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All Textbook Solutions for Precalculus
Suppose that $10,000 is invested and at the end of 5 yr, the value of the account is $13,771.28. Use the model A=Pert to determine the average rate of return r under continuous compounding.On January 1, 2000, the population of Texas was approximately 21 million. On January 1, 2010, the population was 25.2 million. a. Write a function of the form Pt=P0ekt to represent the population Pt of Texas t years after January 1, 2000. Round k to 5 decimal places. b. Use the function in part (a) to population on January 1, 2020. Round to 1 decimal place. c. Use the function in part (a) to determine the year during which the population of Texas will reach 40 million if this trend continues.a. Given Pt=10,00020.4t, write the rule for this function using base e. b. Find the function value for t=10 for both forms of the function from part (a).Use the function Qt=Q0e0.000121t to determine the age of a piece of wood that has 42 of its carbon-14 remaining. Round to the nearest 10 yr.The score on a test of dexterity is by Pt=1001+19e0.354x , where x is the number of times the is taken. a. Determine the initial score. b. Use the function to determine the minimum number of items required for the score to exceed 90. c. What is the limiting of the scores?For the given data, a. Graph the data points. b. Use a graphing utility to find a model form y=abx to fit the data.For the given data, a. Graph the data points. b. Use a graphing utility to find a model of the form y=a+blnx to fit the data.If k0, the equation y=y0ekt is a model for exponential (growth/decay), whereas if k0, the equation is model for exponential (growth/decay).A function defined by y=abx can be written in terms of an exponential function base e as .A function defined by y=c1+aebt is called a growth model and imposes a limiting value on y.Given a logistic function y=c1+aebt , the limiting value of y is .For Exercises 5-14, solve for the indicated variable. (See Example 1) Q=Q0ektforkusedinchemistryFor Exercises 5-14, solve for the indicated variable. (See Example 1) N=N0e0.025tfortusedinchemistryFor Exercises 5-14, solve for the indicated variable. (See Example 1) M=8.8+5.1logDforDusedinastronomyFor Exercises 5-14, solve for the indicated variable. (See Example 1) logE12.2=1.44MforEusedingeologyFor Exercises 5-14, solve for the indicated variable. (See Example 1) pH=logH+forH+usedinchemistryFor Exercises 5-14, solve for the indicated variable. (See Example 1) L=10logII0forIusedinmedicineFor Exercises 5-14, solve for the indicated variable. (See Example 1) A=P1+rtfortusedinfinanceFor Exercises 5-14, solve for the indicated variable. (See Example 1) A=PertforrusedinfinanceFor Exercises 5-14, solve for the indicated variable. (See Example 1) InkA=ERTforkusedinchemistry14PESuppose that $12,000 is invested in a bond fund and the account grows to $14,309.26 in 4 yr. (See Example 2) a. Use the model A=Pert to determine the average rate of return under continuous compounding. Round to the nearest tenth of a percent. b. How long will it take the investment to reach $20,000 if the rate of return continues? Round to the nearest tenth of a year.Suppose that $50,000 from a retirement account is invested in a large cap stock fund. After 20 yr. the value is $194.,809.67. a. Use the model A=Pert to determine the average rate of return under continuous compounding. Round to the nearest tenth of a percent. b. How long will it take the investment to reach one-quarter million dollars? Round to the nearest tenth of a year.Suppose that P dollars in principal is invested in an account earning 3.2 interest compounded continuously. At the end of 3 yr, the amount in the account has earned $806.07 in interest. a. Find the original principal. Round to the nearest dollar. b. Using the original principal from part (a) and the model A=Pert , determine the time required for the investment to reach $10,000 . Round to the nearest year.Suppose that P dollars in principal is invested in an account earning 2.1 interest compounded continuously. At the end of 2 yr, the amount in the account has earned $193.03 in interest. a. Find the original principal. Round to the nearest dollar. b. Using the original principal from part (a) and the model A=Pert , determine the time required for the investment to reach $6000 . Round to the nearest year.The populations of two countries are given for January 1, 2000, and for January 1, 2010. a. Write a function of the form Pt=P0ekt to model each population Pt (in millions) t years after January 1, 2000. (See Example 3) b. Use the models from part (a) to approximate the population on January 1, 2020, for each country. Round to the nearest hundred thousand. c. Australia had fewer people than Taiwan in the year 2000, yet from the result of part (b), Australia would have more people in the year 2020? Why? d. Use the models from part (a) to predict the year during which each population would reach 30 million if this trend continues.The populations of two countries are given for January 1, 2000, and for January 1, 2010. a. Write a function of the form Pt=P0ekt to