 5b a ln 2 ln 5 4 4 4 2 Figure 5.6-2 4 4 4 2 Figure 5.6-3 Solve the equation 5x 2. Confirm your solution with a graph. Solution 5x 2 ln 5x ln 2 x ln 5 ln 2 x ln 2 ln 5 x 0.6931 1.6094 0.4307 take logarithms on each side use the Power Law The intersection of the graphs of confirms the solution. Y1 5x and Y2 2, shown at the left, ■ Example 3 Powers of Different Bases Solve the equation 24x1 31x. Confirm your solution with a graph. Solution 24x1 31x 24x1 ln 2 ln 2 2 1 31x 1 1 x ln 1 4x 1 1 2 ln 3 2 4x ln 2 ln 2 ln 3 x ln 3 4 ln 2 ln 3 4x ln 2 x ln 3 ln 3 ln 2 ln 3 ln 2 x 1 2 x ln 3 ln 2 4 ln 2 ln 3 x 0.4628 Take logarithms on each side Power Law Distributive Property Rearrange terms and isolate x The intersection of the graphs of left, confirms the solution. Y1 24x1 and Y2 31x, shown at the ■ When you multiply each side of an equation by the same expression, extraneous solutions may be introduced, as shown in Example 4. Example 4 Using Substitution Solve the equation ex e x 4. Confirm your solution with a graph. Section 5.6 Solving Exponential and Logarithmic Equations 381 Solution First multiply each side by ex to eliminate negative exponents. x 4 ex e x e x e e x ex 4 1 2 x 4ex ex ex ex e e2x 1 4ex Product Law. 2 1 e2x 4ex 1 0 Let u ex and substitute. u2 4u 1 0 4 – 2 2 4 1 1 |
21 – 220 u 4 – 225 u 2 – 25 ex 2 25 is negative, ex 2 25 2 6 ex Replace u with to get 2 25 be positive and ex 2 25. or ex 2 25 Because has no solution. ex can only 7 7 ln ex ln x ln e ln 2 25 2 25 A A x 1.4436 B B ln e 1 6 Figure 5.6-4 and The intersection of the graphs of ure 5.6-4, confirms that there is exactly one solution Y1 ex e Y2 x 4, shown in Fig- ■ Applications of Exponential Equations When a living organism dies, its carbon-14 decays. The half-life of carbon-14 is 5730 years, so the amount of carbon-14 remaining at time t is given by where P is the mass of carbon-14 that was present initially. The function M can be used to determine the age of fossils and some relics. P t 5730, 0.5 M t 2 2 1 1 Example 5 Radiocarbon Dating The skeleton of a mastodon has lost 58% of its original carbon-14. When did the mastodon die? Solution If the mastodon has lost 58% of its original carbon-14, then 42% of the iniM tial amount, or 0.42P, remains and To determine when the 0.5 mastodon died, solve 0.42P. for t. 0.42P P 2 1 t 5730 t 1 2 382 Chapter 5 Exponential and Logarithmic Functions 10,000 Figure 5.6-5 1 0 0 14,000 0.5 t 5730 2 t 5730 2 0.5 1 0.5 0.42P P 0.42 1 ln 0.42 ln 1 ln 0.42 t t 5730 2 ln 0.5 2 ln 0.42 5730 1 t 5730 1 ln 0.5 t 7171.3171 2 Therefore, the mastodon died approximately 7200 years ago. The interY1 shown in Figure section of the graphs of 5.6-5, confirms the solution. 0.42 x 5730, and 0.5 Y2 1 2 ■ Example 6 Compound Interest If $3000 is to be invested at 8% per year, compounded quarterly, in how many years will the investment be worth $10,680? Solution The interest rate per quarter r is 0.08 4 |
will take the investment to be worth $10,680 use the compound interest formula or 0.02. To find the time t that it.02 t 1.02 t 2 10,680 3000 1 10,680 3000 1 1.02 2 1.02 2 1 ln 1.02 t ln 3.56 ln 1.02 3.56 1 ln 3.56 ln ln 3.56 t 1 t 2 64.1208 quarters 0 0 Figure 5.6-6 Therefore, it will take 64.12 quarters, or 100 10,680 section of the graphs of Figure 5.6-6, confirms the solution. Y1 and 16.03 64.12 4 3000 Y2 years. The inter- 1 0.02 x, 2 1 shown in ■ Example 7 Population Growth A biologist knows that if there are no inhibiting or stimulating factors, the population of a certain type of bacteria will increase exponentially. The population at time t is given by the function Pert, S t 2 1 Section 5.6 Solving Exponential and Logarithmic Equations 383 where P is the initial population and r is the continuous growth rate. The biologist has a culture that contains 1000 bacteria, and 7 hours later there are 5000 bacteria. a. Write the function for this population. b. When will the population reach 1 billion? Solution a. The initial population P is 1000. To find the growth rate r, use the fact that S 7 1 2 5000. 1 2 t 1000ert S 5000 1000er 1 5 e7r ln 5 ln e7r ln 5 7r ln e ln 5 7r 7 2 ln e 1 r ln 5 7 0.2299 Therefore, the function for this population is S t 2 1 1000e0.2299t 5000 Y1 The intersection of the graphs of in Figure 5.6-7, confirms the value of r. b. Find the value of t when S(t) is 1 billion. 1000e0.2299t 1,000,000,000 e0.2299t 1,000,000 ln e0.2299t ln 1,000,000 0.2299t ln e ln 1,000,000 0.2299t ln 1,000,000 t ln 1,000,000 0.2299 60.0936 hours and Y2 1000e7x, shown ln e 1 |
The bacteria population will reach 1 billion after about 60 hours. The 1000e0.2299x 1,000,000,000, intersection of the graphs of shown in Figure 5.6-8, confirms the solution. and Y1 Y2 ■ Example 8 Inhibited Population Growth A population of fish in a lake at time t months is given by the function F. F t 2 1 20,000 t 1 24e 4 How long will it take for the fish population to reach 15,000? 6,000 0 0 Figure 5.6-7 0.5 1.5109 0 108 Figure 5.6-8 100 384 Chapter 5 Exponential and Logarithmic Functions Solution Find the value of t when F(t) is 15,000. 15,000 20,000 1 24e t 4 15,000 1 1 24e 1 24e t 4 20,000 2 4 20,000 t 15,000 24e 4 4 t 3 4 1 t e 3 ln e t 4 ln 1 1 24 1 72 t 4 ln e ln 1 ln 72 0 ln 72 t 4 t 4 ln 72 17.1067 ln e 1 and ln 1 0 Therefore, it will take a little more than 17 months for the population to and reach 15,000. The intersection of the graphs of 15,000 Y1 30 Y2 20,000 x 1 24e 4 Figure 5.6-9 shown in Figure 5.6-9 confirms the solution. ■ 25,000 0 5,000 Logarithmic Equations Properties of one-to-one functions are useful when solving logarithmic equations, as shown in Example 9. Example 9 Equations with Only Logarithmic Terms Solve the equation tion with a graph. ln 1 x 3 ln 1 2 2x 1 2 ln x. 2 1 2 Confirm your solu- Solution First use the Product and Power Laws to rewrite the equation. ln 1 x 3 ln 2 x 3 3 1 ln 21 ln 2 2x 1 ln x 1 2 1 ln x2 2x 1 2 4 2 ˛ ln x2 2x2 5x 3 2x2 5x 3 x2 x2 5x 3 0 1 2 y ln x is a one-to-one function 6 0 6 6 0 6 Figure 5.6-10 Figure 5.6-11 10 30 Section 5.6 Sol |
ving Exponential and Logarithmic Equations 385 Use the Quadratic Formula to solve for x 237 2 5 1 1 2 2 1 2 2 21 5.5414 or x 5 237 1 0.5414 2 x 3, x 5 237 2 0.5414 can- Because ln x 3 1 2 is undefined for not be a solution. Therefore, the only solution of the original equation is x 5 237 2 2x 1 ln and The intersection of the graphs of 5.5414. 2 shown in Figure 5.6-10, confirms the solution. ■ x 3 ln ln x Y1 Y2, 1 2 1 1 2 2 Equations that involve both logarithmic and constant terms may be solved by using the basic property of logarithms. 10log v v and eln v v Example 10 Equations with Logarithmic and Constant Terms Solve the equation a graph. ln 1 x 3 2 5 ln x 3 1 2. Confirm your solution with Solution First get all the logarithmic terms on one side of the equal sign and the constants on the other. Then rewrite the side that contains the logarithms as a single logarithm. ln 1 x 3 2 ln ln 2 ln ln ln 5 5 5 x 3 2 2 5 x3 2 e eln 2 5 x 3 e 2 5 x e ln 1 2 3 15.1825 x 3 The intersection of the graphs of shown in Figure 5.6-11, confirms the solution. and Y2 Y1 1 2 5 ln 1 x 3, 2 ■ Example 11 Equations with Logarithmic and Constant Terms Solve the equation with a graph. log 1 x 16 2 2 log x 1. 2 1 Confirm your solution 386 Chapter 5 Exponential and Logarithmic Functions Solution log 1 x 1 2 1 x 16 log 3 1 log 1 log 1 log 2 log x 16 16 2 4 21 2 x2 17x 16 2 x217x16 2 102 10log 1 x2 17x 16 100 x2 17x 84 0 x 4 0 x 21 x 4 0 or x 21 0 x 4 x 21 x 1 log 21 1 2 x 16 1 log Because and 1 be a solution. Therefore, the only solution is x 16 Y1 the graphs of 5.6-12, confirms the solution. 2 log and Y2 2 2 1 2 log 1 x 21. x |
1 2 are not defined for x 4, it cannot The intersection of shown in Figure, ■ 3 0 3 30 Figure 5.6-12 Exercises 5.6 In Exercises 1–8, solve the equation without using logarithms. 1. 3x 81 2. 3x 3 30 3. 3x1 95x 4. 45x 162x1 5. 35x9x 2 27 6. 2x 25x 1 16 24. 4x 6 2x 8 25. e2x 5e x 6 0 Hint: Let u ex. 26. 2e 2x 9ex 4 0 27. 6e 2x 16e x 6 28. 8e 2x 8e x 6 29. 4x 6 4 x 5 7. 9x 2 3 5x2 8. 4x 21 8x In Exercises 9 – 29, solve the equation. Give exact answers (in terms of natural logarithms). Then use a calculator to find an approximate answer. 9. 3x 5 11. 2x 3x1 13. 312x 5x5 15. 213x 3x1 17. e2x 5 19. 1.4x 21 6e 21. 2.1e x 2 ln 3 5 10. 5x 4 12. 4x2 2x1 14. 43x1 3x2 16. 3z3 2z 18. 3x 2 e 20. 3.4e x 3 5.6 22. 7.8e x 3 ln 5 14 In Exercises 30–32, solve the equation for x. 30. ex e ex e x x t 31. x ex e 2 t 32. ex e ex e x x t 33. Prove that if ln u ln v, basic property of inverses u v. then e ln v v. Hint: Use the 34. a. Solve 7 x 3 using natural logarithms. Give an exact answer, not an approximation. b. Solve 7 x 3 using common logarithms. Give an exact answer, not an approximation. c. Use the change-of-base formula in Excursion 5.5.A to show that your answers in parts a and b are the same. In Exercises 35–44, solve the equation. (See Example 9.) 23. 9x 4 3x 3 0 u 3x. Hint: Note that 9x 2; |
let 3x 1 2 35. ln 1 3x 5 2 4x 1 ln 11 ln 2 log x 1 2 1 2 log 2 36. log 1 Section 5.6 Solving Exponential and Logarithmic Equations 387 2 59. Krypton-85 loses 6.44% of its mass each year. What is its half-life? 37. log 3x 1 log 2 log 4 log x 2 1 x 6 2 ln 10 ln 38. ln 1 2 x 1 1 2 1 ln 2 39. 2 ln x ln 36 40. 2 log x 3 log 4 41. ln x ln x 1 1 2 ln 3 ln 4 42. ln 1 6x 1 ln x 1 2 2 ln 4 43. ln x ln 3 ln x 5 44. ln 1 2x 3 2 2 1 ln x ln e In Exercises 45–52, solve the equation. ln x 1 45. ln 46. ln x 9 2 2x 1 1 1 2 47. log x log 48. log x 1 1 2 1 ln log 1 1 2 60. Strontium-90 loses 2.5% of its mass each year. What is its half-life? 61. The half-life of a certain substance is 3.6 days. How long will it take for 20 grams to decay to 3 grams? 62. The half-life of cobalt-60 is 4.945 years. How long will it take for 25 grams to decay to 15 grams? Exercises 63–68 deal with the compound interest formula which was discussed in Section 5.3 and used in Example 6 of this section. A P(1 r)t, 63. At what annual rate of interest should $1000 be invested so that it will double in 10 years, if interest is compounded quarterly? 64. Find how long it takes $500 to triple if it is invested at 6% in each compounding period. a. annually b. quarterly c. daily 49. log 2x2 1 2 50. log23 x2 21x 2 3 51. ln x2 1 ln 1 2 x 1 2 1 1 ln x 1 1 2 52. ln ln 1 1 x 1 x 1 2 2 2 Exercises 53 – 62 deal with the half-life function M(x) c(0.5) which was discussed in |
Section 5.3 and used in Example 5 of this section. x h, 53. How old is a piece of ivory that has lost 36% of its carbon-14? 54. How old is a mummy that has lost 49% of its carbon-14? 55. Find when part of the Pueblo Benito ruins was built if the doorway timbers have 89.14% of their original carbon-14. (See the image on the first page of this chapter.) 56. How old is a wooden statue that has only one- third of its original carbon-14? 57. A quantity of uranium decays to two-thirds of its original mass in 0.26 billion years. Find the halflife of uranium. 58. A certain radioactive substance loses one-third of its original mass in 5 days. Find its half-life. 65. a. How long will it take to triple your money if you invest $500 at a rate of 5% per year compounded annually? b. How long will it take at 5% compounded quarterly? 66. At what rate of interest compounded annually should you invest $500 if you want to have $1500 in 12 years? 67. How much money should be invested at 5% interest compounded quarterly so that 9 years later the investment will be worth $5000? This answer is called the present value of $5000 at 5% interest. 68. Find a formula that gives the time needed for an investment of P dollars to double, if the interest rate is r% compounded annually. Hint: Solve the A 2P. compound interest formula for t, when Exercises 69 – 76 deal with functions of the form f(x) Pekx, where k is the continuous exponential growth rate. See Example 7. 69. The present concentration of carbon dioxide in the atmosphere is 364 parts per million (ppm) and is increasing exponentially at a continuous yearly k 0.004 rate of 0.4% (that is, will it take for the concentration to reach 500 ppm? ). How many years 70. The amount P of ozone in the atmosphere is currently decaying exponentially each year at a 388 Chapter 5 Exponential and Logarithmic Functions continuous rate of % that is, k 0.0025. How 1 long will it take for half the ozone to disappear 2 1 4 that is, when will the amount be a answer is the half-life of ozone. P 2 b? Your 71. The population of Brazil increased exponentially from 151 million in 1990 |
to 173 million in 2000. a. At what continuous rate was the population growing during this period? b. Assuming that Brazil’s population continues to increase at this rate, when will it reach 250 million? 72. Outstanding consumer debt increased exponentially from $781.5 billion in 1990 to $1765.5 billion in 2002. (Source: Federal Reserve Bulletin) a. At what continuous rate is consumer debt growing? b. Assuming this rate continues, when will consumer debt reach $2500 billion? 73. The probability P percent of having an accident while driving a car is related to the alcohol level of the driver’s blood by the formula where k is a constant. Accident statistics show that the probability of an accident is 25% when the blood alcohol level is P 25, a. Find k. Use b. At what blood alcohol level is the probability t 0.15. not 0.25. P e kt, of having an accident 50%? 74. Under normal conditions, the atmospheric pressure (in millibars) at height h feet above sea P level is given by positive constant. a. If the pressure at 18,000 feet is half the pressure where k is a 1015e kh, h 1 2 at sea level, find k. b. Using the information from part a, find the atmospheric pressure at 1000 feet, 5000 feet, and 15,000 feet. 75. One hour after an experiment begins, the number of bacteria in a culture is 100. An hour later there are 500. a. Find the number of bacteria at the beginning of the experiment and the number 3 hours later. b. How long does it take the number of bacteria at any given time to double? 76. If the population at time t is given by ce kt, find a formula that gives the time it takes for the population to double. S t 1 2 77. The spread of a flu virus in a community of 45,000 people is given by the function f t 2 1 45,000 1 224e 0.889t, 1 t is the number of people infected in f where week t. a. How many people had the flu at the outbreak 2 of the epidemic? after 3 weeks? b. When will half the town be infected? 78. The beaver population near a certain lake in year t is approximately p t. 0.5544t 2000 1 199e 1 2 a. When will the beaver population reach 1000? b. Will the population ever |
reach 2000? Why? 79. Critical Thinking According to one theory of N c learning, the number of words per minute N that a person can type after t weeks of practice is kt where c is an upper limit given by, that N cannot exceed and k is a constant that must be determined experimentally for each person. a. If a person can type 50 wpm (words per 1 e 2 1 minute) after 4 weeks of practice and 70 wpm after 8 weeks, find the values of k and c for this person. According to the theory, this person will never type faster than c wpm. b. Another person can type 50 wpm after 4 weeks of practice and 90 wpm after 8 weeks. How many weeks must this person practice to be able to type 125 wpm? 80. Critical Thinking Wendy has been offered two jobs, each with the same starting salary of $24,000 and identical benefits. Assuming satisfactory performance, she will receive a $1200 raise each year at the company A, whereas the company B will give her a 4% raise each year. a. In what year (after the first year) would her salary be the same at either company? Until then, which company pays better? After that, which company pays better? b. Answer the questions in part a assuming that the annual raise at company A is $1800. Section 5.7 Exponential, Logarithmic, and Other Models 389 5.7 Exponential, Logarithmic, and Other Models Objectives • Model real data sets with power, exponential, logarithmic, and logistic functions Many data sets can be modeled by suitable exponential, logarithmic, and related functions. Most calculators have regression procedures for constructing the models described in the table below. Model Power Equation y axr Exponential y ab x or y ae kx Examples y 5x2.7 y 2 1.64 1 x 2 y 3.5x 0.45 y 2e0.4947x Logarithmic y a b ln x y 5 4.2 ln x y 2 3 ln x Logistic y a 1 be kx y 20,000 1 24e 0.25x y 650 1 6e 0.3x Exponential Models In the table of values for the exponential model examine the patterns in the ratios of successive y-values. y 3 2x that follows, x y 0 3 4 48 8 768 12 16 12,288 196, |
608 48 3 16 768 48 16 12,288 768 16 196,608 12,288 16 At each step, x changes from x to y changes from and the ratio of successive y-values is always the same. x 4, 3 2x to 3 2x4, 3 2x4 3 2x 3 2x 24 3 2x 24 16 A similar argument applies to any exponential model and shows that if x changes by a fixed amount k, then the ratio of the corresponding y-values is the constant above, b is 2 and k is 4. This fact identifies the model that would best represent the data. In the exponential model y 3 2x bk. y ab x When the ratio of successive entries in a table of data is approximately constant, an exponential model is appropriate. Example 1 U.S. Population Before the Civil War In the years before the Civil War, the population of the United States grew rapidly, as shown in the following table. Find a model for this growth. 390 Chapter 5 Exponential and Logarithmic Functions Year 1790 1800 1810 1820 Population in millions 3.93 5.31 7.24 9.64 Year 1830 1840 1850 1860 Population in millions 12.86 17.07 23.19 31.44 [Source: U.S. Bureau of the Census] 50 Solution x 0 The data points, with corresponding to 1790, are shown in Figure 5.7-1. Their shape suggests either a polynomial graph of even degree or an exponential graph. Since populations generally grow exponentially, an exponential model is likely to be a good choice. This can be confirmed by looking at the successive entries in the table. Year 1790 Population 3.93 Year 1830 Population 12.86 5.31 3.93 1.351 1800 5.31 1840 17.07 1810 7.24 7.24 5.31 9.64 7.24 1.363 1.331 1850 23.19 1820 9.64 1860 31.44 12.86 9.64 1.334 1830 12.86 17.07 12.86 1.327 23.19 17.07 1.359 31.44 23.19 1.356 Because the ratios are almost constant, as they would be in an exponential model, use regression to find an exponential model. The procedure is the same as for linear and polynomial regression. An exponential regression produces this model. y 3.9572 1.0299x 1 2 The graph of the exponential model in |
