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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 th... |
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 par... |
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?... |
(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 ... |
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... |
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 Logarith... |
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 Si... |
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 g... |
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 ... |
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. t... |
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.... |
. 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 ... |
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 Exam... |
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 ... |
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 p... |
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,... |
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 log... |
. 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 l... |
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... |
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... |
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 10,68... |
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... |
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 ... |
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.... |
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 lo... |
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 o... |
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. (Sourc... |
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... |
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 ... |
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 o... |
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... |
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 f... |
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 str... |
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 det... |
.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 ... |
. 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, Loga... |
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 ... |
............ 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 ex... |
.......... 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... |
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 simplif... |
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 ... |
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... |
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 −... |
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 de... |
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... |
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 tange... |
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. ■ Exp... |
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 ... |
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... |
° 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-Triangl... |
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 hypotenus... |
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 ... |
° 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 value... |
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 2... |
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 kn... |
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... |
, 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 ha... |
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 safet... |
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 lo... |
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 1... |
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 min... |
°. 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 wate... |
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 f... |
, 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 an... |
. 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 cir... |
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 2... |
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... |
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... |
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 second... |
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 mea... |
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 ... |
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 ang... |
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... |
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 ... |
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 ... |
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 − c... |
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.... |
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 ... |
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,... |
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 ... |
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 pa... |
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 ... |
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 identi... |
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 ... |
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 a... |
. 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 ... |
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 ... |
.......... 413 Degree................................. 414 Hypotenuse............................. 415 Opposite............................... 415 Adjacent............................... 415 Trigonometric ratios...................... 416 Sine................................... 416 Cosine................................... |
.................. 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............. |
........ 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 ... |
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... |
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... |
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. Exp... |
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 ... |
.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 si... |
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 ve... |
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 ... |
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-c... |
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 an... |
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 ... |
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