model each population Pt (in millions) t years after January 1, 2000. b. Use the models from part (a) to approximate the population on January 1, 2020 for each country. Round to the nearest hundred thousand. c. Israel had fewer people than Switzerland in the year 2000, yet from the result of part (b), Israel would have more people in the year 2020? Why? d. Use the models from part (a) to predict the year during which each population would reach 10 million if this trend continues.A function of the form Pt=abt represents the population (in millions) of the given country t years after January 1, 2000. (See Example 4) a. Write an equivalent function using base e; that is, write a function of the form Pt=P0ekt . Also, determine the population of each country for the year 2000. b. The population of the two given countries is very close for the year 2000, but their growth rates are different. Use the model to approximate the year during which the population of each country reached 5 million. c. Costa Rica had fewer people in the year 2000 than Norway. Why would Costa Rica reach a population of 5 million sooner than Norway?A function of the form Pt=abt represents the population (in millions) of the given country t years after January 1, 2000. a. Write an equivalent function using base e; that is, write a function of the form Pt=P0ekt . Also, determine the population of each country for the year 2000. b. The population of the two given countries is very close for the year 2000, but their growth rates are different. Use the model to approximate the year during which the population of each country would reach 10.5 million. c. Haiti had fewer people in the year 2000 than Sweden. Why would Haiti reach a population of 10.5 million sooner?23PEFor Exercises 23-24, refer to the model Qt=Q0e0.000121t used in Example 5 for radiocarbon dating. At the "Marmes Man� archeological site in southeastern Washington State, scientists uncovered the oldest human remains yet to be found in Washington State. A sample from a human bone taken from the site showed that 29.4 of the carbon-14 still remained. How old is the sample? Round to the nearest year.The isotope of plutonium 238Pu is used to make thermoelectric power sources for spacecraft. Suppose that a space probe was launched in 2012 with 2.0 kg of 238Pu . a. If the half-life of 238Pu is 87.7 yr, write a function of the form Qt=Q0ekt to model the quantity Qtof238Pu left after t years. b. If 1.6 kg of 238Pu is required to power the spacecraft’s data transmitter, for how long after launch would scientists be able to receive data? Round to the nearest year.Technetium-99 99mTc is a radionuclide used widely in nuclear medicine.99mTc is combined with another substance that is readily absorbed by a targeted body organ. Then. special cameras sensitive to the gamma rays emitted by the technetium are used to record pictures of the organ. Suppose that a technician prepares a sample of 99mTc -pyrophosphate to image the heart of a patient suspected of having had a mild heart attack. a. At noon, the patient is given 10 mCi (millicuries) of 99mTc . If the half-life of 99mTc is 6 hr, write a function of the form Qt=Q0ekt to model the radioactivity level Qt after t hours. b. At what time will the level of radioactivity reach 3 mCi? Round to the nearest tenth of an hour.Fluorodeoxyglucose is a derivative of glucose that contains the radionuclide fluorine-18 18F. A patient is given a sample of this material containing 300 MBq of 18F (a megabecquerel is a unit of radioactivity). The patient then undergoes a PET scan (positron emission tomography) to detect areas of metabolic activity indicative of cancer. After 174 min, one-third of the original dose remains in the body. a. Write a function of the form Qt=Q0ekt to model the radioactivity level Qt of fluorine-18 at a time t minutes after the initial dose. b. What is the half-life of 18F ? Round to the nearest minute.Painful bone metastases are common in advanced prostate cancer. Physicians often order treatment with strontium-89 89Sr , a radionuclide with a strong affinity for bone tissue. A patient is given a sample containing 4 mCi of 89Sr . a. If 20 of the 89Sr remains in the body after 90 days, write a function of the form Qt=Q0ekt to model the amount Qt of radioactivity in the body t days after the initial dose. b. What is the biological half-life of 89Sr under this treatment? Round to the nearest tenth of a day.Two million E. coli bacteria are present in a laboratory culture. An antibacterial agent is introduced and the population of bacteria Pt decreases by half every 6 hr. The population can be represented by Pt=2,000,00012t/6. a. Convert this to an exponential function using base e. b. Verify that the original function and the result from part (a) yield the same result for P0,P6,P12,andP60. (Note: There may be round-off error.)The half-life of radium-226 is 1620 yr. Given a sample of 1 g of radium-226, the quantity left Qt (in g) after t years is given by Qt=12t/1620. a. Convert this to an exponential function using base e. b. Verify that the original function and the result from part (a) yield the same result for