Figure 5.7-2 appears to fit the data well. In fact, you can readily verify that the model has an error of less than 1% for each of the data points. Furthermore, as discussed before this example, when x changes by 10, the value of y changes by approximately 1.029910 1.343, which is very close to the successive ratios of the data. ■ −5 0 Figure 5.7-1 100 NOTE Throughout this section, coefficients are rounded for convenient reading, but the full expansion is used for calculations and graphs. 50 −5 0 100 Figure 5.7-2 Section 5.7 Exponential, Logarithmic, and Other Models 391 Logistic Models A logistic model represents growth that has a limiting factor, such as food supplies, war, new diseases, etc. Logistic models are often used to model population growth, as shown in Example 2. Example 2 U.S. Population After the Civil War After the Civil War, the population of the United States continued to increase, as shown in the following table. Find a model for this growth. Year 1870 1880 1890 1900 1910 Population in millions 38.56 50.19 62.98 76.21 92.23 Solution Year 1920 1930 1940 1950 Population in millions 106.02 123.20 132.16 151.33 Year 1960 1970 1980 1990 2000 Population in millions 179.32 202.30 226.54 248.72 281.42 The model from Example 1 does not remain valid, as can be seen in Figure 5.7-3, which shows its graph together with all the data points from 1790 through 2000, where corresponds to 1790. x 0 500 −5 5 220 Figure 5.7-3 The rate of growth has steadily decreased since the Civil War. For instance, 50.19 38.56 the ratio of the first two entries is and the ratio of the last 1.302 two is 281.42 248.72 1.131. So an exponential model may not be the best choice. Other possibilities are polynomial models, which grow at a slower rate, or logistic models, in which the growth rate decreases with time. Figure 5.7-4 on the next page shows these models compared to an exponential model. 392 Chapter 5 Exponential and Logarithmic Functions Exponential Model y 6.0662 1.02039x y Quartic Model 2.76 10 x4 7.94 10 1 0.0093x2 0.1621 |
x 5.462 8 2 1 5 Logistic Model x3 2 y 442.10 1 56.329e 0.022x 500 500 500 −5 5 250 −5 5 250 −5 5 250 Figure 5.7-4 The quartic and logistic models fit the data better than does the exponential model. The quartic model indicates unlimited future growth, but the logistic model has the population growing more slowly in the future. ■ Exponential versus Power Models In Example 1, the ratios of successive entries of the data table were used to determine that an exponential model was appropriate. Another way to determine if an exponential model might be appropriate is to consider the exponential function as shown below. y abx, y ab x ln y ln 1 ln y ln a ln b x ln y ln a x ln b ab x 2 Take the natural logarithm of each side Product Law Power Law Because ln a and ln b are constants, let k ln a and m ln b. ln y ln a x ln b ln y x ln a ln b 1 ln y mx k 2 The points (x, line with slope m and y-intercept k. Consequently, a guideline for determining if an exponential model is appropriate is as follows. lie on a straight ln y) (x, y) are data points and if the points If approximately linear, then an exponential model may be appropriate for the data. (x, ln y) are Similarly, consider the power function y axr y axr ln y ln 1 ln y ln a ln xr ln y ln a r ln x axr 2 Take the natural logarithm of each side Product Law Power Law Section 5.7 Exponential, Logarithmic, and Other Models 393 Because r and ln a are constants, let Then: k ln a. ln y ln a r ln x ln y r ln x ln a ln y r ln x k lie on the straight line with slope r and Thus, the points y-intercept k. Consequently, a guideline for determining if a power model is appropriate is as follows ln x, ln y 1 2 (x, y) (ln x, ln y) If are data points and if the points approximately linear, then a power model may be appropriate for the data. |
are Example 3 Different Planet Years The length of time that a planet takes to make one complete rotation around the sun is that planet’s “year.’’ The table below shows the length of each planet’s year, relative to an Earth year, and the average distance of that planet from the Sun in millions of miles. Find a model for this data in which x is the length of the year and y is the distance from the Sun. Planet Year Distance Planet Year Distance Mercury Venus Earth Mars 0.24 0.62 1.00 1.88 Jupiter 11.86 Solution 36.0 67.2 92.9 141.6 483.6 Saturn Uranus 29.46 84.01 886.7 1783.0 Neptune 164.79 2794.0 Pluto 247.69 3674.5 Figure 5.7-5 shows the data points for the five planets with the shortest years. Figure 5.7-6 shows all of the data points, but on this scale, the first four points look like a single large point near the origin. 600 5,000 −1 0 15 −25 300 Figure 5.7-5 0 Figure 5.7-6 Technology Tip Suppose the x- and ycoordinates of the data points are stored in lists L1 L2 and Keying in, respectively. LN L2 STO S L4 L4 produces the list, whose entries are the natural logarithms of the numbers in list, and stores it in the statistics editor. You can L4 then use lists plot the points (x, ln y). and L1 L2 to 394 Chapter 5 Exponential and Logarithmic Functions 1 x, ln y Plotting the point produces the graph for each data point shown in Figure 5.7-7. Its points do not form a linear pattern, so an exponential model is not appropriate. The points shown in Figure 5.7-8 do form a linear pattern, which suggests that a power model will work. ln x, ln y x, y 1 2 2 1 2 10 −5 0 10 0 250 −3 7 Figure 5.7-7 Figure 5.7-8 A power regression produces this model: y 92.8932x0.6669 Its graph in Figures 5.7-9 and 5.7-10 show that it fits the original data points well. 600 5,000 −1 0 15 −25 0 Figure 5.7- |
9 Figure 5.7-10 300 ■ Logarithmic Models Consider the logarithmic function Because a and b are constants, let y m and k ln x 1 2 y b ln x a m b k a. Then: ln x, y line with slope m and The points y-intercept k. Consequently, a guideline for determining if a logarithmic model is appropriate is as follows. lie on the straight 2 1 (x, y) If are approximately linear, then a logarithmic model may be appropriate for the data. are data points and if the points (ln x, y) Section 5.7 Exponential, Logarithmic, and Other Models 395 Example 4 Logarithmic Population Growth Find a model for population growth in El Paso, Texas, given the information in the following table. Year 1950 1970 1980 1990 2000 Population 130,485 322,261 425,259 515,342 563,662 [Source: U.S. Bureau of the Census.] Solution The scatter plot of the data, where Figure 5.7-11, suggests a logarithmic curve with a very slight bend. corresponds to 1950, shown in x 50 700,000 700,000 40 0 Figure 5.7-11 120 3 0 Figure 5.7-12 6 To determine whether a logarithmic model is appropriate for this data, ln 50, 130,485 plot points (ln 100, 563,662). Because these points, shown in Figure 5.7-12, appear to be approximately linear, a logarithmic model seems appropriate. that is ln x, y p ; 1 2 2 1 Using logarithmic regression, the model is: y 2,382,368.345 640,666.815 ln x 700,000 CAUTION When using logarithmic models, you must have data points with positive first coordinates because logarithms of negative numbers and 0 are not defined. 40 0 Figure 5.7-13 120 The graph of this model, shown in Figure 5.7-13, shows that it is a good fit for the data. ■ 396 Chapter 5 Exponential and Logarithmic Functions Exercises 5.7 4. y In Exercises 1–10, state which of the following models might be appropriate for the given scatter plot of data. More than one model may be appropriate. Model A. Linear Corresponding function y ax b B. Quad |
ratic y ax2 bx c 5. y C. Power D. Cubic y axr y ax 3 bx 2 cx d E. Exponential y ab x F. Logarithmic y a b ln x G. Logistic y a 1 be kx 6. y 7. y 8. y 1. y 2. y 3 Section 5.7 Exponential, Logarithmic, and Other Models 397 9. y 10. y x x In Exercises 11 and 12, compute the ratios of successive entries in the table to determine whether an exponential model is appropriate for the data. 11. 12. x y 0 3 2 4 6 8 10 15.2 76.9 389.2 1975.5 9975.8 x y 1 3 3 21 5 55 7 9 11 105 171 253 13. a. Show algebraically that in the logistic model for the U.S. population in Example 2, the population can never exceed 442.10 million people. b. Confirm your answer in part a by graphing the logistic model in a window that includes the next three centuries. 14. According to estimates by the U.S. Bureau of the Census, the U.S. population was 287.7 million in 2002. Based on this information, which of the models in Example 2 appears to be the most accurate predictor? 15. Graph each of the following power functions in a window with 1.5 a. x x f 0 x 20. b. x g 1 2 x0.75 c. h x 2 1 x2.4 1 2 16. Based on your graphs in Exercise 15, describe the a 7 0 y axr, when general shape of the graph of and r is as described below. 0 6 r 6 1 a. r 6 0 b. c. r 7 1 In Exercises 17–20, determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear, where the given data set consists of the points (x, y). 5 6 17. 18. 19. 20. (x, ln y) 5 6 5 (ln x, ln y) 6 5 (ln x, y 25 5 81 7 9 11 175 310 497 3 385 5 17 6 74 10 27 9 14 12 2.75 15 35 20 40 15 0 |
.5 25 43 18 0.1 30 48 x y 5 2 10 15 20 25 30 110 460 1200 2500 4525 21. The table shows the number of babies born as twins, triplets, quadruplets, etc., in recent years. Year Multiple births 1989 1990 1991 1992 1993 1994 1995 92,916 96,893 98,125 99,255 100,613 101,658 101,709 398 Chapter 5 Exponential and Logarithmic Functions a. Sketch a scatter plot of the data, with x 1 corresponding to 1989. b. Plot both of the following models on the same screen with the scatter plot: 93,201.973 4,545.977 ln x x f 1 2 and g x 2 1 102,519.98 1 0.1536e 0.4263x c. Use the table feature to estimate the number of multiple births in 2000 and 2005. d. Over the long run, which model do you think is the better predictor? 22. The graph shows the Census Bureau’s estimates of future U.S. population. Infant Year mortality rate* Year Infant mortality rate* 1920 1930 1940 1950 1960 1970 76.7 60.4 47.0 29.2 26.0 20.0 1980 1985 1990 1995 2000 12.6 10.6 9.2 7.6 6.9 *Rates are infant (under 1 year) deaths per 1000 live births. a. Sketch a scatter plot of the data, with 425 400 375 350 325 300 275 250 U.S. Population Projections: 2000–2050 403.687 377.350 351.070 324.927 299.862 281.422 2000 2010 2020 2030 2040 2050 Year a. How well do the projections in the graph compare with those given by the logistic model in Example 2? b. Find a logistic model of the U.S. population, using the data given in Example 2 for the years from 1900 to 2000, with corresponding to 1900. x 0 c. How well do the projections in the graph compare with those given by the model in part b? 23. Infant mortality rates in the United States are shown in the following table. corresponding to 1900. b. Verify that the set of points x, ln y, 2 are the original data points, is 1 x, y 1 approximately linear. 2 where c. Based on part b, what type of model would be appropriate for this data? Find such a model |
. 24. The average number of students per computer in the U.S. public schools (elementary through high school) is shown in the table below. Fall of school year Students per computer 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 32 25 22 20 18 16 14 10.5 10 7.8 6.1 5.7 5.4 Section 5.7 Exponential, Logarithmic, and Other Models 399 a. Sketch a scatter plot of the data, with x 1 corresponding to 1987. b. Find an exponential model for the data. c. Use the model to estimate the number of students per computer in 2003. d. In what year, according to this model, will each student have his or her own computer in school? e. What are the limitations of this model? 25. The number of children who were home-schooled in the United States in selected years is shown in the table below. Fall of school year Number of children (in 1000s) 1985 1988 1990 1992 1993 1994 1995 1996 1997 1999 2000 183 225 301 470 588 735 800 920 1100 1400 1700 [Source: National Home Education Research Institute] a. Sketch a scatter plot of the data, with x 0 corresponding to 1980. b. Find a quadratic model for the data. c. Find a logistic model for the data. d. What is the number of home-schooled children predicted by each model for the year 2003? e. What are the limitations of each model? 26. a. Find an exponential model for the federal debt, based on the data in the following table. Let x 0 correspond to 1960. Accumulated gross federal debt Amount (in billions of dollars) 284.1 313.8 370.1 533.2 907.7 1823.1 3233.3 4974.0 5674.2 Year 1960 1965 1970 1975 1980 1985 1990 1995 2000 b. Use the model to estimate the federal debt in 2003. 27. The table gives the life expectancy of a woman born in each given year. Life Expectancy (in years) 51.8 54.6 61.6 65.2 71.1 Year 1910 1920 1930 1940 1950 Life Expectancy (in years) 73.1 74.7 77.5 78.8 79.4 Year 1960 1970 1980 1990 2000 [Source: National Center for Health Statistics] a. Find a logarithmic model for the data, with x 10 corresponding to 1910. b. |
Use the model to find the life expectancy of a woman born in 1986. For comparison, the actual expectancy is 78.3 years. c. Assume the model remains accurate. In what year will the life expectancy of a woman born in that year be at least 81 years? 400 Chapter 5 Exponential and Logarithmic Functions 28. The table gives the death rate in motor vehicle accidents, per 100,000 population, in selected years. Year 1970 1980 1985 1990 1995 2000 Death Rate 26.8 23.4 19.3 18.8 16.5 15.6 a. Find an exponential model for the data, with x 0 corresponding to 1970. b. Use the model to predict the death rate in 1998 and in 2002. c. Assuming the model remains accurate, when will the death rate drop to 13 per 100,000? 29. Worldwide production of computers has grown dramatically, as shown in the first two columns of the following table. a. Sketch a scatter plot of the data, with x 1 corresponding to 1985. b. Find an exponential model for the data. c. Use the model to complete column 3 of the table. d. Fill in column 4 of the table by dividing each entry in column 2 by the preceding one. e. What does column 4 tell you about the appropriateness of the model? Worldwide shipments (in thousands) Predicted number of shipment (in thousands) Ratio 14.7 15.1 16.7 18.1 21.3 23.7 27 32.4 38.9 47.9 60.2 70.9 84.3 Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 [Source: Dataquest Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 Section 5.5.A Section 5.6 Section 5.7 nth root................................ 328 Rational exponents....................... 330 Laws of exponents....................... 330 Rationalizing numerators and denominators... 332 Irrational exponents...................... 333 Graphs of exponential functions |
............ 336 Exponential growth and decay.............. 339 The number e and its exponential function.... 341 Other exponential functions (logistic models).. 341 Compound interest....................... 345 Continuous compounding................. 348 Constructing exponential growth functions.... 349 Constructing exponential decay functions..... 351 Radioactive decay........................ 352 Common logarithms...................... 356 Natural logarithms....................... 358 Graphing logarithmic functions............. 359 Solving logarithmic equations by graphing.... 360 Basic properties of logarithms.............. 364 Product Law of Logarithms................ 365 Quotient Law of Logarithms............... 366 Power Law of Logarithms................. 366 Richter scale............................ 368 Logarithms to base b..................... 371 Basic properties of logarithms to base b...... 372 Laws of logarithms to base b............... 373 Change-of-base formula................... 374 Exponential equations.................... 379 Logarithmic equations.......... |
.......... 384 Exponential models...................... 389 Logistic models......................... 391 Power models.......................... 392 Logarithmic models...................... 394 401 402 Chapter Review Important Facts and Formulas • Rational Exponents: 1 n 2n c c 1 k ct • Laws of Exponents: crcs crs cr cs crs s crs cr 2 1 a cd 1 2 r cr dr r cr c dr db r 1 cr 10x log x is the inverse function of f 10log v v for all v 7 0 and log 10u u for all u ln x is the inverse function of f ex: x 2 1 eln v v for all v 7 0 and ln eu u for all u logb x is the inverse function of blogb v v for all v 7 0 and logb v, w 7 0 and any k: x k 1 2 bx: • Logarithm Laws: For all u for all u bu 1 2 vw ln 1 2 ln v ln w logb vw 2 1 logb v logb w logb a logb v wb vk 2 1 logb v logb w k logb v v wb vk ln a ln ln v ln w k ln v 1 2 • Exponential Growth Functions: P 1 Pax Pekx x x x f f f • Exponential Decay Functions: P 1 Pax Pekx • Logistic Function: x f 1 2 a 1 be kx • Compound Interest Formula: • Continuous Compounding: 1 r A P 1 A Pert P • Radioactive Decay Function: 1 • Change of Base Formula: logb v ln v ln b x f 1 2 0.5 t 2 x h 2 Review Exercises In Exercises 1–6, simplify the expression. Section 5.1 1. 323 c12 4. 3 5 1 5 2 2 1 1 3c 4c 1 1 2d 2d 2 4c 1 2 |