Q0,Q1620,andQ3240. (Note: There may be round-off error.)The population of the United States Pt (in millions) since January 1, 1900, can be approximated by Pt=7251+8.295e0.0165t where t is the number of year since January 1, 1900. (See Example 6) a. Evaluate P0 and interpret its meaning in the context of this problem. b. Use the function to approximate the U.S. population on January 1, 2020. Round to the nearest million. c. Use the function to approximate the U.S. population on January 1, 2050. d. From the model, during which year would the U.S. population reach 500 million? e. What value will the term 8.295e0.0165t approach as t? f. Determine the limiting value of Pt.The population of Canada Pt (in millions) since January 1, 1900, can be approximated by Pt=55.11+9.6e0.02515t where t is the number of years since January 1, 1900. a. Evaluate P0 and interpret its meaning in the context of this problem. b. Use the function to approximate the Canadian population on January 1, 2015. Round to the nearest tenth of a million? c. Use the function to approximate the Canadian population on January 1, 2040. d. From the model, during which year would the Canadian population reach 45 million? e. What value will the term 9.6e0.02515t approach as t? f. Determine the limiting value of Pt.The number of computers Nt (in millions) infected by a computer virus can be approximated by Nt=2.41+15e0.72t where t is the time in months after the virus was first detected. a. Determine the number of computers initially infected when the virus was first detected. b. How many computers were infected after 6 months? Round to the nearest hundred thousand. c. Determine the amount of time required after initial detection for the virus to affect 1 million computers. Round to the nearest tenth of a month. d. What is the limiting value of the number of computers infected according to this model?After a new product is launched the cumulative sales Stin$1000t weeks after launch is given by St=721+9e0.36t a. Determine the cumulative amount in sales 3 weeks after launch. Round to the nearest thousand. b. Determine the amount of time required for the cumulative sales to reach $70,000 . c. What is the limiting value in sales?For Exercises 35-38, a graph of data is given. From visual inspection, which model would best fit the data? Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogisticFor Exercises 35-38, a graph of data is given. From visual inspection, which model would best fit the data? Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogisticFor Exercises 35-38, a graph of data is given. From visual inspection, which model would best fit the data? Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogisticFor Exercises 35-38, a graph of data is given. From visual inspection, which model would best fit the data? Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogisticFor Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.For Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+bInxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.For Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.For Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.For Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.For Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.For Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.For Exercises 39-46, a table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from y=mx+blineary=abxexponentialy=a+blnxlogarithmicy=c1+aebxlogistic b. Use a graphing utility to find a function that fits the data.During a recent outbreak of Ebola in western Africa, the cumulative number of cases y was reported t months after April 1. (See Example 7) a. Use a graphing utility to find a model of the form y=abt. Round a to 1 decimal place and b to 3 decimal places. b. Write the function from part (a) as an exponential function of the form y=aebt. c. Use either model to predict the number of Ebola cases 8 months after April 1 if this trend continues. Round to the nearest thousand. d. Would it seem reasonable for this trend to continue indefinitely? e. Use a graphing utility to find a logistic model y=c1+aebt . Round a and c to the nearest whole number and b to 2 decimal places. f. Use the logistic model from part (e) to predict the number of Ebola cases 8 months after April 1. Round to the nearest thousand.The monthly costs for a small company to do business has been increasing over time due in part to inflation. The table gives the monthly cost yin$ for the month of January for selected years. The variable t represents the number of years since 2016. a. Use a graphing utility to find a model of the form y=abt . Round a to the nearest whole unit and b to 3 decimal places. b. Write the function from part (a) as an exponential function with base e. c. Use either model to predict the monthly cost for January in the year 2023 if this trend continues. Round to the nearest hundred dollars.The age of a tree t (in yr) and its corresponding height Ht are given in the table. (See Example 8) a. Write a model of the form Ht=a+bInt. Round a and b to 2 decimal places. b. Use the model to predict the age of a tree if it is 25 ft high. Round to the nearest year. c. Is it reasonable to assume that this logarithmic trend will continue indefinitely? Why or why not?The sales of a book tend to increase over the short-term as word-of-mouth makes the book “catch on.� The number of books sold Nt for a new novel t weeks after release at a certain book store is given in the table for the first 6 weeks. a. Find a model of the form Nt=a+bInt . Round a and b to 1 decimal place. b. Use the model to predict the sales in week 7. Round to the nearest whole unit. c. Is it reasonable to assume that this logarithmic trend will continue? Why or why not?A van is purchased new for $29,200. a. Write a linear function of the form y=mt+b to represent the value y of the vehicle t years after purchase. Assume that the vehicle is depreciated by $2920 per year. b. Suppose that the vehicle is depreciated so that it holds only 80 of its value from the previous year. Write an exponential function of the form y=V0bt, where V0 is the initial value and t is the number of years after purchase. c. To the nearest dollar, determine the value of the vehicle after 5 yr and after 10 yr using the linear model. d. To the nearest dollar, determine the value of the vehicle after 5 yr and after 10 yr using the exponential model.A delivery truck is purchased new for $54,000 . a. Write a linear function of the form y=mt+b to represent the value y of the vehicle t years after purchase. Assume that the vehicle is depreciated by $6750 per year. b. Suppose that the vehicle is depreciated so that it holds 70 of its value from the previous year. Write an exponential function of the form y=V0bt, where V0 is the initial value and t is the number of years after purchase. c. To the nearest dollar, determine the value of the vehicle after 4 yr and after 8 yr using the linear model. d. To the nearest dollar, determine the value of the vehicle after 4 yr and after 8 yr using the exponential model.Why is it important to graph a set of data before trying to find an equation or function to model the data.54PE55PEExplain how to convert an exponential expression bt to an exponential expression base e.57PESuppose that a population follows a logistic growth pattern, with a limiting population N. If the initial population is denoted by P0,andt is the amount of time elapsed, then the population P can be represented by P=P0NP0+NP0ekt. where k is a constant related to the growth rate. a. Solve for t (note that there are numerous equivalent algebraic forms for the result). b. Interpret the meaning of the resulting relationship.Convert 9628 to decimal degrees. Round to 2 decimal places.Convert 225.24 to DMS (degree-minute-second) form. Round to the nearest second if necessary.Convert 124 to radians. Give the answer in exact form in terms of .Convert 73.8 to radians. Round to 2 decimal places.Convert 58 radians to decimal degrees. Round to 1 decimal place if necessary.For Exercises 6-8, find a positive angle coterminal with the given angle. 48For Exercises 6-8, find a positive angle coterminal with the given angle. 110For Exercises 6-8, find a positive angle coterminal with the given angle. 35For Exercises 9-11, find a negative angle coterminal with the given angle. 74For Exercises 9-11, find a negative angle coterminal with the given angle. 56For Exercises 9-11, find a negative angle coterminal with the given angle. 126Find an angle between 0 and 360 that is coterminal to 745 .13REA municide with a wheel diameter of 24in . moves through an angle of 140 . What distance does a point on the edge of the wheel move? Round the answer to the nearest tenth of an inch.A pulley is 1.5ft in diameter. Find the distance the load will move if the pulley is rotated 750 . Find the exact distance in terms of and then round the answer to the nearest tenth of a foot.Seattle, Washington 47.61N,122.33W , and San Francisco, California 37.78N,122.42W , have approximately the same longitude, which means that they are roughly due north-south of each other. Use the difference in latitude to approximate the distance between the cities assuming that the radius of the Earth is 3960mi . Round the answer to the nearest mile.A spinning disk has radius of 10in . and rotates at 2800rpm . For a point at the edge of the disk a. Find the exact value of the angular speed. b. Find the linear speed. Round the answer to the nearest inch per minute.A bicycle has wheels 26 inches in diameter. If the wheels turn at 90rpm , what is the linear speed in inches per minute? Give the exact speed and an approximation to the nearest inch per minute.A sprinkler rotates through an angle of 75 spraying water outward for a distance of 6ft . Find the exact area watered, then round the result to the nearest tenth of a square foot.A round pie 10in. in diameter is cut into a slice with a 30 angle. Find the exact area of the slice, then round the result to the nearest tenth of a square inch.The real number t corresponds to the point P357,27 on the unit circle. Evaluate each expression. a. sint b. cost c. tant d. csct e. sect f. cottIdentify the ordered pair on the unit circle corresponding to each real number t . a. t=23 b. t=54 c. t=116Identify