2 3 2 1 2c 2 2 4 1 2 23 4c3 d2 2. A c2d 3 A B 2 B 5 Chapter Review 403 3. A 2 a 2 3 b 5 a3 b6 B A 4 3 B 6. 3 2 c A 1 2c 2 3c 3 2 B In Exercises 7 and 8, simplify and write the expression without radicals or negative exponents. 7. 23 6c4 d14 2 d2 23 48c 8u5 2 8. 1 3 1u 1 4 2 2u8 9. Rationalize the numerator and simplify: 22x 2h 1 22x 1 h 10. Rationalize the denominator: 5 2x 3 Section 5.2 In Exercises 11–16, list the transformations needed to transform the graph of f(x) 5x into the graph of the given function. 11. 14 5x 52x 12. 15. h h x x 1 1 2 2 53x 5x 4 13. 16 2x 5x2 In Exercises 17 and 18, find a viewing window (or windows) that shows a complete graph of the function. 17. x f 1 2 2x 3x2 18. g x 1 2 850 1 5e 0.4x 19. Compunote offers a starting salary of $60,000 with $1000 yearly raises. Calcuplay offers a starting salary of $30,000 with a 6% raise each year. a. Complete the following table for each company. Year Compunote Year Calcuplay 1 2 3 4 5 $60,000 $61,000 $30,000 $31,800 1 2 3 4 5 b. For each company write a function that gives your salary in terms of years employed. c. If you plan on staying with the company for only five years, which job should you take to earn the most money? d. If you plan on staying with the company for 20 years, which is your best choice? 404 Chapter Review 20. A computer software company claims that the following function models the “learning curve” for their software. P t 2 1 100 1 48.2e 0.52t where t is measured in months and P(t) is the average percent of the software program’s capabilities mastered after t months. a. Initially what percent of the program is mastered? b. After 6 months what percent of the program is mastered? c. Roughly, when can a |
person expect to “learn the most in the least amount of time”? d. If the company’s claim is true, how many months will it take to have completely mastered the program? Section 5.3 21. Phil borrows $800 at 9% annual interest, compounded annually. a. How much does he owe after 6 years? b. If he pays off the loan at the end of 6 years, how much interest will he owe? 22. If you invest $5000 for 5 years at 9% annual interest, how much more will you make if interest is compounded continuously than if it is compounded quarterly? 23. Mary Karen invests $2000 at 5.5% annual interest, compounded monthly. a. How much is her investment worth in 3 years? b. When will her investment be worth $12,000? 24. If a $2000 investment grows to $5000 in 14 years, with interest compounded annually, what is the interest rate? 25. Company sales are increasing at 6.5% per year. If sales this year are $56,000, write the rule of a function that gives the sales in year x (where x 0 corresponds to the present year). 26. The population of Potterville is decreasing at an annual rate of 1.5%. If the population is 38,500 now, what will be the population x years from now? 27. The half-life of carbon-14 is 5730 years. How much carbon-14 remains from an original 16 grams after 12,000 years? 28. How long will it take for 4 grams of carbon-14 to decay to 1 gram? Section 5.4 In Exercises 29–34, translate the given exponential statement into an equivalent logarithmic one. 29. e6.628 756 32. eab c 30. e5.8972 364 31. er 21 u v 33. 102.8785 756 34. 10cd t In Exercises 35–38, translate the given logarithmic statement into an equivalent exponential one. 35. ln 1234 7.118 38. log 1234 3.0913 39. Find log 0.01. 2 1 36. ln 1 ax b y 2 37. ln rs 2 1 t Chapter Review 405 In Exercises 40–43, describe the transformation from to the given function. Give the domain and range of the given function. or f (x) log x g(x) ln x |
40. h 42 log 1 x 3 2 ln 3x 1 2 41. k 43. k x x 2 2 1 1 log 4 x 1 2 3 ln x 5 44. You are conducting an experiment about memory. The people who 2 1 t M 91 14 ln participate agree to take a test at the end of your course and every month thereafter for a period of two years. The average score for the group is given by the model months after the first test. a. What is the average score on the initial exam? b. What is the average score after three months? c. When will the average drop below 50%? d. Is the magnitude of the rate of memory loss greater in the first month t 1 t 12 ) or after the first year (from 0 t 24, where t is time in t 1 t 0 to, 1 2 after the course (from to t 13 )? e. Hypothetically, if the model could be extended past would it be possible for the average score to be 0%? t 24 months, Section 5.5 In Exercises 45–48, evaluate the given expression without using a calculator. 45. ln e3 46. ln e 47. eln 3 4 48. eln 1 x2y 2 49. Simplify: 3 ln 2x 1 2 ln x 50. Simplify: ln 1 4e e4e 1 2 In Exercises 51 and 52, write the given expression as a single logarithm. 51. ln 3x 3 ln x ln 3y 52. 4 ln x 2 ln x3 4 ln x 2 1 53. Which of the following statements is true? a. ln 10 c. ln 1 7b a ln 2 ln 5 1 21 ln 7 0 2 b. ln d. ln e a 6b e 1 2 ln e ln 6 1 e. None of the above is true. 54. Which of the following statements is false? log 50 a. c. log 5 10 1 2 log 1 ln 1 b. d. log 100 3 log 105 log 6 log 3 log 2 e. All of the above are false. 55. What is the domain of the function f ln x x 1b? a x 1 2 Section 5.5.A In Exercises 56 and 57, translate the given logarithmic statement into an equivalent exponential one. |
56. log5 cd k u 2 1 57. logd uv 1 2 w 58. Write log7 7x log7 y 1 as a single logarithm. 59. log20 400? 60. If log3 9x2 4, what is x? 406 Chapter Review Use the following six graphs for Exercises 61 and 62. y 3 2 1 −2 −1 −1 1 2 Figure 2 −1 −1 1 2 Figure II 2 −1 −1 1 2 Figure III y 1 −2 −1 −1 1 2 −2 −3 −2 −1 −1 1 2 −2 −1 −1 1 2 Figure IV Figure V Figure VI 61. If b 7 1, then the graph of a. I b. IV c. V d. VI x logb x f 1 e. none of these 2 could possibly be 62. If 0 6 b 6 1 a. II b. III then the graph of c. IV d. VI x g b x 1 2 e. none of these 1 could possibly be Section Section 5.6 5.6 In Exercises 63–71, solve the equation for x. 63. 8x 4x 23 64. e3x 4 65. 2 4x 5 4 66. 725e 4x 1500 67. u c d ln x 68. 2x 3x3 69. ln x ln 3x 5 ln 2 1 x2 1 2 2 log 2 x 1 1 2 71. log 1 70. ln x 8 2 1 ln x 1 72. At a small community college the spread of a rumor through the population of 500 faculty and students can be modeled by ln n ln 1 1000 2n 2 0.65t ln 998, where n is the number of people who have heard the rumor after t days. a. How many people know the rumor initially (at b. How many people have heard the rumor after four days? c. Roughly, in how many weeks will the entire population have heard the t 0 )? rumor? d. Use the properties of logarithms to write n as a function of t; in other words solve the model above for n in terms of t. Chapter Review 407 e. Enter the function you found in part d into your calculator and use the table feature to check your answers to parts a, b, and c. Do they agree? f. Graph the function. Over what time interval does the |
rumor seem to “spread” the fastest? 73. The half-life of polonium milligrams, how much will be left at the end of a year? 210Po 1 2 is 140 days. If you start with 10 74. An insect colony grows exponentially from 200 to 2000 in 3 months. How long will it take for the insect population to reach 50,000? 75. Hydrogen-3 decays at a rate of 5.59% per year. Find its half-life. 76. The half-life of radium-88 is 1590 years. How long will it take for 10 grams to decay to 1 gram? 77. How much money should be invested at 8% per year, compounded quarterly, in order to have $1000 in 10 years? 78. At what annual interest rate should you invest your money if you want to double it in 6 years? 79. One earthquake measures 4.6 on the Richter scale. A second earthquake is 1000 times more intense than the first. What does it measure on the Richter scale? Section 5.7 80. The table below gives the population of Austin, Texas. Year 1950 1970 1980 1990 2000 Population 132,459 253,539 345,890 465,622 656,562 a. Sketch a scatter plot of the data, with b. Find an exponential model for the data. c. Use the model to estimate the population of Austin in 1960 and 2005. corresponding to 1950. x 0 81. The wind-chill factor is the temperature that would produce the same cooling effect on a person’s skin if there were no wind. The table shows the wind-chill factors for various wind speeds when the temperature is 25 °F. Wind speed (mph) 0 5 10 15 20 25 30 35 40 45 Wind chill temperature (°F) 25 19 15 13 11 9 8 7 6 5 [Source: National Weather Service] a. What does a 20-mph wind make 25 °F b. Sketch a scatter plot of the data, with c. Explain why an exponential model would be appropriate. d. Find an exponential model for the data. e. According to the model, what is the wind-chill factor for a feel like? x 0 corresponding to 0 mph. 23-mph wind 12 y f (x, f(x)) x (b, f(b)) b Figure 5.C-1 Tangents to Exponential Functions ctions Tangent lines to a |
curve are important in calculus—where they are used to approximate function values close to a specific point and used for finding the zeros of general functions. The procedure developed in the Can Do Calculus for Chapter 3 will be used here to develop the equations of the tangent lines to exponential functions. Slopes of Secant Lines and Tangent Lines Recall that the difference quotient of f at the specific value x b is given by, where h is the amount of change in the x values from one point to another. The difference quotient can be interpreted as the slope of the secant line As h gets very b that passes through the points b h, f b h and. 1 22 approaches the value of the slope of 22 1 small, the value of f 1 the tangent line at b, f. b 1 22 1 1 f b Also, if f is any function, then the slope of the secant line through and any other point on the graph of f is given by x, f x b, f b 1 22 1 1 1 22. As the point x, f x 1 22 approaches the point 1 b, f b 1 22 1, the value of f 1 approaches the value of the slope of the tangent line to the curve at Figure 5.C-1 shows four secant lines that pass through the point The tangent line to f at is shown in red. b, f b b b, f 1 1 b 22 22 2.. 1 1 22 f Consider the function 0 0, e that pass through Tangent Lines to the Exponential Function e x 2 1 that is Find the values of e 0 and the values of the slopes of secant lines when x is near 0. x e x 1 x 0.01 0.001 0.001 0.99502 0.9995 1.0005 0.01 1.005 408 y 4 2 −4 −2 0 2 4 x −2 −4 Figure 5.C-2 The table suggests that the slope of the tangent line to is 1. The tangent line to the curve at has slope x 0 at m 1. can be found by using the point-slope form of a line. x 0 Therefore, the equation of the tangent line to x 0 x 2 contains the point (0, 1) and e x e at x f f 1 1 2 x m y y0 y 1 1 y 1 x x x02 1 x 0 2 1 Point-slope form of a |
line y x 1 Equation of tangent line e x and the tangent line to the curve at 0, 1 1 2 is shown x f The graph of 1 in Figure 5.C-2. 2 Example 1 Tangent Line to the Exponential Function Find the tangent line to line. x f 1 2 Solution e x when x 1. Graph f and the tangent x 1, f e 1 e, 1 When is the point where the tangent line will touch the graph. To find the slope of the tangent line, look at values of the difference quotient near 2 1 x 1. so the point 1 Alternately, you may use the numerical derivative feature of your calculator to find an approximate value of the slope of the tangent line for f You should find that the value of the slope of the tangent line is approximately 2.718282282. x 1. e at x 1 2 x Recall that e f at 1 tangent line’s equation is e 2.71828. 1, e x 2 1 2 x It appears that the slope of the tangent line to is e, which can be proved using calculus. Therefore, the Figure 5.C-3a 10 y e e x 1 1 2 or equivalently y ex and the graphs of f and the tangent line are in Figure 5.C-3b. ■ Calculator Exploration –5 5 –10 Figure 5.C-3b 2, 3, 4. Plot the points Find the slope of the tangent lines to 1, line, along with the corresponding points for tion would represent the graph of these points? 2, where y is the slope of the tangent x 0, 1. What func- x 3, when e x 409 In the Calculator Exploration, you should have found the values and plotted the points shown in Figure 5.C-4. Also shown in Figure 5.C-4 are all the points on the graph of f e x. x 1 2 Slope of the Tangent line to y e x The slope of the tangent line at any point on same value as the y coordinate, at that point. x, e Figure 5.C-4 y e x has the The exponential function is the only function with this characteristic. Example 2 Slope of y e x a. Find the x-value where the slope of b. Write the equation of the tangent line at c. Graph and the tangent line at y e x x 3 y ex |
3. e is x 3. on the same screen. Solution y e x a. The slope of b. Using the point-slope form of a line with x 3 tion of the tangent line to is when y e e x x 3. 3 3 m e is and 3 3, e 2 1, the equa- 5 y e 3 e y e at x 3 3 2 1 3x 2e 3 e x 2 3 1 2 c. The graphs of y e x and y e 3 x 2 2 1 are shown in Figure 5.C-5. ■ Exponential Functions with Bases Other Than e The procedure for finding the equation of the tangent line at a specific value of x for exponential functions with bases other than e is the same as that for finding the equation of the tangent line at a specific value of x for any function. 1. Find the values of slopes of secant lines by using the difference quo- tient and several values of x near the point in question. 2. Find the slope of the tangent line by determining the value of the slope suggested by the values found in Step 1. 3. Write the equation of the tangent line using the point-slope form of a linear equation. 4. Confirm your finding by graphing the function and the tangent line. 50 –5 Figure 5.C-5 –5 410 Calculator Exploration x 2, 1, Find the values of the slope of the tangent line for 0, 1, 2, 3. Graph the ordered pairs (x, slope of tangent) and find an equation to represent the graph by using exponential regression. y 2 at x The points to be graphed in the Calculator Exploration are 2, 0.17329 1, 0.34657 1 1 1 2 and the regression equation is 2 0, 0.69315 1, 1.3863 2, 2.7726 3, 5.5452 2 2 1 2 1 2 1 y 0.6931471806 x 2. 2 1 ln 2 0.6931471806. c, f on the graph of c In fact, the slope of the tangent line at or is given by ln 2 f 2 x f c, x 1 22 1 2 1 2 Notice that any point 2 ln 2. 1 c 1 2 Example 3 Tangent Line of y 2 x Find the equation of the tangent line to the curve confirm your result by graphing. y 2 x at x |
4, and 30 Solution –1 6 –5 Figure 5.C-6 is 4, 4, 2 1 11.09, 2 or (4, 16). The slope of the tanso the equation of the tangent The point on the curve at gent line at that point is line of x 4 ln 2 16 at (4, 16) is 2 21 x f 1 x 2 1 2 y 16 11.09 y 11.09 16 The graph of ure 5.C-6. x f 1 2 2 x and y 11.09 16 2 are shown in Fig- ■ Exercises In Exercises 1–4, write the equation of the tangent line f (x) ex. at the following values of x for the function Graph the function along with the tangent at each point. 5. x 0 7. x 2 6. x 1 8. x 2 1. x 0 3. x 2 2. x 1 4. x 2 In Exercises 9–12, write the equation of the tangent line at the following values of x for the function f (x) 3 Graph the function along with the tangent at each point. x. In Exercises 5–8, write the equation of the tangent line at the following values of x for the function f (x) e Graph the function along with the tangent at each point. x 2. 9. x 0 11. x 2 10. x 1 12. x 2 411 C H A P T E R 6 Trigonometry Where are we? Navigators at sea must determine their location. Surveyors need to determine the height of a mountain or the width of a canyon when direct measurement is not feasible. A fighter plane’s computer must set the course of a missile so that it will hit a moving target. Many phenomena such as the tides, seasonal change, and radio waves, have cycles that repeat. All of the situations can be described mathematically using trigonometry. 412 Chapter Outline 6.1 Right-Triangle Trigonometry 6.2 Trigonometric Applications 6.3 Angles and Radian Measure 6.4 Trigonometric Functions Interdependence of Sections 6.1 > 6.2 > 6.3 > 6.4 > > 6.5 6.5 Basic Trigonometric Identities Chapter Review can do calculus Optimization with Trigonometry T rigonometry, which means “triangle measurement,” was developed in ancient |
times for determining the angles and sides of triangles in order to solve problems in astronomy, navigation, and surveying. With the development of calculus and physics in the 17th century, a different viewpoint toward trigonometry arose, and trigonometry was used to model all kinds of periodic behavior, such as sound waves, vibrations, and planetary orbits. In this chapter, you will be introduced to both types of trigonometry, beginning with right-triangle trigonometry. 6.1 Right-Triangle Trigonometry Objectives Angles and Degree Measure • Define the six trigonometric ratios of an acute angle in terms of a right triangle • Evaluate trigonometric ratios, using triangles and on a calculator Recall from geometry that an angle is a figure formed by two rays with a common endpoint, called the vertex. The rays are called the sides of the angle. An angle may be labeled by the angle symbol and the vertex. The angle in Figure 6.1-1 may be labeled A. 1 2 side angle vertex A side Figure 6.1-1 413 414 Chapter 6 Trigonometry Angles may be measured in degrees, where 1 degree 2 angle is an entire circle, a 1 90° 360° angle is a quarter of a circle. A (See Figure 6.1-2.) Thus, a angle is half angle is also called of a circle, and a a right angle. A right angle is indicated on a diagram by a small square, is called an acute angle. as shown in Figure 6.1-3. An angle of less than The measure of an angle is indicated by the letter m in front of the angle symbol, such as mA 36°. 180° 90° 90° is 1 360 of a circle. 1° right-angle symbol Figure 6.1-2 Figure 6.1-3 Minutes and Seconds Fractional parts of a degree are usually expressed in decimal form or in 1 60 minutes and seconds. A minute of a degree, and a second 1 60 is is 2 2 1 1 of a minute, or 1 3600 of a degree. This form is often called DMS form, for degrees, minutes, seconds. Example 1 Converting Between Decimal Form and DMS Form a. Write b. Write 35° 15¿ 27– 48.3625° in decimal form. in DMS form. Solution a. 35° 15¿ 27– 35° ° a 15 27 3600b 60b 35° 0.25 |
° 0.0075° 35.2575° a ° b. First, convert the entire decimal part to minutes by writing it in terms of 1 60 of a degree. 48.3625° 48° 0.3625° 48° a 0.3625° 60 60b 21.75 a 60 b 48° ° 48° 21.75¿ Second, convert the decimal part of the minutes to seconds by writing it in terms of 1 60 of a minute. Section 6.1 Right-Triangle Trigonometry 415 48° 21.75¿ 48° 21¿ 48° 21¿ 48° 21¿ 45– 0.75¿ ¿ 60 60b 45 60b a a Similar Triangles and Trigonometric Ratios Examine the following right triangles. B a C c 34° b f e 34° D Figure 6.1- Since all right triangles that contain a angle are similar, the corresponding ratios would be the same. Thus, the ratio is dependent only on the measure of the angle. These ratios, which can be determined for any angle between are the basis of trigonometry. and 90°, 34° 0° The hypotenuse (hyp) of a right triangle is the side across from the right angle. The hypotenuse is always the longest side of the triangle. The remaining sides are labeled by their relationship to the given angle, as shown in Figure 6.1-5. The adjacent (adj) side is the side of the given angle that is not the hypotenuse, and the opposite (opp) side is the side of the triangle that is across from the given angle. C opp ∠B adj ∠A b opp ∠A adj ∠B a A c hypotenuse Figure 6.1-5 B In the figure, if site side is a. If side is b. A B is the given angle, the adjacent side is b and the oppois the given angle, the adjacent side is a and the opposite There are six possible ratios for the three sides of a triangle. These ratios are called trigonometric ratios. 416 Chapter 6 Trigonometry Trigonometric Ratios For a given acute angle U in a right triangle: The sine of written as sin U, is the ratio U, sin U opposite hypotenuse hypotenuse opposite adjacent Figure 6.1-6 NOTE The Greek letter (theta) is commonly used u to label the measure of an angle in trigon |
ometry. 13 12 Figure 6.1-7 U, The cosine of written as cos is the ratio U, cos U adjacent hypotenuse U, The tangent of written as tan U, is the ratio tan U opposite adjacent In addition, the reciprocal of each ratio above is also a trigonometric ratio. cosecant of U csc U hypotenuse opposite secant of U sec U hypotenuse adjacent cotangent of U cot U adjacent opposite 1 sin U 1 cos U 1 tan U Example 2 Evaluating Trigonometric Ratios 5 Evaluate the six trigonometric ratios of the angle u shown in Figure 6.1-7. Solution The opposite side has length 5, the adjacent side has length 12, and the hypotenuse has length 13. sin u opposite hypotenuse cos u adjacent hypotenuse 5 13 12 13 13 5 13 12 1.0833 0.9231 0.3846 2.6 csc u hypotenuse opposite sec u hypotenuse adjacent cot u adjacent opposite tan u opposite adjacent 5 12 0.4167 12 5 2.4 ■ Example 3 Evaluating Trigonometric Ratios Evaluate the six trigonometric ratios of ure 6.1-8. (Side lengths given are approximate.) 62° by using the triangle in Fig- 3 3.4 Figure 6.1-8 1.6 62° Section 6.1 Right-Triangle Trigonometry 417 Solution sin 62° opposite hypotenuse cos 62° adjacent hypotenuse 3 3.4 1.6 3.4 0.8824 0.4706 tan 62° opposite adjacent 3 1.6 1.8750 csc 62° hypotenuse opposite sec 62° hypotenuse adjacent cot 62° adjacent opposite 3.4 3 3.4 1.6 1.1333 2.1250 1.6 3 0.5333 ■ Evaluating Trigonometric Ratios Using a Calculator If the measure of an angle is given without a corresponding triangle, it may be difficult to accurately evaluate the trigonometric ratios of that angle. For example, to find sin it would be possible to draw a right and measure its sides. However, there may triangle with an angle of be inaccuracies in drawing and measuring the triangle. Tables of trigonometric ratios are available, but it is usually most convenient to use a calculator. 20°, 20° Technology Tip The following facts will be helpful in evaluating trigonometric ratios |
on a calculator. • Scientific and graphing calculators have modes for different units of angle measurements. When using degrees, make sure that your calculator is set in degree mode. • The functions on a calculator do not indicate the reciprocal functions. These functions will be discussed in Section 8.2. and tan sin 1, cos 1, 1 • Some calculators automatically insert an opening parenthesis “(” after sin, cos, or tan. Be sure to place the closing parenthesis “)” in the appropriate place. In trigonometry, many of the values used are approximate, and answers are usually rounded to 4 decimal places. However, in calculations involving trigonometric ratios, the values should not be rounded until the end of the problem. Example 4 Evaluating Trigonometric Ratios on a Calculator Evaluate the six trigonometric ratios of 20°. Solution Your calculator should have buttons for sine, cosine, and tangent. To find the cosecant, secant, and cotangent, take the reciprocal of each answer. 418 Chapter 6 Trigonometry sin 20° 0.3420 cos 20° 0.9397 tan 20° 0.3640 Figure 6.1-9 Special Angles csc 20° 1 sec 20° cot 20° sin 20° 1 cos 20° 1 tan 20° 1 0.3420 1 0.9397 1 0.3640 2.9238 1.0642 2.7475 ■ Properties of 30-60-90 and 45-45-90 triangles can be used to find exact val30°, 60°, These angles are called ues of the trigonometric ratios for special angles. and 45°. NOTE For a review of the properties of 30-60-90 and 45-45-90 triangles, see the Geometry Review in the Appendix. Example 5 Evaluating Trigonometric Ratios of Special Angles Evaluate the six trigonometric ratios of 30°, 60°, and 45°. Solution A 30-60-90 and a 45-45-90 triangle are shown below. 60° 1 2 3 30° Figure 6.1-10 45° 1 2 45° 1 In the 30-60-90 triangle: In the 45-45-90 triangle: • the hypotenuse is 2 • for the angle, 30° 1 is opposite and • for the 60° angle, 23 23 • the hypotenuse is 22 • for either 45 |
° angle, is adjacent 1 is both opposite and adjacent is opposite and 1 is adjacent The following table summarizes the values of the trigonometric ratios for the special angles. NOTE The exact values of the trigonometric ratios of the special angles will be needed regularly. You should memorize the sine and cosine values for all three angles. The other ratios are easily derived from the sine and cosine. Section 6.1 Right-Triangle Trigonometry 419 U sin u opposite hypotenuse cos u adjacent hypotenuse 30° 1 2 23 2 tan u opposite adjacent 1 23 23 3 csc u hypotenuse opposite 2 2 1 sec u hypotenuse adjacent cot u adjacent opposite 2 23 223 3 23 1 23 45 1 22 1 22 22 2 22 2 1 1 1 22 1 22 1 22 22 1 1 1 60 23 2 1 2 23 1 23 2 23 223 3 2 2 1 1 23 23 3 ■ Exercises 6.1 In Exercises 1–4, write the DMS degree measurement in decimal form. 10. 1. 47° 15¿ 36– 3. 15° 24¿ 45– 2. 38° 33¿ 9– 4. 20° 51¿ 54– In Exercises 5–8, write the decimal degree measurement in DMS form. 5. 23.16° 7. 4.2075° 6. 50.3625° 8. 85.655° In Exercises 9–14, find the six trigonometric ratios for U. 9. 11 3 θ 2 11. 12 17 15 θ 8 420 13. 14. Chapter 6 Trigonometry In Exercises 15–20, use a calculator in degree mode to find the following. Round your answers to four decimal places. 15. sin 32° 16. cos 68° 17. tan 6° 18. csc 25° 19. sec 47° 20. cot 39° In Exercises 21–26, use the exact values of the trigonoU metric ratios for the special angles to find a value of that is a solution of the given equation. (See Example 5.) 21. sin u 1 2 22. tan u 1 23. csc u 22 24. cot u 23 25. cos u 1 2 26. sec u 2 In Exercises 27–32, refer to the figure below. Find the exact value of the trigonometric ratio for the given values of a, b, and c. |
A b c C a B 27. a 4, b 2, tan B? 28. a 5, c 7, sin A? 29. b 3, c 8, cos A? 30. a 12, b 15, cot A? 31. a 7, c 16, sec B? 32. b 2, c 3, csc B? In Exercises 33–38, use a calculator in degree mode to determine whether the equation is true or false, and explain your answer. 33. sin 50° 2 sin 25° 34. sin 50° 2 sin 25° cos 25° 35. 36. 37. 38. cos 28° 2 1 2 1 sin 28° 1 cos 28° 2 1 sin 28° 2 1 1 tan 75° tan 30° tan 45° 2 2 2 2 tan 75° tan 30° tan 45° 1 tan 30° tan 45° 39. Critical Thinking Complete the table below. U 1° 0.1° 0.01° 0.001° sin U cos U???????? Based on the values in the table, what do you think would be a reasonable value for sin Verify your answers with a calculator. cos Why can’t these values be found by using the definition on page 416? 0°? 0° and 40. Critical Thinking Complete the table below. U 89° 89.9° 89.99° 89.999° sin U cos U???????? Based on the values in the table, what do you 90° think would be a reasonable value for sin Verify your answers with a calculator. cos Why can’t these values be found by using the definition on page 416? 90°? and Section 6.2 Trigonometric Applications 421 41. Critical Thinking Use the diagram below to show that the area of a triangle with acute angle that has sides a and b is u A 1 2 ab sin u. b θ h a In Exercises 42–45, use the result of Exercise 41 to find the area of the given triangle. 42. 10 25° 14 43. 44. 45. 59° 72 140 44 30° 20 12 38° 9 6.2 Trigonometric Applications Objectives • Solve triangles using trigonometric ratios • Solve applications using triangles Solving Right Triangles Many applications in trigonometry involve solving a triangle. This means finding the lengths of all three sides and the measures of all three angles when only some of these |
quantities are known. Solving right triangles by using trigonometric ratios involves two theorems from geometry: Triangle Sum Theorem: The sum of the measures of the angles in a triangle is Pythagorean Theorem: In a right triangle with legs a and b and hypotenuse c, a2 b2 c 2. 180. If the measures of two angles are known, the Triangle Sum Theorem can be used to find the measure of the third. If the lengths of two sides of a right triangle are known, the Pythagorean Theorem can be used to find the length of the third. Trigonometric ratios are used to solve right triangles when the measure of an angle and the length of a side or when the lengths of two sides are given. The underlying idea is that the definition of each trigonometric ratio involves three quantities: • the measure of an angle • the lengths of two sides of the triangle 422 Chapter 6 Trigonometry When two of the three quantities are known, the third can always be found, as illustrated in the next two examples. Example 1 Finding a Side of a Triangle Find side x of the right triangle in Figure 6.2-1. 8 65° x Figure 6.2-1 Solution 65° The be found, so use the cosine ratio. angle and the hypotenuse are known. The adjacent side x must cos 65° adjacent hypotenuse x 8 Solve this equation for x and then use a calculator to evaluate cos 65°. cos 65° x 8 x 8 cos 65° 3.3809 Multiply both sides by 8 Use a calculator ■ Example 2 Finding an Angle of a Triangle Find the measure of angle u in Figure 6.2-2. 3 4 θ 5 Figure 6.2-2 Solution Note that sin u opposite hypotenuse 3 5 0.6. Section 6.2 Trigonometric Applications 423 u Before calculators were available, was found by using a table of sine values. You can do the same thing by having your calculator generate a table for by using the settings shown in Figure 6.2-3a and Figure 6.2-3b. sin Y1 X 1 2 View the table, shown in Figure 6.2-3c, and look through the column of sine values for the closest one to 0.6. Then look in the first column for the corresponding value of The closest entry to 0.6 in the sine column is 0.60042, which corresponds to |
an angle of Hence, u 36.9°. 36.9°. u. Figure 6.2-3a Figure 6.2-3b Figure 6.2-3c A faster and more accurate method of finding SIN1 on your calculator. SIN1 key in angle whose sine is 0.6, namely, key is labeled ASIN on some models.) When you (0.6), as in Figure 6.2-3d, the calculator produces an acute u 36.8699°. is to use the 1 u SIN1 NOTE Make sure your calculator is in degree mode. Figure 6.2-3d SIN1 Thus, the sine table, without actually having to construct the table. key provides the electronic equivalent of searching the ■ NOTE In this chapter, the and TAN1 u. keys will be used as they were in Example 2 to find an angle The other uses of these keys are discussed in Section 8.2, which deals with inverse trigonometric functions. key and the analogous COS1 SIN1 Here is a summary of how these techniques can be used to solve any right triangle. 424 Chapter 6 Trigonometry Solving Right Triangles A right triangle can be solved if the following information is given. Case 1: an acute angle and a side Case 2: two sides Sketch the triangle and label the acute angle, the right angle, and the given side. Sketch the triangle and label the right angle and the two given sides. Find the remaining acute angle by subtracting the 180°. known angles from Write a trigonometric equation that has an unknown side as the variable, and solve it with a calculator to evaluate the trigonometric ratio of the angle. Repeat the previous step or use the Pythagorean Theorem to find the third side. Find the third side by using the Pythagorean Theorem. Write a trigonometric equation that has an unknown angle as the variable. If the angle is a special angle, you can solve it by recognizing the value of the trigonometric ratio. If the angle is not one of the special angles, use the technique explained in Example 2. Example 3 Solving a Right Triangle θ Solve the right triangle in Figure 6.2-4. Solution 17 a 75° b Figure 6.2-4 One side and one angle are given, so use the first case above. To solve the triangle, it is necessary to find a, and b. u, To find u |
, subtract the measures of the given angles from 180°. u 180° 75° 90° 15° Write a trigonometric equation that has a as the variable. Since a is opposite the given angle and the hypotenuse is given, the sine is used. opposite hypotenuse sin 75° a 17 a 17 sin 75° a 17 0.9659 1 Next, write a trigonometric equation that has b as the variable. Since b is adjacent to the given angle, cosine is used. Evaluate cos by using a calculator, and solve. 16.42 75° 2 θ a 12 β 6 Figure 6.2-5 Section 6.2 Trigonometric Applications 425 cos 75° b 17 b 17 cos 75° b 4.40 adjacent hypotenuse ■ Example 4 Solving a Right Triangle Solve the right triangle in Figure 6.2-5. Solution Two sides are given, so use the second case above. To solve the triangle, it is necessary to find a, and b. u, To find a, use the Pythagorean Theorem. a2 62 122 a2 108 a 2108 623 b, To find use trigonometric ratios. The adjacent side and the hypotenuse are given, so cosine is used. cos b 6 12 1 2 adjacent hypotenuse From the table of trigonometric ratios of special angles on page 419, cos 60° 1 2 is the only acute angle with a cosine of Since 60° 1 2,. b 60° and u 180° 60° 90° 30° ■ Applications The following examples illustrate a variety of applications of the trigonometric ratios. Example 5 Height Above Sea Level A straight road leads from an ocean beach at a constant upward angle of How high above sea level is the road at a point 1 mile from the beach? 3°. ocean 5 2 8 0 f t 3° r o a d Figure 6.2-6 h = height above sea level sea level 426 Chapter 6 Trigonometry Technology Tip When using a calculator to evaluate trigonometric ratios, do not round your answer until the end of the problem. The rounding error can be increased significantly as other operations are performed. It may help to store the value in the calculator’s memory, or to use the entire trigonometric expression in each calculation. Solution Figure 6.2-6 shows a right triangle with the road as the hypotenuse 1 mi 5280 ft and the opposite side h as |
the height above sea level. 1 Write a trigonometric equation that uses sine. 2 sin 3° h 5280 h 5280 sin 3° h 5280 0.0523 1 h 276.33 ft 2 opposite hypotenuse Solve for h Use a calculator to evaluate sin 3° Simplify At one mile, the road is about 276 feet above sea level. ■ Example 6 Ladder Safety According to the safety sticker on a 20-foot ladder, the distance from the bottom of the ladder to the base of the wall on which it leans should be one-fourth of the length of the ladder: 5 feet. a. How high up the wall will the ladder reach? b. If the ladder is in this position, what angle does it make with the ladder 20 ft wall h ground? Solution θ ground 5 ft Figure 6.2-7 Draw the right triangle formed by the ladder, the wall, and the ground. Label the sides and angles as shown in Figure 6.2-7. a. Since the length of the ladder and the distance from the wall are known, find the third side by using the Pythagorean Theorem. h2 52 202 h2 375 h 19.36 The ladder will safely reach a height of a little more than 19 feet up the wall. b. The hypotenuse and the side adjacent to angle are known, so use u the cosine ratio. Figure 6.2-8 cos u adjacent hypotenuse 5 20 1 4 Use the COS1 key to find that u 75.5°, as shown in Figure 6.2-8. ■ Angles of Elevation and Depression In many applications the angle between a horizontal line and another line is used, such as the line of sight from an observer to a distant object. If the line is above the horizontal, the angle is called the angle of elevation. Section 6.2 Trigonometric Applications 427 If the line is below the horizontal, the angle is called the angle of depression. Horizontal Angle of elevation Angle of depression Figure 6.2-9 Example 7 Indirect Measurement A flagpole casts a 60-foot shadow when the angle of elevation of the sun as shown in Figure 6.2-10. Find the height of the flagpole. is 35°, h 35° 60 ft Figure 6.2-10 Solution A right triangle is formed by the flagpole and its shadow. The opposite side is unknown and the adjacent side is given, so the tangent is |
used. tan 35° h 60 h 60 tan 35° h 42.012 opposite adjacent Solve for h Use a calculator Thus, the flagpole is about 42 feet high. ■ Example 8 Indirect Measurement A wire needs to reach from the top of a building to a point on the ground. The building is 10 m tall, and the angle of depression from the top of the How long should the wire be? building to the point on the ground is 22°. 428 Chapter 6 Trigonometry α 22° Angle of depression 10 Wire Figure 6.2-11 Solution Figure 6.2-11 shows that the sum of the angle of depression and the angle a formed by the wall of the building and the wire is 90°. a 90° 22° 68° The wall, wire, and ground form a right triangle where the wall is the side adjacent to and the wire is the hypotenuse. Thus, a adjacent hypotenuse cos 68° 10 w w 10 cos 68° w 26.7 m Thus, the wire should be about 27 m long. ■ Example 9 Indirect Measurement A person on the edge of a canal observes a lamp post on the other side to the top of the lamp post and an angle with an angle of elevation of of depression of to the bottom of the lamp post from eye level. The person’s eye level is 152 cm (about 5 ft). 12° 7° a. Find the width of the canal. b. Find the height of the lamp post. 152 12° 7° Figure 6.2-12 Section 6.2 Trigonometric Applications 429 Solution The essential information is shown in Figure 6.2-13 below. Note that is parallel to so CD is also 152 cm. DE, AC 152 A E 12° 7° Figure 6.2-13 B C D 152 a. The width of the canal AC is adjacent to the 7° angle, and 152 is opposite the 7° angle. tan 7° 152 AC opposite adjacent AC 152 tan 7° 1237.94 cm, or about 12.38 m wide b. The height BC can be represented in terms of the width of the canal found in part a. tan 12° BC AC opposite adjacent BC AC tan 12° tan 12° 2 2 1 263.13 cm 1 1237.94 BC CD. The height of the lamp post is BC CD 263.13 152 415.13 cm ■ Exercises 6.2 In Exercises 1–6, find side |