the domain for each pair of functions. a. ft=sint,gt=cost b. ft=tant,ft=sect c. ft=cott,ft=csctFor Exercises 24-29, use the unit circle and the period of the function to evaluate the function or state that the function is undefined at the given value. cos152For Exercises 24-29, use the unit circle and the period of the function to evaluate the function or state that the function is undefined at the given value. cot540For Exercises 24-29, use the unit circle and the period of the function to evaluate the function or state that the function is undefined at the given value. csc240For Exercises 24-29, use the unit circle and the period of the function to evaluate the function or state that the function is undefined at the given value. sin143For Exercises 24-29, use the unit circle and the period of the function to evaluate the function or state that the function is undefined at the given value. tan236For Exercises 24-29, use the unit circle and the period of the function to evaluate the function or state that the function is undefined at the given value. sec480For Exercises 30-31, use the even -odd and periodic properties of the trigonometric functions to simplify. tantcos2tFor Exercises 30-31, use the even -odd and periodic properties of the trigonometric functions to simplify. sint+sint+2Write cost in terms of sint for t in Quadrant IV.Write cott in terms of csct for t in Quadrant II.Suppose that a right triangle C has legs of length 5 cm and 2 cm. Evaluate the six trigonometric functions for angle , where angle 0 is the larger acute angle.Determine the values of the six trigonometric functions of for the given right triangle.For Exercises 36-37, construct a right triangle to find the indicated values. Assume is an acute angle. If cos=57, find csc and tan .For Exercises 36-37, construct a right triangle to find the indicated values. Assume is an acute angle. If tan=3, find sin and sec .For Exercises 38-39, evaluate the expression without the use of a calculator. sin3+tan6For Exercises 38-39, evaluate the expression without the use of a calculator. cos45csc60Given sin=99101 and cos=20101 , use the reciprocal and quotient identities to find the values of the other trigonometric functions of .For Exercises 41-43, use an appropriate Pythagorean identity to find the indicated value for an acute angle . Given cos=4041 , find the value of sin .For Exercises 41-43, use an appropriate Pythagorean identity to find the indicated value for an acute angle . Given sec=2920 , find the value of tan .For Exercises 41-43, use an appropriate Pythagorean identity to find the indicated value for an acute angle . Given cot=1384 , find the value of csc .For Exercises 44-45, given the function value, find a cofunction of another angle with the same function value. cos15=6+24For Exercises 44-45, given the function value, find a cofunction of another angle with the same function value. tan12=23Use a calculator to approximate the function values. Round to 4 decimal places. a. tan23.8 b. cos58 c. sin8An observer at the top of a 48-ft building measures the angle of depression from the top of the building to a point on the ground to be 27 . What is the distance from the base of the building to the point on the ground? Round to the nearest foot.During the first quarter moon, the Earth, Sun, and Moon form a right triangle. The distance between the Sun and the Earth is approximately 92,900,000 mi and the measure of SEM is approximately 89.85 . Determine the distance between the Earth and the Moon. Round to the nearest thousand miles.Let P3,5 be a point on the terminal side of angle drawn in standard position. Find the values of the six trigonometric functions of .For Exercises 50-55, find the reference angle for the given angle. 56For Exercises 50-55, find the reference angle for the given angle. 260For Exercises 50-55, find the reference angle for the given angle. 200For Exercises 50-55, find the reference angle for the given angle. 5For Exercises 50-55, find the reference angle for the given angle. 74For Exercises 50-55, find the reference angle for the given angle. 750For Exercises 56-61, use reference angles to find the exact value. cos116For Exercises 56-61, use reference angles to find the exact value. sin53For Exercises 56-61, use reference angles to find the exact value. tan176For Exercises 56-61, use reference angles to find the exact value. cot34For Exercises 56-61, use reference angles to find the exact value. csc120For Exercises 56-61, use reference angles to find the exact value. sec240Identify which expressions are undefined. a. sec270 b. tan2 c. cot180 d. csc2Given tan=23 and sin0 . Find sec .Given cos=37 and cot0 , find sin .Given sin=6061 and Quadrant III, find cot .For Exercises 66-69, determine the amplitude and period. y=4sin2xFor Exercises 66-69, determine the amplitude and period. y=2cosx2For Exercises 66-69, determine the amplitude and period. y=13cosxFor Exercises 66-69, determine the amplitude and period. y=sin37xFor Exercises 70-72, graph one period of the function y=cos3xFor Exercises 70-72, graph one period of the function y=3sin2xFor Exercises 70-72, graph one period of the function