c in the figure below by using the given conditions. b c A C a B 2. 3. sin C 3 4 tan A 5 12 4. sec A 2 5. cot A 6 6. csc C 1.5 b 12 a 15 b 8 a 1.4 b 4.5 1. cos A 12 13 b 39 In Exercises 7–12, find the exact value of h without using a calculator. 430 7. 8. 9. 10. 11. 12. Chapter 6 Trigonometry 25 h h 45° 30° h 60° 72 150 h 45° 12 h 30° 100 20 h 60° Use the figure below for Exercises 13–24. A c B b a C In Exercises 13–16, find the indicated value without using a calculator. 13. a 4 14. c 5 15. c 10 16. a 12 mA 60° mA 60° mA 30° mA 30° Find c. Find a. Find a. Find c. In Exercises 17–24, solve the triangle with the given conditions. 17. b 10 18. c 12 19. a 6 20. a 8 21. c 5 22. c 4 23. b 3.5 24. a 4.2 mC 50° mC 37° mA 14° mA 40° mA 65° mC 28° mA 72° mC 33° In Exercises 25–28, find angle U. 25. 26. 27. 4 3 θ 12 θ 10 2 θ 3 28. θ 200 144 In Exercises 29–36, use the figure for Exercises 13–24 to find angles A and C under the given conditions. 29. a 4 and c 6 30. b 14 and c 5 31. a 7 and b 10 32. a 5 and c 3 33. b 18 and c 12 34. a 4 and b 9 35. a 2.5 and c 1.4 36. b 3.7 and c 2.2 37. A 24-ft ladder positioned against a wall forms an 75° with the ground. angle of a. How high up the wall does the ladder reach? b. How far is the base of the ladder from the wall? 38. A guy wire stretches from the top of an antenna tower to a point on level ground 18 feet from the base of the tower. The angle between the wire and the ground is How high is the |
tower? 63°. 39. A plane takes off at an angle of 5°. After traveling 1 mile along this flight path, how high (in feet) is the plane above the ground? 1 mi 5280 ft 1 2 5° 40. A plane takes off at an angle of 6° traveling at the rate of 200 feet/second. If it continues on this flight path at the same speed, how many minutes will it take to reach an altitude of 8000 feet? 41. The Ohio Turnpike has a maximum uphill slope 3°. of How long must a straight uphill segment of the road be in order to allow a vertical rise of 450 feet? 42. Ruth is flying a kite. Her hand is 3 feet above ground level and is holding the end of a 300-ft Section 6.2 Trigonometric Applications 431 kite string, which makes an angle of horizontal. How high is the kite above the ground? 57° with the 43. Suppose that a person with a reach of 27 inches and a shoulder height of 5 feet is standing upright on a mountainside that makes a the horizontal, as shown in the figure below. Can the person touch the mountain? angle with 62° 62° 44. A swimming pool is 3 feet deep in the shallow end. The bottom of the pool has a steady downward drop of 12° the pool is 50 feet long, how deep is the deep end? toward the deep end. If 45. A wire from the top of a TV tower makes an angle 49.5° of with the ground and touches the ground 225 feet from the base of the tower. How high is the tower? 46. A plane flies a straight course. On the ground directly below the flight path, observers 2 miles apart spot the plane at the same time. The plane’s angle of elevation is point and plane? from one observation from the other. How high is the 71° 46° 71° 46° 2 miles 2 miles 432 Chapter 6 Trigonometry 47. A buoy in the ocean is observed from the top of a 40-meter-high radar tower on shore. The angle of depression from the top of the tower to the base How far is the buoy from the of the buoy is base of the radar tower? 6.5°. 48. A plane passes directly over your head at an 33° QP altitude of 500 feet. Two seconds later you observe that its angle of elevation is plane travel during those 2 seconds? How far did the 42 |
°. 49. A man stands 12 feet from a statue. The angle of elevation from eye level to the top of the statue is and the angle of depression to the base of the 30°, statue is How tall is the statue? 15°. 50. Two boats lie on a straight line with the base of a lighthouse. From the top of the lighthouse, 21 meters above water level, it is observed that the angle of depression of the nearest boat is 53° the angle of depression of the farthest boat is How far apart are the boats? and 27°. 53° 27° 54. A drinking glass 5 inches tall has a 2.5-inch diameter base. Its sides slope outward at a angle as shown. What is the diameter of the top of the glass? 4° 4° 4° 55. In aerial navigation, directions are given in 90°, degrees clockwise from north, called headings. Thus east is shown below. A plane travels from an airport for 200 miles at a heading of the airport is the plane? How far west of and so on, as south is 180°, 300°. 51. A rocket shoots straight up from the launch pad. Five seconds after lift-off, an observer 2 miles away notes that the rocket’s angle of elevation is 3.5° is 4 seconds?. Four seconds after that, the angle of elevation How far did the rocket rise during those 41°. 0° North 300° 270° West 90° East 52. From a window 35 meters high, the angle of depression to the top of a nearby streetlight is The angle of depression to the base of the streetlight is. How tall is the streetlight? 57.8° 55°. Distance west of airport 180° South 53. A 60-foot drawbridge is 24 feet above water level when closed. When open, the bridge makes an with the horizontal. angle of a. How high is the tip P of the open bridge above 33° the water? b. When the bridge is open, what is the distance from P to Q? 56. A plane travels from an airport at a constant 300 65° mph at a heading of a. How far east of the airport is the plane after. (See Exercise 55.) half an hour? b. How far north of the airport is the plane after 2 hours and 24 minutes? 57. A car on a straight road passes under a bridge. Two seconds later an observer on the bridge, 20 feet above |
the road, notes that the angle of How fast, in miles depression to the car is per hour, is the car traveling? (Note: 60 mph is equivalent to 88 feet/second.) 7.4°. 58. A pedestrian overpass is shown in the figure below. If you walk on the overpass from one end to the other, how far have you walked? 15° 18 ft 21° 200 ft Section 6.3 Angles and Radian Measure 433 59. Critical Thinking A 50-ft flagpole stands on top of a building. From a point on the ground the angle of elevation to the top of the pole is and the angle of elevation to the bottom of the pole is How high is the building? 40°. 43° 60. Critical Thinking Two points on level ground are 500 meters apart. The angles of elevation from and these points to the top of a nearby hill are 52° respectively. The two points and the ground67°, level point directly below the top of the hill lie on a straight line. How high is the hill? 6.3 Angles and Radian Measure Objectives Extending Angle Measure • Use a rotating ray to extend the definition of angle measure to negative angles and angles greater than 180° • Define radian measure and convert angle measures between degrees and radians In geometry and triangle trigonometry, an angle is a static figure consisting of two rays that meet at a point. But in modern trigonometry, which will be introduced in the next section, an angle is thought of as being formed dynamically by rotating a ray around its endpoint, the vertex. The starting position of the ray is called the initial side and its final position after the rotation is called the terminal side. The amount the ray is rotated is the measure of the angle. Counterclockwise rotations have positive measure and clockwise rotations have negative measure. T er m in al Vertex T e r m i n a l Initial Initial Initial 40° 830° Figure 6.3-1 T e r m i n a l 43° Terminal Initial 312° 434 Chapter 6 Trigonometry An angle in the coordinate plane is said to be in standard position if its vertex is at the origin and its initial side is on the positive x-axis. y y Positive angle x Figure 6.3-2 x Negative angle Angles formed by different rotations that have the same initial and terminal sides are called coterminal. (See Figure 6.3-3.) For example |
, and 360° are coterminal angles. 0° Example 1 Coterminal Angles Find three angles coterminal with an angle of 60° in standard position. Solution To find an angle that is coterminal with a given angle, add or subtract a complete revolution, or To find additional angles, add or subtract any multiple of Three possible angles are shown below. 360°. 360°. 60° 360° 420° 60° 360° 300° 60° 2 360° 2 1 780° y y y 60° 60° x x 420° −300° Figure 6.3-4 Arc Length 60° 780° x ■ Recall from geometry that an arc is a part of a circle and that a central angle is an angle whose vertex is the center of the circle. The length of an arc depends on the radius of the circle and the measure of the central angle that it intercepts, as shown in Figure 6.3-5 Initial Figure 6.3-3 arc θ r center Figure 6.3-5 Section 6.3 Angles and Radian Measure 435 Arc length can be calculated by considering an arc as a fraction of the entire circle. Suppose an arc in a circle of radius r has a central angle meas- ure of u. Since there are 360° in a full circle, the arc is The circumference of the circle is 2pr, so, the length of the arc is / u 360 of the circle. / u 360 2pr upr 180 Example 2 Finding an Angle Given an Arc Length / An arc in a circle has an arc length which is equal to the radius r. Find the measure of the central angle that the arc intercepts. Solution r / r upr 180 180r upr 180 up 180 p u The central angle measure is 180 p b °, a or about 57.3°. ■ Radian Measure The angle found in Example 2 leads to another unit used in finding angle measure called a radian. Because it simplifies many formulas in calculus and physics, radians are used as a unit of angle measurement in mathematical and scientific applications. Angle measurement in radians can be described in terms of the unit circle, which is the circle of radius 1 centered at the origin, whose equation When an angle is in standard position, its initial side lies is on the x-axis and passes through Its terminal side intersects the unit circle at some point P, as shown in Figure 6.3-6. x2 y2 1 |
. 1, 0. 2 1 y P x (1, 0) Figure 6.3-6 436 Chapter 6 Trigonometry Definition of Radian Measure NOTE Generally, measurements in radians are not labeled with units, although the word radian or the abbreviation rad may sometimes be used for clarity. The radian measure of an angle is the distance traveled along the unit circle in a counterclockwise direction by the point P, as it moves from its starting position on the initial side to its final position on the terminal side of the angle. 1 radian 180 P b a 57.3 If the vertex of an angle is the center of a circle of radius r, then an angle of 1 radian intercepts an arc of length r. Movement along the unit circle is counterclockwise for positive measure and clockwise for negative measure. y distance = 3.75 y 3.75 radians x (1, 0) P (1, 0) x −2 radians P distance = 2 Figure 6.3-7 Consider an angle in standard position, with its terminal side rotating 360° around the origin. In degree measure, one full revolution produces a angle. The radian measure of this angle is the circumference of the unit circle, namely Other angles can be considered as a fraction of a full revolution, as shown in Figure 6.3-8. 2p. 1 revolution 2p radians y 1 3 4 3/4 revolution 2p 3p 2 y radians 1/2 revolution 2p p radians 1 2 −1 x 1 −1 −1 1 −1 x 1 −1 Figure 6.3-8 y 1 −1 1 4 1/4 revolution 2p p 2 y radians x 1 −1 1 −1 x 1 Section 6.3 Angles and Radian Measure 437 Radian Measure of Special Angles The special angles of tion of a full revolution. Note that and 30°, 60°, 45° can also be considered as a frac- • • • 360 12 360 6 360 8 30, so 30° is 1 12 of a complete revolution: 1 6 60, so 60° 45, so 45° is of a complete revolution: is of a complete revolution: 1 8 rad 2p p 6 30° 1 12 2p p 3 2p p 4 60° 1 6 45° 1 8 rad rad NOTE p; Radian measurements are usually given in terms of however, it is useful to know the decimal equivalents for common measurements when |
using a calculator. p 3.14 2p 6.28 p 2 p 6 p 1.57 4 p 0.52 3 1.05 0.79 y x 16π 3 = 2π + 2π + 4π 3 Figure 6.3-10 Figure 6.3-9 shows a unit circle with radian and degree measures for important values. The radian measures for the angles shown in the first quadrant and on the x- and y-axes should be memorized. 90° = π 2 2π 3 120° = 3π 4 135° = 150° = 5π 6 60° = π 3 45° = π 4 30° = π 6 180° = ππ 0° = 360° = 2ππ 210° = 7π 6 225° = 5π 4 240° = 4π 3 11π 6 330° = 7π 4 315° = 5π 3 300° = 270° = 3π 2 Figure 6.3-9 As shown in Figure 6.3-9, radians corresponds to a full revolution of the terminal side of an angle in standard position. So an angle of radian measure t is coterminal with the angles whose radian measures are t ± 2p, t ± 4p, and so on, as shown in Figure 6.3-10. 2p Increasing or decreasing the radian measure of an angle by an integer multiple of 2P results in a coterminal angle. Converting Between Degrees and Radians As shown in Figure 6.3-9, p radians 180°. 438 Chapter 6 Trigonometry y Dividing both sides by p shows that 1 1 radian 180 p b a ° 57.3°, 1 rad x which agrees with the definition of radian. (1, 0) Similarly, both sides of the original equation can be divided by 180. Figure 6.3-11 Radian/Degree Conversion p 180 ˛ radians 1° These two equations give the conversion factors for radians to degrees and degrees to radians. To convert radians to degrees, multiply by To convert degrees to radians, multiply by 180 P. P 180. Example 3 Converting From Radians to Degrees Convert the following radian measurements to degrees. a. p 5 Solution b. 4p 9 c. 6p a. p 5 180 p 36° b. 4p 9 180 p 80° c. 6 |
p 180 p 1080° Example 4 Converting From Degrees to Radians Convert the following degree measurements to radians. a. 75° Solution b. 220° c. 400° a. 75° p 180 5p 12 b. 220° p 180 11p 9 c. 400° p 180 20p 9 Arc Length and Angular Speed The formula for arc length can also be written in terms of radians. ■ ■ NOTE One radian, which is illustrated in Figure 6.3-11, is close to 60°. There are about 2p 6.28 6 radians complete circle. 1 2 in a NOTE To help you remember which conversion factor to use, it may be helpful to notice that radians are usually written in terms of p. p Degrees to radians: to get in final answer, multiply by p 180 Radians to degrees: to cancel in final answer, p multiply by 180 p Section 6.3 Angles and Radian Measure 439 Arc Length An arc with central angle measure rU U radians has length In other words, the arc length is the radius times the radian measure of the central angle of the arc. 10 9 8 12 11 7 6 1 π 2 5 2 4 Figure 6.3-12 Example 5 Arc Length The second hand on a clock is 6 inches long. How far does the tip of the second hand move in 15 seconds? 3 Solution The second hand makes a full revolution every 60 seconds, that is, it moves through an angle of radians. During a 15-second interval it will make 15 60 (Figure 6.3-12), so the tip of the second hand travels along an arc with a of a revolution, moving through an angle of p 2 1 4 radians 1 4 ˛1 2p 2p 2 central angle measure of p 2 15 seconds is the arc length. Therefore, the distance that the tip moves in ru 6˛a p 2 b 3p 9.4 inches. ■ Example 6 Central Angle Measure Find the central angle measure (in radians) of an arc of length 5 cm on a circle with a radius of 3 cm. y 5 cm 3 cm Solution x Solve the arc length formula for u. / ru / r 5 3 u radians Figure 6.3-13 This is a little more than one-quarter of a complete revolution, as shown in Figure 6.3-13. ■ Linear and Angular Speed Suppose that a wheel is rotating at a constant rate |
around its center, O, and P is a point on the outer edge of the wheel. There are two ways to measure the speed of point P, in terms of the distance traveled or in terms of the angle of rotation. The two measures of speed are called linear speed and angular speed. 440 Chapter 6 Trigonometry P Recall that the speed of a moving object is distance time. If the object is trav- O eling in a circular path with radius r, the linear speed is given by linear speed arc length ru t time and the angular speed is given by angular speed angle time u t Figure 6.3-14 u where is the radian measure of the angle through which the object travels in time t. Notice the relationship between linear speed and angular speed: NOTE The angular speed of an object traveling in a circular path is the same, regardless of its distance from the center of the circle. When the angular speed of the object stays the same, the linear speed increases as the object moves farther from the center. linear speed r angular speed Example 7 Linear and Angular Speed A merry-go-round makes 8 revolutions per minute. a. What is the angular speed of the merry-go-round in radians per minute? b. How fast is a horse 12 feet from the center traveling? c. How fast is a horse 4 feet from the center traveling? Solution a. Each revolution of the merry-go-round corresponds to a central angle of of 2p 8 2p 16p radians in one minute. radians, so the merry-go-round travels through an angle angular speed u t 16p 1 16p radians per minute b. The horse 12 feet from the center travels along a circle of radius 12. From part a, linear speed r angular speed 12 16p 192p ftmin which is about 6.9 mph. c. The horse 4 feet from the center travels along a circle of radius 4. From part a, linear speed r angular speed 4 16p 64p ftmin which is about 2.3 mph. ■ NOTE follows: The units ft/min in Example 7 can be converted to mph as 192p ft 1 min 60 min 1 hr 1 mi 5280 ft 6.9 mi 1 hr Section 6.3 Angles and Radian Measure 441 Exercises 6.3 In Exercises 1–10, find the degree and radian measure of the angle in standard position formed by rotating the terminal side by the given amount. In Exerc |