y=2cosx4Determine the amplitude, period, and phase shift for each function. a. y=13cos2x+ b. y=5sin2xFor Exercises 74-75, graph one period of the function. y=13cosx4For Exercises 74-75, graph one period of the function. y=2sinx+3For Exercises 76-77, determine the amplitude, period, phase shift, and vertical shift for each function. y=4sin3x+2For Exercises 76-77, determine the amplitude, period, phase shift, and vertical shift for each function. y=14cos3x2+5For Exercises 78-79, graph one period of the function. y=cosx3+2For Exercises 78-79, graph one period of the function. y=2sin2x+31The depth of water along a coastal inlet varies with the tides. On a summer day, the water depth at the end of a pier is 23ft at 6:30A.M. , 21ft at 12:30P.M , and 23ft at 6:30P.M. Assuming that this pattern continues indefinitely and behaves like a cosine wave, write a function of the form ht=AcosBtC+D . The value ht is the water depth (in feet), t hours after 6:30A.M.The data in the table represent the percentage of the moon illuminated for selected days in January fora recent year. The value 0.0=0 represents a new moon and 1.0=100 represents a full moon. a. Enter the data in a graphing utility and use the sinusoidal regression tool (SinReg) to find a model of the form y=asinbx+c+d . Round a,b,c , and d to 1 decimal place. b. Graph the data and the resulting function.For Exercises 82-85, give the period of the function and the equations of two asymptotes. y=csc3xFor Exercises 82-85, give the period of the function and the equations of two asymptotes. y=tan3xFor Exercises 82-85, give the period of the function and the equations of two asymptotes. y=cotx+2For Exercises 82-85, give the period of the function and the equations of two asymptotes. y=2secx+For Exercises 86-89, graph one period of the function. y=5sec3xFor Exercises 86-89, graph one period of the function. y=2cscx2For Exercises 86-89, graph one period of the function. y=2secx+For Exercises 86-89, graph one period of the function. y=3csc4x+2For Exercises 90-91, graph two periods of the function. y=tan4xFor Exercises 90-91, graph two periods of the function. y=2tanx3y=cot2xy=cot2x+3For Exercises 94-109, evaluate the expression. cos122For Exercises 94-109, evaluate the expression. sin122For Exercises 94-109, evaluate the expression. arctan33For Exercises 94-109, evaluate the expression. arccos32For Exercises 94-109, evaluate the expression. arcsin32For Exercises 94-109, evaluate the expression. cos1cos76For Exercises 94-109, evaluate the expression. tantan17For Exercises 94-109, evaluate the expression. coscos10.35For Exercises 94-109, evaluate the expression. arcsinsin76For Exercises 94-109, evaluate the expression. sinsin123For Exercises 94-109, evaluate the expression. cossin132For Exercises 94-109, evaluate the expression. sinarccos32For Exercises 94-109, evaluate the expression. sinarctan32For Exercises 94-109, evaluate the expression. cossin125For Exercises 94-109, evaluate the expression. tanarccos34For Exercises 94-109, evaluate the expression. sincos114Use a calculator to approximate the function values in both radians and degrees. a. sin10.35 b. cos125 c. arctan10For Exercises 111-114, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle 6 subject to the given conditions. tan=419 and 180270For Exercises 111-114, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle 6 subject to the given conditions. sin=511 and 270360For Exercises 111-114, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle 6 subject to the given conditions. sin=511 and 32For Exercises 111-114, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle 6 subject to the given conditions. tan=4 and 32Write the given expression as an algebraic expression. It is not necessary to rationalize the denominator. a.cossin1x for x1 b. sinarctanx c. tanarcsinxx2+9 for x0 .Find the lengths of sides a,b , and c to the nearest tenth of a meter and the measure of angle a to the nearest tenth of a degree.The length of the perpendicular line segment BP from a rotating beacon to a straight shoreline is 100ft . The beam of light emitted from the beacon strikes the shoreline at a point Q , a distance of x feet from point P . Let represent QBP . a. Write as a function of x. b. Find for x=100,200 , and 300ft . Round to the nearest degree, c. What happens to as x ?Find the angle of repose for the pile of coarse gravel. Round to the nearest degreeConvert 15.36 to DMS (degree-minute-second) form. Round to the nearest second if necessary.Convert 130.3 to radians. Round to 2 decimal places.Find the exact length of the arc intercepted by a central angle of 27 on a circle of radius 5 ft.A skateboard designed for rough surfaces has wheels with diameter 60 mm. If the wheels turn at 2200 rpm, what is the linear speed in mm per minute?A pulley is 20 in. in diameter. Through how many degrees should the pulley rotate to lift a load 3 ft? Round to the nearest degree.A circle has a radius of 9 yd. Find the area of a sector with a central angle of 120 .For an acute angle if sin=56 evaluate cos and tan .Evaluate tan 6cot6 without the use of a calculator.Given sin=513 , use a Pythagorean identity to find cos .Given