ises 47 – 52, determine the positive radian measure of the angle that the second hand of a clock travels through in the given time. of a circle 10. 55. If the radius of the circle in the figure is 20 cm 47. 40 seconds 48. 50 seconds 49. 35 seconds 50. 2 minutes 15 seconds 51. 3 minutes 25 seconds 52. 1 minute 55 seconds 53. The second hand on a clock is 6 cm long. How far does its tip travel in 40 seconds? 54. The second hand on a clock is 5 cm long. How far does its tip travel in 2 minutes and 15 seconds? 85 cm, and angle u? what is the radian measure of the y θ x 1 9 of a circle 1 18 1 36 of a circle of a circle of a circle 1. 3. 5. 7. 9. 2 3 4 5 2. 4. 6. 8. 1 24 1 72 of a circle of a circle 1 5 of a circle 7 12 5 36 of a circle of a circle In Exercises 11–22, convert the given radian measure to degrees. 11. 15. p 5 3p 4 12. p 6 16. 5p 3 19. 5p 12 20. 7p 15 13. p 10 14. 2p 5 17. p 45 21. 27p 5 18. p 60 22. 41p 6 In Exercises 23–34, convert the given degree measure to radians. Write your answer in terms of P. 23. 6° 27. 75° 24. 10° 25. 12° 26. 36° 28. 105° 29. 135° 30. 165° 31. 225° 32. 252° 33. 930° 34. 585° 56. Find the radian measure of the angle u in the preceding figure if the diameter of the circle is 150 cm and 360 cm. In Exercises 35–42, state the radian measure of an angle in standard position between 0 and that is coterminal with the given angle in standard position. 2P In Exercises 57–60, assume that a wheel on a car has radius 36 cm. Find the angle (in radians) that the wheel turns while the car travels the given distance. 35. p 3 39. 7p 5 36. 3p 4 40. 45p 8 37. 19p 4 38. 16p 3 41. 7 42. 18.5 In Exercises 43–46, find the rad |
ian measure of four angles in standard position that are coterminal with the given angle in standard position. 43. p 4 44. 7p 5 45. p 6 46. 9p 7 57. 2 meters (200 cm) 58. 5 meters 59. 720 meters 60. 1 kilometer (1000 meters) In Exercises 61–64, find the length of the circular arc with the central angle whose radian measure is given. Assume that the circle has diameter 10. 61. 1 radian 62. 2 radians 63. 1.75 radians 64. 2.2 radians 442 Chapter 6 Trigonometry The latitude of a point P on Earth is the degree measure of the angle between the point and the plane of the equator, with Earth’s center as the vertex, as shown in the figure below. U P θ Equator In Exercises 65–68, the latitudes of a pair of cities are given. Assume that one city is directly south of the other and that the earth is a perfect sphere of radius 4000 miles. Use the arc length formula in terms of degrees to find the distance between the two cities. 65. The North Pole: latitude 90° Springfield, Illinois: latitude north 40° north 66. San Antonio, Texas: latitude Mexico City, Mexico: latitude 29.5° 20° north north 67. Cleveland, Ohio: latitude Tampa, Florida: latitude 41.5° north 28° north 68. Rome, Italy: latitude 42° north Copenhagen, Denmark: latitude 54.3° north In Exercises 69–76, a wheel is rotating around its axle. Find the angle (in radians) through which the wheel turns in the given time when it rotates at the given number of revolutions per minute (rpm). Assume t 77 0 k 77 0. and 69. 3.5 minutes, 1 rpm 70. t minutes, 1 rpm 78. A circular saw blade has an angular speed of 15,000 radians per minute. a. How many revolutions per minute does the saw make? b. How long will it take the saw to make 6000 revolutions? 79. A circular gear rotates at the rate of 200 revolutions per minute (rpm). a. What is the angular speed of the gear in radians per minute? b. What is the linear speed of a point on the gear 2 inches from the center in inches per minute? in feet per minute? 80. A wheel in a large machine is |
2.8 feet in diameter and rotates at 1200 rpm. a. What is the angular speed of the wheel? b. How fast is a point on the circumference of the wheel traveling in feet per minute? in miles per hour? 81. A riding lawn mower has wheels that are 15 inches in diameter, which are turning at 2.5 revolutions per second. a. What is the angular speed of a wheel? b. How fast is the lawn mower traveling in miles 71. 1 minute, 2 rpm 72. 3.5 minutes, 2 rpm per hour? 73. 4.25 minutes, 5 rpm 74. t minutes, 5 rpm 82. A bicycle has wheels that are 26 inches in 75. 1 minute, k rpm 76. t minutes, k rpm 77. One end of a rope is attached to a circular drum of radius 2 feet and the other to a steel beam. When the drum is rotated, the rope wraps around it and pulls the object upward (see figure). Through what angle must the drum be rotated in order to raise the beam 6 feet? diameter. If the bike is traveling at 14 mph, what is the angular speed of each wheel? 83. A merry-go-round horse is traveling at 10 feet per second when the merry-go-round is making 6 revolutions per minute. How far is the horse from the center of the merry-go-round? 84. The pedal sprocket of a bicycle has radius 4.5 inches and the rear wheel sprocket has radius 1.5 inches (see figure). If the rear wheel has a radius of 13.5 inches and the cyclist is pedaling at the rate of 80 rpm, how fast is the bicycle traveling in feet per minute? in miles per hour? Section 6.4 Trigonometric Functions 443 away on the western horizon. (The figure is not to scale.) Assuming that the radius of the earth is 3950 miles, how high was the plane when the picture was taken? Hint: The sight lines from the plane to the horizons are tangent to the earth and a tangent line to a circle is perpendicular to the radius at that point. The arc of the earth between St. Louis and Cleveland is 520 miles long. Use this fact and the arc length formula to find angle Your answers will be in radians. Note that a u 2 why? u.. 2 1 85. A spy plane on a practice run over the Midwest takes a picture that shows Cleveland, Ohio, on the eastern |
horizon and St. Louis, Missouri, 520 miles St. Louis θ α Cleveland 6.4 Trigonometric Functions Objectives Extending the Trigonometric Ratios • Define the trigonometric ratios in the coordinate plane • Define the trigonometric functions in terms of the unit circle NOTE P can be any point on the terminal side of the angle, except for the origin, since different choices for P generate similar right triangles. Thus, the value of a trigonometric ratio depends only on the angle. Trigonometric ratios were defined for acute angles in Section 6.1. The next step is to develop a definition of these ratios that applies to angles of any measure. u To do this, first consider an acute angle in standard position. Choose a point P, with coordinates (x, y), on the terminal side, and draw a right triangle, as shown in Figure 6.4-1. The side adjacent to has length x and the side opposite has length y. The length of the hypotenuse, r, is the distance from the origin, which may be the Pythagorean Theorem. found by using u u x2 y2 r2 r 2x2 y2 y hypotenuse r θ adjacent x opposite y x Figure 6.4-1 444 Chapter 6 Trigonometry The trigonometric ratios can now be written in terms of x, y, and r. For example, sin u opposite hypotenuse y r and cos u adjacent hypotenuse x r Thus, the trigonometric ratios can be described without triangles by using a point on the terminal side of the angle. More importantly, this process can be carried out for any angle, not just acute angles. Therefore, the following definition applies to any angle and agrees with the previous definition when the angle is acute. Trigonometric Ratios in the Coordinate Plane y θ (−3, −2) x Figure 6.4-2 U Let be an angle in standard position and let point on the terminal side of Let r be the distance from (x, y) to the origin: P (x, y) U. be any r 2x2 y2 U Then the trigonometric ratios of are defined as follows: sin U y r cos U x r tan U y x csc U r y sec U r x cot U x y P(x, y) θ y r x Example 1 Trigonometric Ratios in the Coordinate Plane Find the sine, |
cosine, and tangent of the angle whose terminal side passes through the point 3, 2. u, 1 2 Solution Using the values sin u x 3, y 2, 2 213 cos u and 3 213 r 2 3 2 2 1 1 tan u 2 2 2 3 2 213, 2 3 ■ Trigonometric Functions Trigonometric ratios have been defined for all angles. But modern applications of trigonometry deal with functions whose domains consist of real numbers. The basic idea is quite simple: If t is a real number, then sin t is defined to be the sine of an angle of t radians; cos t is defined to be the cosine of an angle of t radians; Section 6.4 Trigonometric Functions 445 and so on. Instead of starting with angles, as was done up until now, this new approach starts with a number and only then moves to angles, as summarized below. Trigonometric Functions of Real Numbers ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ Begin with a number t > Form an angle of t radians > Determine sin t, cos t, tan t ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ Trigonometric Ratios of Angles Adapting earlier definitions of ratios to this new viewpoint produces the following definition of trigonometric functions of real numbers. Use Figure 6.4-3 for reference. Trigonometric Functions of a Real Variable Let t be a real number. Choose any point side of an angle of t radians in standard position. Then cos t x r sec t r x tan t y x cot t x y sin t y r csc t r y (x, y) on the terminal y where r 2x 2 y 2 is |
the distance from (x, y) to the origin. (x, y) r t x y x Although this definition is essential for developing various facts about the trigonometric functions, the values of these functions are usually approximated by a calculator in radian mode, as shown in Figure 6.4-4. Figure 6.4-3 NOTE Unless stated otherwise, use radian mode when evaluating trigonometric functions of real numbers. Figure 6.4-4 Trigonometric Functions and the Unit Circle Recall that the unit circle is the circle of radius 1 centered at the origin, whose equation is The unit circle is the basis for the most useful description of trigonometric functions of real numbers. x2 y2 1. Let t be any real number. Construct an angle of t radians in standard posibe the point where the terminal side of this angle meets tion. Let the unit circle, as shown in Figure 6.4-5. x, y P 2 1 446 Chapter 6 Trigonometry NOTE arc from 1 The length of the to P is t. 1, 0 2 y 1 t P(x, y) −1 x 1 −1 Figure 6.4-5 P, r 1, The distance from P to the origin is 1 because the unit circle has radius 1. Using the point and the definition of trigonometric functions of real numbers shows the following: sin t y r y and cos t x r x y 1 x 1 Unit Circle Description of Trigonometric Functions Let t be a real number and let P be the point where the terminal side of an angle of t radians in standard position meets the unit circle. Then P has coordinates (cos t, sin t) and tan t y x sec t 1 x sin t cos t 1 cos t cot t x y csc t 1 y cos t sin t 1 sin t Graphing Exploration With your calculator in radian mode and parametric graphing mode, set the range values as follows: 0 t 2p 1.8 x 1.8 1.2 y 1.2 Then, graph the curve given by these parametric equations: x cos t y sin t The graph is the unit circle. Use the trace to move around the circle. At each point, the screen will display three numbers: the values of t, x, and y. For each t, the cursor is on the point where the terminal side of an angle of t radians meets the unit circle, so the corresponding x |
is the number cos t and the corresponding y is the number sin t. Section 6.4 Trigonometric Functions 447 The coordinates of points on a circle of radius r that is centered at the origin can be written by using r and t with the definition of the trigonometric ratios in the coordinate plane. See Exercise 61. Domain and Range By the domain convention in Section 3.1, the domain of a function is all real numbers for which the function is defined. For any real number t, an appropriate angle of t radians and its intersection point with the unit circle are always defined, so the domain of the sine function and of the cosine function is the set of all real numbers. The range of a function is the set of all possible outputs. Because sin t and cos t are the coordinates of a point on the unit circle, they take on all values between and 1 and no other values. Thus, 1 the range of the sine function and of the cosine function is the set of all real numbers between and 1, that is, the interval tan t y x, all points on the unit circle except (0, 1) and The tangent function is defined as whenever 0, 1 x 0,. 1 [1, 1]. that is, for 1 2 The point (0, 1) is on the terminal side of an angle of p 2 radians or any angle obtained by adding integer multiples of 2p p (a complete circle) to it, that is,, 7p 2, 3p 2 9p 2 5p 2, p p 2,,, The point (0, 1) is on the terminal side of an angle of radians or any 3p 2 to it, that is, p angle obtained by adding integer multiples of 2 p, 5p 2, p 2 11p 2 3p 2 7p 2,,,, p Combining these facts shows that the domain of the tangent function consists of all real numbers except P 2 2kP, where k 0, 1, 2, 3, p. In contrast to sine and cosine, the range of the tangent function is the set of all real numbers. A proof of this fact is found in Exercise 60. Figure 6.4-6 shows that values of the tangent can be very large positives, very large negatives, or in between. Signs of the Trigonometric Functions It is often important to know whether the value of a trigonometric function is |
positive or negative. For any real number t, the point (cos t, sin t) is on the terminal side of an angle of t radians in standard position. The Figure 6.4-6 448 Chapter 6 Trigonometry π 2 < t < π sin t + cos t − tan t − 3π 2 π < t < sin t − cos t − tan t + y π 0 < t < 2 sin t + cos t + tan t + 3π 2 < t < 2π sin t − cos t + tan t − Figure 6.4-7 quadrant in which this point lies determines the signs of sine and cosine, as well as those of the other trigonometric functions, as summarized in Figure 6.4-7. x Exact Values of Trigonometric Functions Although a calculator is used to evaluate trigonometric functions approximately, there are a few special numbers for which exact values can be found. Recall that 30°, 45°, and 60° are the same as p 6, p 4, and p 3, respec- tively. Therefore, the chart on page 419 can be translated as follows. t sin t cos t tan t csc t sec t cot t P 6 1 2 23 2 1 23 23 3 2 2 1 2 23 223 3 23 1 23 P 4 22 2 1 22 1 22 22 2 1 22 1 22 1 22 22 P 3 23 2 1 2 23 2 23 223 3 2 2 1 1 1 1 1 23 23 3 The exact values of the trigonometric functions can also be found for any number that is an integer multiple of p 6, p 4, and p 3. The technique for doing this depends on the concept of a reference angle. Example 2 Exact Values of Trigonometric Functions Find the exact value of the sine, cosine, and tangent functions when p 2 3p 2 and, p, 2p., t 0, Solution If the measure of an angle is a multiple of p 2, then its terminal side lies on an axis. Thus, the only possible values of the sine and cosine functions NOTE The diagram of of such angles are 0 and 1. The following chart shows the value of the sine, cosine, and tangent functions for these angles between 0 and 2p. 1, Section 6.4 Trigonometric Functions 449 the unit circle shown in Figure 6.4-8 will help you memorize the values of sin |
t p 2 and cos t for multiples of. Definition of Reference Angle t 0 p 2 p 3p 2 2p sin t cos t tan undefined 0 undefined 0 Reference Angles y (0, 1) (−1, 0) x (1, 0) (0, −1) Figure 6.4-8 ■ U For an angle in standard position, the reference angle is the positive acute angle formed by the terminal side of and the x-axis. U y t′ = t y In the following figure, the reference angle standard position is shown in two ways. t¿ for an angle of t radians in Definition of Reference Angle y t t = t′ x t′ y t x y t x t′ x t′ t′ = − tπ t′ = t − π t′ = 2 − t π Unit Circle with Reference Angle Placed in Quadrant I P(x, y) t = t′ x Q(−x, y) t′ y t P(x, y) t′ x y t P(x, y) t′ x t′ Q(−x, −y) Figure 6.4-9 y t t′ t′ P(x, y) x Q(x, −y) 450 Chapter 6 Trigonometry In every case, the figure that references the unit circle illustrates the following fact that can be proved by using congruent triangles: x-coordinate of Q ± y-coordinate of Q ± x-coordinate of P y-coordinate of P 2 2 1 1 By definition, the values of the trigonometric functions for t are given by the coordinates of Q and the values of these functions for are given by the coordinates of P. So, these values will be the same, except for a plus or minus sign. The correct sign is determined by the quadrant in which the terminal side of an angle of t radians lies, as shown in Figure 6.4-7 on page 448. t¿ Finding Trigonometric Function Values To find the sine, cosine, or tangent of t radians, • Sketch an angle of t radians in standard position and determine the quadrant in which the terminal side lies. • Find the reference angle, which has measure • Find the sine, cosine, and tangent of t appropriate sign. t and append the radians. Example 3 Using Reference Angles Use reference angles to find the exact value of sin |
t, cos t, and tan t. a. t 3p 4 b. t 4p 3 c. t 11p 6 x Solution a. Sketch the angle, as shown in Figure 6.4-10. The terminal side is in the second quadrant, so the reference angle is p 3p 4 p 4 p t. y 3π 4 π 4 Figure 6.4-10 Because the terminal side of the angle of 3p 4 radians lies in the second quadrant, sin 3p 4 is positive, and cos 3p 4 and tan 3p 4 are negative. sin tan 3p 4 4p 3 sin p 4 22 2 cos 3p 4 cos p 4 22 2 tan p 4 1 b. Sketch the angle, as shown in Figure 6.4-11. The terminal side is in quadrant III, so the reference angle is t p. 4p 3 p p 3 y 4π 3 x π 3 Figure 6.4-11 Section 6.4 Trigonometric Functions 451 Thus, the sine, cosine, and tangent functions are sin tan 4p 3 3p 4 sin p 3 23 2 cos 4p 3 cos p 3 1 2 tan p 3 23 c. Sketch the angle, as shown in Figure 6.4-12. The terminal side is in quadrant IV, so the reference angle is 2p 11p 6 p 6 2p t. Thus, the sine, cosine, and tangent functions are sin tan 11p 6 11p 6 sin tan p 6 p 6 11p 6 cos p 6 23 2 cos 1 2 23 3 ■ y 11π 6 x π 6 Figure 6.4-12, it is possible to find a coterIf an angle is less than 0 or greater than 2p. minal angle between 0 and Thus, the trigonometric functions of a real variable have the following property. by adding or subtracting multiples of 2p 2p Trigonometric Ratios of Coterminal Angles Any trigonometric function of a real number t is equal to the same trigonometric function of all numbers where k is an integer. t 2kP, Example 4 Trigonometric Functions Where t 7 2P Find the sine, cosine, and tangent of 7p 3. y 7π 3 π 3 x Figure 6.4-13 Solution 7p 3 can be written as p 3 2p. Therefore, is coterminal with |