sec75=2+6 , find a cofunction of another angle with the same function value.Use a calculator to approximate sin sin211 to 4 decimal places.A flag pole casts a shadow of 20 ft when the angle of elevation of the Sun is 40 . How tall is the flag pole? Round to the nearest foot.A newly planted tree is anchored by a covered wire running from the top of the tree to a post in the ground 5 ft from the base of the tree. If the angle between the wire and the top of the tree is 20 , what is the length of the wire? Round to the nearest foot.For Exercises 14-21, evaluate the function or state that the function is undefined at the given value. sin34For Exercises 14-21, evaluate the function or state that the function is undefined at the given value. tan930For Exercises 14-21, evaluate the function or state that the function is undefined at the given sec116For Exercises 14-21 evaluate the function or state that the function is undefined at the given value. csc150For Exercises 14-21 evaluate the function or state that the function is undefined at the given value. cot20For Exercises 14-21 evaluate the function or state that the function is undefined at the given value. cos690For Exercises 14-21 evaluate the function or state that the function is undefined at the given value. tan73For Exercises 14-21, evaluate the function or state that the function is undefined at the given value. sec630Given sin=58 andcos0 , findtan .Given sec=43 and sin0 find csc .Given tan=43 and is in Quadrant IV, find sec .For Exercises 25-28, select the trigonometric function, ft=sint,gt=cost,ht=tantorrt=cott , with the given properties. The function is odd, with period 2 , and domain of all real numbers.For Exercises 25-28, select the trigonometric function, ft=sint,gt=cost,ht=tantorrt=cott , with the given properties. The function is odd, with period , and domain of all real numbers excluding odd multiples of 2 .For Exercises 25-28, select the trigonometric function, ft=sint,gt=cost,ht=tantorrt=cott , with the given properties. The function is even, with period, 2 and domain of all real numbers.For Exercises 25-28, select the trigonometric function, ft=sint,gt=cost,ht=tantorrt=cott , with the given properties. The function is odd, with period , and domain of all real numbers excluding multiples of .Use the even-odd and periodic properties of the trigonometric functions to simplify cos2sin2cot+Suppose that a rectangle is bounded by the x-axis and the graph of y=sinx on the interval 0, . a. Write a function that represents the area Ax of the rectangle for 0x2 . b. Determine the area of the rectangle for x=6 and x=4 .For Exercises 31-32, determine the amplitude, period, phase shift, and vertical shift for each function. y=34sin2x6For Exercises 31-32, determine the amplitude, period, phase shift, and vertical shift for each function. y=5cos5x++7For Exercises 33-36, graph one period of the function. y=2sinxFor Exercises 33-36, graph one period of the function. y=3cos2xFor Exercises 33-36, graph one period of the function. y=sin2x4For Exercises 33-36, graph one period of the function. y=32cos2xWrite a function of the form fx=AcosBx for the given graph.Identify each statement as true or false. If a statement is false, explain why. a. The relative maxima of the graph of y=sinx correspond to the relative minima of the graph of y=cscx . b. The period of y=tan2x is . c. The amplitude of y=2cotx is 2 . d. The period of y=sec2x is . e. The vertical asymptotes of the graph of y=cotx occur where the graph of y=cosx has x-intercepts.For Exercises 39-40, graph one period of the function. y=4sec2xFor Exercises 39-40, graph one period of the function. y=csc2x4For Exercises 41-42, graph two periods of the function. y=tan3xFor Exercises 41-42, graph two periods of the function. y=4cotx+4Simplify each expression. a. sin132+cos122 b. cosarctan3Use a calculator to approximate the degree measure (to 1 decimal place) of the angle 0 subject to the given conditions. sin=38and90180Find the exact value of sin1sin116 .Find the exact value of tancos17847TWrite the expression tancos1xx2+25 for x0 as an algebraic expression.A radar station tracks a plane flying at a constant altitude of 6mi on a path directly over the station. Let be the angle of elevation from the radar station to the plane. a. Write as a function of the plane’s ground distance x0 from the station. b. Without the use of a calculator, will the angle of elevation be less than 45 or greater than 45 when the plane’s ground distance is 3.2mi away? c. Use a calculator to find to the nearest degree for x=3.2,1.6 , and 0.5mi .For Exercises 1-9, solve the equations and inequalities. Write the solution set to inequalities in interval notation 3x5+36 .2CREFor Exercises 1-9, solve the equations and inequalities. Write the solution set to inequalities in interval notation x+2x32For Exercises 1-9, solve the equations and inequalities. Write the solution set to inequalities in interval notation 3x4x210=05CRE6CREFor Exercises 1-9, solve the equations and inequalities. Write the solution set to inequalities in interval notation e3x=28CREFor Exercises 1-9, solve the equations and inequalities. Write the solution