p 3. 7p 3 23 2 1 2 sin cos tan 7p 3 7p 3 7p 3 sin cos tan p 3 p 3 p 3 23 ■ 452 Chapter 6 Trigonometry Exercises 6.4 Note: Unless stated otherwise, all angles are in standard position. In Exercises 1–6, find sin t, cos t, and tan t when the terminal side of an angle of t radians passes through the given point. 1. 4. 2, 7 2 4, 3 1 1 2. 5. 2 3, 2 2 23, 10 1 1 3. 6. 5, 6 1 2 p, 2 2 1 2 In Exercises 7–10, find sin t, cos t, and tan t when the terminal side of an angle of t radians passes through the given point on the unit circle. 7. 9. 2 25 a, 1 25b 3 5, 4 5b a 8. 1 210 a, 3 210b 10. 1 0.6, 0.8 2 In Exercises 11–14, identify an angle that is coterminal with the given angle, and find the sine and cosine of the given angle. 0 t P 11. 13p 6 12. 9p 2 13. 16p 14. 7p 4 In Exercises 15–23, a. Use a calculator in radian mode to find the sine, cosine, and tangent of each number. Round your answers to four decimal places. b. Use the signs of the functions to identify the quadrant of the terminal side of an angle of t radians. If the terminal side lies on an axis, identify which axis and whether it is on the positive or negative side of the axis. Explain your reasoning. 15. 7p 5 18. 23p 21. 9.5p 16. 11 19. 10p 3 22. p 17 17. 14p 9 20. 6.4p 23. 17 In Exercises 24 – 29, sketch each angle whose radian measure is given and find its reference angle. 24. 7p 3 25. 17p 6 26. 6p 5 27. 1.75p 28. 3p 4 29. p 7 In Exercises 30–47, find the exact value of the sine, cosine, and tangent of the number without using a calculator. 30. 34. 7p 6 5p 4 31. 7p 3 35. 3p 2 |
32. 17p 3 33. 11p 4 36. 3p 37. 23p 6 38. 11p 6 39. 19p 3 40. 10p 3 41. 15p 4 42. 25p 4 46. p 43. 5p 6 47. 4p 44. 17p 2 45. 9p 2 In Exercises 48 – 53, write the expression as a single real number. Do not use decimal approximations. 48. sin p 6 b a cos p 2 b a cos 49. cos p 2 b a cos p 4 b a sin p 6 b a p 2 b a sin sin p 2 b a p 4 b a 50. cos 2p 3 b a cos p sin 2p 3 b a sin p 51. sin 3p 4 b a cos 5p 6 b a cos 3p 4 b a sin 5p 6 b a 52. sin 7p 3 b a cos 5p 4 b a cos 7p 3 b a sin 5p 4 b a 53. sin p 3 b a cos p sin p cos p 3 b a In Exercises 54–59, the terminal side of an angle of t radians lies in the given quadrant on the given line. Find sin t, cos t, and tan t. (Hint: Find a point on the terminal side of the angle.) 54. Quadrant III; line 2y 4x 0 55. Quadrant IV; line through 3, 5 1 and 1 2 9, 15 2 56. Quadrant III; line through the origin parallel to 7x 2y 6 57. Quadrant II; line through the origin parallel to 2y x 6 58. Quadrant I; line through the origin perpendicular to 3y x 6 59. Quadrant IV; line y 3x 60. The terminal side of an angle of t radians lies on a straight line through the origin, and therefore, has an equation of the form slope of the line. where m is the y mx, y y = mx t x m tan t. a. Prove that Hint: a point on the terminal side of the angle has coordinates x, mx b. Explain why tan t approaches infinity as t 1 2 approaches p 2 from below. Hint: What happens to the slope of the terminal side when t is close to p 2? c. Explain why tan t approaches negative infinity as t approaches from above. Hint: When t is p 2 p 2 Section 6.4 Tr |
igonometric Functions 453 61. The figure below shows an angle of t radians. Use trigonometric functions to write the coordinates of point P in terms of r and t. y r P t x 62. Complete the following table by writing each value as a fraction with denominator 2 and a radical in the numerator. You may find the resulting pattern an easy way to remember these function values. t sin t cos t 0 2? 2 2? 2 P 6 2? 2 2? 2 P 4 2? 2 2? 2 P 3 2? 2 2? 2 P 2 2? 2 2? 2 63. Find the domain and range of the cosecant function. 64. Find the domain and range of the secant function. 65. Find the domain and range of the cotangent a bit larger than, is the slope of its terminal function. side positive or negative? d. Use parts b and c to show that the range of the tangent function is the set of all real numbers. 66. Critical Thinking Using only the definition and no calculator, determine which number is larger: sin(cos 0) or cos(sin 0). 454 Chapter 6 Trigonometry 6.5 Basic Trigonometric Identities Objectives • Develop basic trigonometric identities The algebra of trigonometric functions is just like that of other functions. They may be added, subtracted, composed, etc. However, two notational conventions are normally used with trigonometric functions. NOTE Most calculators automatically insert an opening parenthesis when a trigonometric function key is pushed. The display cos is interpreted as 1 cos. If you want cos 1 5 3, parenthesis after the 5: cos 1 you must insert Technology Tip Calculators do not use the convention of writing an exponent between the trigonometric function and its argument. In order to obtain you must enter sin sin3 4, 4 ^3. 1 2 Parentheses can be omitted whenever no confusion can result. Figure 6.5-1 shows, however, that parentheses are needed to distinguish cos 1 t 3 2 and cos t 3. Figure 6.5-1 When dealing with powers of trigonometric functions, exponents (other than 1 ) are written between the function symbol and the variable. For example, Furthermore, cos t 2 1 3 is written cos3t. sin t3 means sin 1 t3 2 not sin t 2 1 3 or sin3 t, as illustrated in Figure 6.5-2. |
Figure 6.5-2 Identities Trigonometric functions have numerous relationships that can be expressed as identities. An identity is an equation that is true for all val- Section 6.5 Basic Trigonometric Identities 455 ues of the variables for which every term of the equation is defined. For example, a b 1 2 2 a2 2ab b2 is an identity because it is true for all possible values of a and b. The unit circle description of trigonometric functions (see the box on page 446) leads to the following quotient identities. Quotient Identities tan t sin t cos t cott cos t sin t Example 1 Quotient Identities Simplify the expression below. tan t cos t Solution By the quotient identity, tan t cos t sin t cos t ˛cos t sin t ■ Reciprocal Identities The reciprocal identities follow immediately from the definitions of the trigonometric functions. sin t 1 csc t csc t 1 sin t cos t 1 sec t sec t 1 cos t tan t 1 cot t cot t 1 tan t Example 2 Reciprocal Identities Given that sin t 0.28 and cos t 0.96, find csc t and sec t. Solution By the reciprocal identities, csc t 1 sin t 1 0.28 3.57 sec t 1 cos t 1 0.96 1.04 ■ Reciprocal Identities CAUTION An identity may not be true for a value of the variable that makes a term of the equation undefined. For example, if while cot t is undefined. tan t 1 cot t tan t 0 t 0, then for Thus, t 0. 456 Chapter 6 Trigonometry y 1 P(cos t, sin t) Pythagorean Identities t x 1 −1 0 −1 Figure 6.5-3 Pythagorean Identities For any real number t, the coordinates of the point P where the terminal side of an angle of t radians meets the unit circle are (cos t, sin t), as shown in Figure 6.5-3. Since P is on the unit circle, its coordinates must satisfy x2 y2 1, which is the equation of the unit circle. That is, cos2 t sin2 t 1 This identity, which is usually written is called the Pythagorean identity. It can be used as follows to derive two other identities, which are also called Pythagorean identities. sin2 t cos2 t |
1, sin2 t cos2 t sin2 t cos2 t 1 1 cos2 t cos2 t cos2 t tan2 t 1 sec2 t Divide by cos2 t Simplify Similarly, dividing both sides of sin2 t cos2 t 1 by sin2 t shows that 1 cot 2 t csc2 t sin2 t cos2 t 1 tan2 t 1 sec2 t 1 cot2 t csc2 t In addition to the version shown above, the following forms of the Pythagorean identity are also commonly used. sin2 t 1 cos2 t cos2 t 1 sin2 t Example 3 Pythagorean Identities Simplify the expression below. tan2 t cos2 t cos2 t Solution By the quotient and Pythagorean identities, tan2 t cos2 t cos2 t sin2 t cos2 t ˛ cos2 t cos2 t sin2 t cos2 t 1 ■ Periodicity Identities Let t be any real number. Construct two angles in standard position of measure t and radians, as shown in Figure 6.5-4. Since both of t 2p Section 6.5 Basic Trigonometric Identities 457 these angles have the same terminal side, the point P where the terminal side intersects the unit circle is the same for both angles. y y P (cos t, sin t) P (cos (t + 2π), sin (t + 2π)) t x t + 2π x Figure 6.5-4 In both cases, the sine is the y-coordinate of P, so sin t sin 1 t 2p. 2 In addition, the terminal side of the angle is the same for measures of t, t ± 2p, t ± 6p, t ± 4p, sin t sin 1 and so on. Thus, t ± 4p sin 1 2 t ± 2p sin 1 2 t ± 6p 2 p Similarly in both cases, the cosine is the x-coordinate of P, so cos 1 cos t cos 1 cos 1 t ± 2p t ± 4p t ± 6p 2 2 2 p The identities above show that sine and cosine functions repeat their values at regular intervals. Such functions are called periodic. A function is said to be periodic if there exists some constant k such that 1 for every number t in the domain of f. The smallest value of k that has this property is called the period of the function f. 2 2 1 f t |
f t k Since the tangent function is the quotient of the sine and cosine functions,. However, there is a number it must also be true that 2 2p that has this property. Figure 6.5-5 shows the angles t and smaller than t p. A rotation of so the image of the point (x, y) is radians is the same as a rotation of tan t tan 1 x, y t 2p Thus, 180°, p y (x, y) t x t + π (−x, −y) Figure 6.5-5 tan tan t Calculator Exploration Use your calculator to verify the following: 3 4p sin sin 3 sin 2 1 1 4 6p cos cos 4 cos 1 2 1 tan 1 tan 1 5p tan 1 1 3 2p 4 2p 1 p 2 2 2 2 458 Chapter 6 Trigonometry Periodicity Identities The sine and cosine functions are periodic with period For every real number t, 2P. sin (t 2P) sin t and cos (t 2P) cos t P. The tangent function is periodic with period number t in the domain of the tangent function, For every tan (t P) tan t Example 4 Periodicity Identities Find the exact value of sin 13p 6. Solution By the periodicity identity for sine, sin 13p 6 sin p 6 a 12p 6 b sin p 6 a 2p b sin p 6 1 2 ■ Negative Angle Identities Let t be any real number and construct two angles in standard position of measure t and radians, as shown in Figure 6.5-6. t y (cos t, sin t) 1 P −1 t −t x 1 (cos (−t), sin (−t)) −1 Q Figure 6.5-6 Since the point Q is the reflection of the point P across the x-axis, the x-coordinates of P and Q are the same, and the y-coordinates are opposites of each other. Thus, cos t cos and sin t sin t t 1 2 1 2 Also, tan t 2 1 sin cos t 2 t 2 1 1 sin t cos t sin t cos t tan t Section 6.5 Basic Trigonometric Identities 459 Negative Angle Identities sin(t) sin t cos(t) cos t tan(t) tan t Example 5 Negative Angle Identities Find the exact value of sin p 6 b a and of cos p |
6 b a. Solution By the negative angle identities, sin p a 6 b sin p 6 1 2 and cos p a 6 b cos p 6 23 2 ■ y Other Identities Q(−x, y) P(x, y) π − t t t x p t Let t be any real number. Figure 6.5-7 shows the angles of t and radians in standard position. The terminal side of the angle of t radians meets the unit circle at P, and the terminal side of the angle of radians meets the unit circle at Q. Congruent triangles can be used to prove what the figure illustrates: p t The y-coordinates of P and Q are the same, and their x-coordinates are opposites. Figure 6.5-7 This leads to the following identities. Identities Involving P t sin t sin(P t) cos t cos(P t) tan t tan(P t) Example 6 Identities Involving P t Find the exact value of sin 5p 6 b. a NOTE The identity sin t sin p t is used in solving basic trigonometric equations. (See Section 8.3.) 1 2 Solution By the identity p t sin t, sin 1 sin 2 6p 6 a sin a 5p 6 b p 6 b sin p p 6 b a sin p 6 b a 1 2 ■ 460 Chapter 6 Trigonometry Summary of Identities Quotient Identities: tan t sin t cos t cot t cos t sin t Reciprocal Identities: sin t 1 csc t csc t 1 sin t cos t 1 sec t sec t 1 cos t tan t 1 cot t cot t 1 tan t Pythagorean Identities: sin2 t cos2 t 1 tan2 t 1 sec2 t 1 cot2 t csc2 t Periodicity Identities: sin (t 2P) sin t cos(t 2P) cos t tan(t P) tan t Negative Angle Identities: sin (t) sin t cos(t) cos t tan (t) tan t Identities Involving sin t sin(P t) : P t cos t cos(P t) tan t tan(P t) Exercises 6.5 In Exercises 1–4, use the quotient and reciprocal identities to simplify the given expression. remaining five trigonometric functions. Round your answers to four decimal places. 1. cot t sin t 3 |
. csc t sin t 2. tan t cot t 4. cot t sec t In Exercises 5–8, use the Pythagorean identities to simplify the given expression. 5. sin 2 ˛t cot 2 ˛t sin 2 t 6. 1 sec2 ˛ t 7. 8. csc 2 ˛t cot 2 sin 2 t ˛t sin 2 ˛t cos 2 sin 2 t ˛t sin 2 t In Exercises 9–14, the value of one trigonometric func- tion is given for 0 6 t 6 p 2 and Pythagorean identities to find the values of the. Use quotient, reciprocal, 9. sin t 0.3251 10. cos t 0.4167 11. tan t 3.6294 12. sec t 2.5846 13. csc t 6.2474 14. cot t 1.8479 In Exercises 15–25, use basic identities and algebra to simplify the expression. Assume all denominators are nonzero. 15. 16. 17. sin t cos t 1 sin t cos t 1 2 sin t cos t 21 2 2 sin t tan t 18. 1 tan t 2 tan t 3 1 2 21 6 tan t 2 2 tan t 19. 4 cos2 t sin2 t b a a sin t 4 cos tb 2 20. 21. 22. 5 cos t sin2 t sin2 t sin t cos t sin2 t cos2 t cos2 t 4 cos t 4 cos t 2 sin2 t 2 sin t 1 sin t 1 23. 1 cos t sin t tan t 24. 1 tan2 t 1 tan2 t 2 sin2 t 25. 2sin3 t cos t 2cos t Recall that a function is even if f(x) f(x) and a function is odd if Section 6.5 Basic Trigonometric Identities 461 39. sin 2p t 2 1 41. tan t 43. tan 2p t 2 1 40. cos t 42. cos 44. sin t 2 1 p t 1 2 In Exercises 45–50, cos t 2 5 and p 66 t 66 3p 2. Use basic identities and the signs of the trigonometric functions in each quadrant to find each value. 45. sin t 46. tan t 47. cos 49. sin 2p t 1 4p t 2 2 1 48. |
cos 50. tan t 2 4p t 1 1 2 In Exercises 51–54, it is given that sin p 8 32 22 2 f(x) f(x) Use basic identities to find each value. for every value of x in the domain of f. In Exercises 26–32, use the negative angle identities to determine whether the function is even, odd, or neither. 26. 28. 30. 32 sin t tan t t sin t t cos t 27. 29. 31 cos t sec t t tan t In Exercises 33–36, use the Pythagorean identities to find sin t for the given value of cos t. Make sure that the sign is correct for the given quadrant. 33. cos t 0.5 34. cos t 3 210 35. cos t 1 2 36. cos t 2 25 p 6 t 6 3p 3p 2 6 t 6 2p In Exercises 37–44, sin t 3 5 identities and the signs of the trigonometric functions in each quadrant to find each value. 0 66 t 66 p 2. Use basic and 37. sin t 2 1 38. sin 1 t 10p 2 51. cos p 8 53. sin 17p 8 52. tan p 8 54. tan 15p 8 In Exercises 55–60, use the Pythagorean identities to determine if it is possible for a number t to satisfy the given conditions. 55. sin t 5 13 and cos t 12 13 56. sin t 2 and cos t 1 57. sin t 1 and cos t 1 58. sin t 1 22 and cos t 1 22 59. sin t 1 and tan t 1 60. cos t 8 17 and tan t 15 8 61. Use the periodicity identities for sine, cosine, and tangent to write periodicity identities for cosecant, secant, and cotangent. 62. Use the negative angle identities for sine, cosine, and tangent to write negative angle identities for cosecant, secant, and cotangent Important Concepts Section 6.1 Section 6.2 Section 6.3 Section 6.4 462 Angle.................................. 413 Vertex....................... |
.......... 413 Degree................................. 414 Hypotenuse............................. 415 Opposite............................... 415 Adjacent............................... 415 Trigonometric ratios...................... 416 Sine................................... 416 Cosine................................. 416 Tangent................................ 416 Cosecant............................... 416 Secant................................. 416 Cotangent.............................. 416 Special angles........................... 418 Solving a triangle........................ 421 Triangle Sum Theorem.................... 421 Pythagorean Theorem..................... 421 Angle of elevation...... |
.................. 426 Angle of depression...................... 426 Initial side of an angle.................... 433 Terminal side of an angle.................. 433 Standard position of an angle............... 434 Coterminal............................. 434 Arc length.............................. 434 Radian................................. 435 Unit circle.............................. 435 Trigonometric ratios in the coordinate plane... 444 Trigonometric functions of a real variable..... 445 Unit circle descriptions of trigonometric functions............................. 446 Reference angle.......................... 449 Trigonometric ratios of coterminal angles...... 451 Chapter Review 463 Section 6.5 Identity................................ 454 Quotient identities....................... 455 Reciprocal identities...................... 455 Pythagorean identities..................... 456 Period of a function.............. |
........ 457 Periodicity identities...................... 458 Negative angle identities.................. 459 Important Facts and Formulas For a given acute angle u: sin u csc u opposite hypotenuse hypotenuse opposite cos u sec u adjacent hypotenuse hypotenuse adjacent tan u cot u opposite adjacent adjacent opposite U sin U cos U tan U csc U sec U cot U 30° 60° 45° 1 2 23 2 22 2 23 2 1 2 22 2 23 3 23 2 223 3 223 3 2 1 22 22 23 23 3 1 To convert radians to degrees, multiply by To convert degrees to radians, multiply by 180 p. p 180. Quotient Identities: tan t sin t cos t cot t cos t sin t Reciprocal Identities: sin t 1 csc t csc t 1 sin t cos t 1 sec t sec t 1 cos t tan t 1 cot t cot t 1 tan t Pythagorean Identities: sin2 t cos2 t 1 tan2 1 sec2 t 1 cot2 t sec2 t 464 Chapter Review Important Facts and Formulas Periodicity Identities: sin t ± 2p sin t 1 2 Negative Angle Identities: cos t ± 2p 1 2 cos t tan 1 t ± p 2 tan t sin t sin t 1 2 Identities Involving p t : cos t 2 1 cos t tan t 2 1 tan t 2 1 sin t sin p t 2 1 cos t cos p t 2 1 tan t tan p t 2 1 Review Exercises Section 6.1 1. Write 41° 6¿ 54– in decimal form. 2. Write 10.5625° in DMS form. 3. Which of the following statements about the angle u is true? a. c. e. sin u 3 4 tan u 3 5 sin u 4 3 b. d. cos u 5 4 sin u 4 5 4 5 3 θ In Exercises 4–9, use the right triangle in the figure to find each ratio. 4. sin u 7. csc u 5. cos u 8. sec u 6. tan u 9. cot u 7 θ 4 10. Find the length of side h in the triangle, given that angle A measures 40° and the |
distance from C to A is 25. B h C A Section 6.2 In Exercises 11–14, solve triangle ABC. Chapter Review 465 11. A 40° b 10 12. C 35° a 12 13. A 56° a 11 14 15. From a point on level ground 145 feet from the base of a tower, the angle of elevation to the top of the tower is 57.3°. How high is the tower? 16. A pilot in a plane at an altitude of 22,000 feet observes that the angle of depression to a nearby airport is the point on the ground directly below the plane? 26°. How many miles is the airport from 17. A lighthouse keeper 100 feet above the water sees a boat sailing in a straight line directly toward her. As she watches, the angle of depression to the boat changes from How far has the boat traveled during this time? 40°. 25° to Section 6.3 18. 9p 5 radians ___? degrees 19. 17p 12 radians ___? degrees 20. 11p 4 radians ___? degrees 21. 36° ___? radians 22. 220° ___? radians 23. 135° ___? radians 24. Find a number between 0 and u 2p in standard position is coterminal with an angle of such that an angle of 23p 3 u radians radians in standard position. 25. Through how many radians does the second hand of a clock move in 2 minutes and 40 seconds? 26. 10 revolutions per minute ___? radians per minute 27. 4p radians per minute ___? revolutions per minute Section 6.4 28. If the terminal side of an angle of t radians in standard position passes through the point 1 2, 3, then 2 tan t ___?. 29. If the terminal side of an angle of t radians in standard position passes through the point 1 6, 8, then 2 cos t ___?. 30. If the terminal side of an angle of t radians in standard position passes through the point (1.2, 3.5), then sin t ___?. In Exercises 31–42, give the exact values. 31. cos 34. cos 47p 2 3p 4 32. sin 13p 1 2 35. tan 8p 3 33. sin 7p 6 36. sin 7p 4 b a 466 Chapter Review 37. cot 4p 3 38. cos p 6 b a 40. |
sin 11p 6 b a 41. sin p 3 39. sec 42. csc 2p 3 5p 2 43. The value of cos t is negative when the terminal side of an angle of t radians in standard position lies in which quadrants? In Exercises 44–46, express as a single real number (no decimal approximations allowed). 44. cos 3p 4 sin 5p 6 sin 3p 4 cos 5p 6 45. sin a p 6 2 1 b 46. sin p 2 b a sin 0 cos 0 Section 6.5 47. Write tan t cot t entirely in terms of sin t and cos t, then simplify. 48. 3 sin S 2 p 5500b T a S 3 cos 2 p 5500b T a? 49. Which of the following could possibly be a true statement about a real number t? a. and cos t sin t 2 and cos t 1 22 sin t 1 2 2 sin t 1 and cos t 1 sin t p 2 sin t 3 5 and cos t 4 5 and cos t 1 p 2 b. c. d. e. 50. If p 2 6 t 6 p and sin t 5 13, then cos t? 51. If sin t 4 5 and the terminal side of an angle of t radians in standard position lies in the third quadrant, then cos t ___?. 52. Simplify 53. If sin tan sin t p 2 1 t 2p 2. 1 101p a 2 b 1, then sin 105p 2 b a? 54. Which of the statements (i)–(iii) are true? sin x cos x tan x x 2 x 2 x 2 (i) sin 1 (ii) cos 1 (iii) tan 1 (i) and (ii) only (ii) only (i) and (iii) only a. b. c. d. all of them e. none of them 55. Suppose is a real number. Consider the right triangle with sides as shown Chapter Review 467 u in the figure. Then: a. b. c. cos u sin u d. e. none of the above sin θ 2 cos θ 56. Determine the following segment lengths in terms of a single trigonometric function of t. a. OR b. PR c. SQ d. OQ Figure 6.C-1 Optimization with Trigonometry Optimization problems involve finding a solution that is either a maximum or a minimum value |
of a function. Calculus is needed to find exact solutions to most optimization problems, but tables or graphs can often be used to find approximate solutions. Example 1 Maximum Area A gutter is to be made from a strip of metal 24 inches wide by bending up the sides to form a trapezoid, as shown in Figure 6.C-1. a. Express the area of the cross-section of the gutter as a function of the angle t. b. For what value of t will this area be as large as possible? Solution a. The cross-section of the gutter is a trapezoid, shown in Figure 6.C-2. b2 8 h x t t 8 The bases are parallel, so these alternate interior angles are equal. = 8 b1 Figure 6.C-2 The area of a trapezoid is b1 h 1 b22 2. The top base b2 8 2x, where The height is h, where. cos t x 8 So x 8 cos t. So h 8 sin t. sin t h 8. Thus, the area of the cross-section is A 8 sin t 1 8 8 2 2 1 8 cos t 22 4 sin t 1 16 16 cos t 64 sin t 1 cos t. 2 1 2 b. To find the value of t that makes the area be as large as possible, first notice that t must be between 0 and p 2 1.57. By examining a table of values, it is possible to estimate the maximum value of A over this interval. 468 NOTE Trigonometric functions often have maxima and minima at p. fractional multiples of It is a good idea to try these values as increments for a table. 6 ft t d1 – t π 2 d2 8 ft A good starting interval for the table is p 12 0.26. The table in Figure 6.C-3 shows the highest value at about 1.05, or 4p 12 p 3. Figure 6.C-3 Figure 6.C-4 It is possible to confirm this value or get a better estimate by using a table with a smaller step size, such as firms that p 3 area, about p 144 appears to be the value of t that corresponds to the largest 83.1 in2. Figure 6.C-4 con- 0.02. ■ Example 2 Maximum Length Two corridors meet at a right angle. One corridor is 6 ft wide, and the other is 8 ft wide. A |
ladder is being carried horizontally along the corridor. What is the maximum length of a ladder that can fit around the corner? Solution The length of the longest ladder that fits around the corner is the same as the shortest length of the red segment in Figure 6.C-5 as it pivots about the corner. Let the part of the segment from the corner to the opposite and the part to the wall of the 8-ft corridor wall of the 6-ft corridor be be Then the desired length is d1, d2. d2. d1 Figure 6.C-5 For the angle t in Figure 6.C-5, sin t 6 d1 6 d1 sin t sin p 2 a t b 8 d2 d2 8 p 2 Q t R sin The function that describes the desired length is L 6 sin t 8 p 2 Q. t R sin 469 To find the minimum of the function, note that t is between 0 and p 2. Construct a table with an increment of p 12 to construct a table with an increment of, then use the minimum value p 144. The minimum appears to be at t 0.74, which corresponds to a length of about 19.7 ft. Figure 6.C-6 ■ In Examples 1 and 2, an area and a length were represented in terms of an angle to find the optimal solution. In the following example, the angle is the quantity to be maximized. Example 3 Maximum Viewing Angle The best view of a statue is where the viewing angle is a maximum. In Figure 6.C-7, the height of the statue is 24 ft and the height of the pedestal is 8 ft. Find the distance from the statue where the viewing angle is optimal. t eye level 5 ft Figure 6.C-7 Solution In Figure 6.C-8, the angle The important quantities are opposite and adjacent to the angles, so the tangent function is used to describe the relationship. t t1 t2. tan t1 27 d tan t2 3 d tan˛ t1 1 27 d t2 tan˛ 1 3 d t tan˛ 1 27 d tan˛ 1 3 d statue 24 ft pedestal 8 ft t2 t1 t d Figure 6.C-8 eye level 5 ft 470 d 7 0. To create a table, first notice that Start with a large increment, such as 5 ft, and narrow the increment to refine your estimate, as shown in Figure 6.C-9. Figure 6 |
.C-9 The best distance to view the statue is at about 9 ft away. The viewing angle at this distance is about 0.93 radians, which is about 53°. ■ Exercises Estimate the maximum value of the given function between 0 and P 2 by using tables with increments of P 12 1. 3. and P 144. f t 2 1 f t 2 1 sin t cos t 2. f t 2 1 sin t 2 cos t 3 sin t sin p 2 a t b 4. f t 2 1 2 cos t 1 1 sin t 5. The cross section of a tunnel is a semicircle with radius 10 meters. The interior walls of the tunnel form a rectangle. y t 10 10 x a. Express the area of the rectangular cross-section of the tunnel opening as a function of angle t. b. For what value of t is the cross-sectional area of the tunnel opening as large as possible? What are the dimensions of the tunnel opening in this case? 6. A 30-ft statue stands on a 10-ft pedestal. Find the best distance to view the statue, assuming eye level is 5 ft (see Example 3). 7. Two towns lie 10 miles apart on opposite sides of a mile-wide straight river, as shown. A road is to be built along one side of the river from town A to point X, then across the river to town B. The cost of building on land is $10,000 per mile, and the cost of building over the water is $20,000 per mile. a. Express the cost of building the road as a function of the angle t. b. Find the minimum cost of the road. A 1 mi road X t 10 mi B 471 C H A P T E R 7 Trigonometric Graphs Stay tuned for more! Radio stations transmit by sending out a signal in the form of an electromagnetic wave that can be described by a trigonometric function. The shape of this signal is modified by the sounds being transmitted. AM radio signals are modified by varying the “height,” or amplitude, of the waves, whereas FM signals are modified by varying the frequency of the waves. The signal displayed in the photo is from an AM radio station found at 900 on the broadcast dial. See Exercise 65 of Section 7.3. 472 Chapter Outline 7.1 Graphs of the Sine, Cosine, and Tangent Functions 7.2 Graphs of the Cosec |
ant, Secant, and Cotangent Functions 7.3 Periodic Graphs and Amplitude 7.4 Periodic Graphs and Phase Shifts 7.4.A Excursion: Other Trigonometric Graphs Chapter Review can do calculus Approximations with Infinite Series Interdependence of Sections > 7.2 > 7.3 7.1 > 7.4 G raphs of trigonometric functions often make it very easy to see the essential properties of these functions, particularly the fact that they repeat their values at regular intervals. Because of the repeating, or peri- odic, nature of trigonometric functions, they are used to model a variety of phenomena that involve cyclic behavior, such as sound waves, elec- tron orbitals, planetary orbits, radio transmissions, vibrating strings, pendulums, and many more.• 7.1 Graphs of the Sine, Cosine, and Tangent Functions Objectives • Graph the sine, cosine, and tangent functions • State all values in the domain of a basic trigonometric function that correspond to a given value of the range • Graph transformations of the sine, cosine, and tangent graphs Although a graphing calculator will quickly sketch the graphs of the sine, cosine, and tangent functions, it will not give you much insight into why these graphs have the shapes they do and why these shapes are important. So the emphasis in this section is the connection between the functions’ definitions and their graphs. Using radians and the unit circle, you learned in Chapter 6 that trigonometric functions can be defined as functions of real numbers. Using this definition, you will see that the graphs of trigonometric functions are directly related to angles in the unit circle. Graph of the Sine Function Consider an angle of t radians in standard position. Let P be the point where the terminal side of the angle meets the unit circle. Then the sin t y-coordinate of P is the number sin t. As t increases, the graph of can be sketched from the corresponding y-coordinates of P. t f 1 2 473 474 Chapter 7 Trigonometric Graphs Change in t from 0 to p 2 Movement of point P sint (y-coordinate of P) Corresponding graph from (1, 0) to (0, 1) increases from 0 to 1 (0, 1) P t (1, 0) y y y y 1 − |
1 1 −1 1 −1 1 −1 t t t t π 2 π 2π 3π 2 π 2 π 2π 3π 2 π 2 π 2 π 2π 3π 2 π 2π 3π 2 from p 2 to p from 0, 1 to 1 2 1 1, 0 2 decreases from 1 to 0 (0, 1) t P (−1, 0) from p to 3p 2 from 1, 0 1 to 1 2 0, 1 2 decreases from 0 to 1 (−1, 0) t P (0, −1) from 3p 2 to 2p from 0, 1 1 to 1 2 1, 0 2 increases from 1 to 0 t (1, 0) P (0, −1) CAUTION Throughout this chapter, the independent variable used for trigonometric functions will be t to avoid any confusion with the x and y that are part of the definition of these functions. However, using a graphing calculator in function mode, you must enter x as the independent variable. Section 7.1 Graphs of the Sine, Cosine, and Tangent Functions 475 Your graphing calculator can provide a dynamic view of the graph of the sine function and its relationship to points on the unit circle. Graphing Exploration With your graphing calculator in parametric mode, set the viewing window as shown below, with a t-step of 0.1. 0 t 2p p 3 x 2p 2.5 y 2.5 On the same screen, graph the two functions given below. sin t cos t, Y1 t, Y2 sin t X1 X2 Use the trace feature to move the cursor along the first graph, which is the unit circle. Stop at a point, and note the values of t and y. Use the up or down key to move the cursor to the second graph, which is the graph of the sine function. The value of t will remain the same. What are the x- and y-coordinates of this point? How does the y-coordinate of the new point compare with the y-coordinate of the original point on the unit circle? To complete the graph of the sine function, note that as t goes from 4p, the point P on the unit circle retraces the path it took from 0 to 2p the same curve will repeat on the graph. This repetition occurs each along the horizontal axis, therefore the sine function has |
a period of That is, for any real number t, 2p to 2p, so units 2p. t ± 2p sin 1 sin t. 2 y 1 h(t) = sin t t −4π −3π −2π −π 0 π 2π 3π 4π −1 Graph of the Cosine Function Graph of the Sine Function Let P be the point where the terminal side of an angle of t radians in standard position meets the unit circle. Then the x-coordinate of P is the the same process as number To obtain the graph of cos cos t. t t f, 1 2 1 2 476 Chapter 7 Trigonometric Graphs that used for the sine function is followed, except the x-coordinate is observed. The following chart illustrates the graph of the cosine function. Movement of point P cos t (x-coordinate of P) Corresponding graph from (1, 0) to (0, 1) decreases from 1 to 0 y Change in t from 0 to p 2 from p 2 to p (0, 1) P t (1, 0) from (0, 1) to 1, 0 2 decreases from 0 to 1 1 (0, 1) P t (−1, 0) from p to 3p 2 from 1, 0 1 to 1 2 0, 1 2 increases from 1 to 0 (−1, 0) t P (0, −1) from 3p 2 to 2p from 0, 1 1 2 to (1, 0) increases from 0 to 1 t (1, 0) P (0, −1) 1 −1 1 −1 1 −1 1 − 3π 2 2π π 2 π 2 π 2 π 3π 2 2π π 3π 2 2π π 3π 2 2π As the value of t increases, the point P on the unit circle retraces its path t along the unit circle, so the graph of repeats the same curve 2 Because the cosine function also has a period of at intervals of length 2p, for any number t, cos 2p. t f 1 2 1 cos 1 t ± 2p 2 cos t. Section 7.1 Graphs of the Sine, Cosine, and Tangent Functions 477 Graph of the Cosine Function y 1 0 −1 h(t) = cos t π 2π 3π 4π t −4π −3π −2 |
π −π The graphs of the sine and cosine functions visually illustrate two basic facts about these functions. Because the graphs extend infinitely to the right and to the left, the domain of the sine and cosine functions is the set of all real numbers. Also, the y-coordinate of every point on these graphs lies between and 1 (inclusive), so that 1 the range of the sine and cosine functions is the interval [1, 1]. You can use the period of the function to state all values of t for which sin t is a given number, as shown in Examples 1 and 2. cos t or Example 1 Finding All t-values State all values of t for which sin t is 1. Solution (i.e., and 1 and has a period of units on the horizontal axis), so there are an These points occur every units on the horizontal axis, and a few are highlighted in red on the graph The sine function oscillates between it repeats the pattern every infinite number of t-values for which sin t is 2p y sin t shown in Figure 7.1.1. 1. 2p 2p 1 y 1 0 −1 3π 2 t π 2π 3π 4π −4π −3π −2π −π 2π 2π 2π Figure 7.1-1 478 Chapter 7 Trigonometric Graphs On the interval y sin t 3 2 has only one point, 0, 2p, highlighted in red on the graph above, the graph of 1. t 3p 2 Therefore, all values of t for which 2kp, where k is any integer. 3p 2 a, 1 b sin t, at which the y-coordinate is is 1 can be expressed as ■ Example 2 Finding All t-values State all values of t for which cos t is 1 2. Solution The cosine function repeats its pattern of y-values at intervals of 2p, so there are an infinite number of t-values for which y cos t 1 2 of. shown in Figure 7.1-2 highlights a few points with a y-coordinate cos t is The graph of. 1 2 1 −4π −3π −2π −π 0 −1 y π 3 y = 1 2 π 2π 3π 4π t 5π 3 On the interval y cos t of 0, 2p 2 3 has two points, Figure 7.1-2, highlighted in red on the graph |
above, the graph p 3, 1 2 b a and 5p 3, 1 2 b a, at which the y-coordinate 1. is 2 t p 3 Therefore, all values of t for which cos t is 1 2 can be expressed as 2kp or 5p 3 2kp, where k is any integer. ■ Graph of the Tangent Function f a connection between To determine the shape of the graph of the tangent function and slope can be used. As shown in Figure 7.1-3, the point P where the terminal side of an angle of t radians in standard position meets the unit circle has coordinates This point and the point (0, 0) can be used to compute the slope of the line containing the terminal side. cos t, sin t t. 2 1 2 1 tan t, y 1 P (cos t, sin t) −1 t x 1 −1 Figure 7.1-3 Section 7.1 Graphs of the Sine, Cosine, and Tangent Functions 479 slope sin t 0 cos t 0 sin t cos t tan t The graph of minal side of an angle of t radians, as t takes different values. can be sketched by noting the slope of the ter- f t 2 1 tan t Change in t from 0 to p 2 Movement of terminal side from horizontal upward toward vertical tan t (terminal side slope) increases from 0 in the positive direction Corresponding graph t from 0 to p 2 from horizontal downward toward vertical decreases from 0 in the negative direction t π− 2 π− 2 π 2 π 2 When t ± p 2, the terminal side of the angle is vertical, so its slope is not defined. The graph of the tangent function has vertical asymptotes at the values of t for which the function is undefined. To complete the graph of the tangent function, note that as t goes from p 2 the terminal side goes from almost vertical with negative slope 3p 2 to, to almost vertical with positive slope, exactly as it does from So the graph repeats this pattern at intervals of length p. p 2 to p 2. 480 Chapter 7 Trigonometric Graphs Graph of the Tangent Function −2π −π − 3π 2 − π 2 h(t) = tan t π 2 π 3π 2 t 2π y 4 2 0 −2 −4 Notice that the domain of the tangent function is all real numbers except odd multi |
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