set to inequalities in interval notation lnx1+lnx=ln210CREGiven y=x2+2x4 a. Identify the vertex. b. Write the domain and range in interval notation.Given fx=x42x34x2+8x, a. Find the x -intercepts of the graph of f b. Determine the end behavior of the graph of f .a. Graph y=3x2+x53x2 . b. Identify the asymptotes.Given y=2sin4x6, identify the a. Domain and range in interval notation. b. Amplitude. c. Period. d. Phase shift. e. Vertical shift.Given fx=tanx and gx=x2 evaluate a. fog2 b. g+f16CREEvaluate tanarcsin38 .Write the logarithm as the sum or difference of logarithms. Simplify as much as possible. logx2y100zDivide. Write the answer in standard form, a+bi.28i35iSuppose that y varies inversely as x and directly as z . If y is 12 when x is 8 and z is 3 , find the constant of variation k .Convert 1311233 to decimal degrees. Round to 4 decimal places.Convert 26.48 to degree, minute, second form.Convert from degrees to radians. a. 300 b. 70Convert from radians to degrees. a. 18 b. 74Find an angle coterminal to between 0 and 360 . a.=1230b.=315Find an angle coterminal to on the interval [0,2) . a. =8 b. =194Find the length of the arc made by an angle of 220 on a circle of radius 9 in.Lincoln, Nebraska, is located at 40.8N,96.7W and Dallas, Texas, is located at32.8N,96.7W . Since the longitudes are the same, the cities are north-south of each other. Using the difference in latitudes, approximate the distance between the cities assuming that the radius of the Earth is 3960 mi. Round to the nearest mile.A bicycle wheel rotates at 2 revolutions per second. a. Find the angular speed. b. How fast does the bicycle travel (in ft/sec) if the wheel is 2.2 ft in diameter? Round to the nearest tenth.A sprinkler rotates through an angle of 120 and sprays water a distance of 30 ft. Find the amount of area watered. Round to the nearest whole unit.1PEAn angle with its vertex at the origin of an xy-coordinate plane and with initial side on the positive x-axis is in position.Two common units used to measure angles are and .One degree is what fractional amount of a full rotation?An angle that measures 360 has a measure of radians.Two angles are called if the sum of their measures is 90 . Two angles are called if the sum of their measures is 180 .An angle with measure or radians is a right angle. A straight angle has a measure of or radians.A(n) angle has a measure between 0 and 90 . whereas a(n) angle has a measure between 90 and180 .One degree is equally divided into 60 parts called .One minute is equally divided into 60 parts called .1='='An angle with its vertex at the center of a circle is called an angle.A central angle of a circle that intercepts an arc equal in length to the radius of the circle has a measure of .Which angle has a greater measure, 2 or 2 radians?To convert from radians to degrees, multiply by To convert from degrees to radians, multiply by .Two angles are if they have the same initial side and same terminal side.The measure of all angles coterminal to 74 differ from 74 by a multiple of .The measure of all angles coterminal to 112 differ from 112 by a multiple of .The length s of an arc made by an angle on a circle of radius r is given by the formula , where is measured in .To locate points on the surface of the Earth that are north or south of the equator, we measure the of the point. To locate points that are east or west of the prime meridian, we measure the of the point.The relationship v=arclengthtime represents the speed of a point traveling in a circular path.The symbol is typically used to denote speed and represents the number of radians per unit time that an object rotates.A wedge of a circle, similar in shape to a slice of pie, is called an of the circle.The area A of a sector of a circle of radius r with central angle is given by the formula , where is measured in .For Exercises 25-26, sketch the angles in standard position. a. 60 b. 225 c. 210 d. 86For Exercises 25-26, sketch the angles in standard position. a. 30 b. 120 c. 135 d. 73For Exercises 27-30, convert the given angle to decimal degrees. Round to 4 decimal places. (see Example 1) 1734For Exercises 27-30, convert the given angle to decimal degrees. Round to 4 decimal places. (See Example 1) 21547For Exercises 27-30, convert the given angle to decimal degrees. Round to 4 decimal places. (See Example 1) 543655For Exercises 27-30, convert the given angle to decimal degrees. Round to 4 decimal places. (See Example 1) 234248For Exercises 31-34, convert the given angle to DMS (degree-minute-second) form. Round to the nearest second if necessary. (See Example 2) 46.418For Exercises 31-34, convert the given angle to DMS (degree-minute-second) form. Round to the nearest second if necessary. (See Example 2) 84.074For Exercises 31-34, convert the given angle to DMS (degree-minute-second) form. Round to the nearest second if necessary. (See Example 2) 84.64For Exercises 31-34, convert the given angle to DMS (degree-minute-second) form. Round to the nearest second if necessary. (See Example 2) 61.4635PE36PEFor Exercises 37-40, convert from degrees to radians. Give the answers in exact form in terms of . (See Example 3) 7538PE