Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Trigonometry (MindTap Course List)
75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98EDocking a Boat A boat is pulled in by means of a winch located on a dock 5 feet above the deck of the boat (see figure). Let be the angle of elevation from the boat to the winch and let s be the length of the rope from the winch to the boat. (a) Write as a function of s. (b) Find when s=40 feet and s=20 feet.A television camera at ground level films the lift-off of a space shuttle at a point 750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let s be the height of the shuttle. (a) Write as a function of s. (b) Find when s=300 meters and s=1200 meters.Granular Angle of Repose Different types of granular substances naturally settle at different angles when stored in cone-shaped piles. This angle is called the angle of repose (see figure). When rock salt is stored in a cone-shaped pile 5.5 meters high, the diameter of the pile’s base is about 17 meters. (a) Find the angle of repose for rock salt. (b) How tall is a pile of rock salt that has a base diameter of 20 meters?102EPhotography A photographer takes a picture of a three-foot-tall painting hanging in an art gallery. The camera lens is 1 foot below the lower edge of the painting (see figure). The angle subtended by the camera lens x feet from the painting is given by =arctan3xx2+4,x0. (a) Use a graphing utility to graph as a function of x. (b) Use the graph to approximate the distance from the picture when is maximum. (c) Identify the asymptote of the graph and interpret its meaning in the context of the problem.104EPolice Patrol A police car with its spotlight on is parked 20 meters from a warehouse. Consider and x as shown in the figure. (a) Write as a function of x. (b) Find when x=5 meters and x=12 meters.106E107E108E109E110E111E112E113E114E115E116E117E118E119EEvaluating an Inverse Trigonometric Function In Exercises 115-120, use the results of Exercises 111-113 to find the exact value of the expression. arccsc233121E122E123E124ECalculators and Inverse Trigonometric Functions In Exercises 121-126, use the results of Exercises 111-113 and a calculator to approximate the value of the expression. Round your result to two decimal places. arccot5.25Calculators and Inverse Trigonometric Functions In Exercises 121-126, use the results of Exercises 111-113 and a calculator to approximate the value of the expression. Round your result to two decimal places. arccot167127EThink About It Use a graphing utility to graph the functions fx=xandgx=6arctanx.Forx0, it appears that gf. Explain how you know that there exists a positive real number a such that gfforxa. Approximate the number a.129E130ESolve the right triangle shown at the right for all unknown sides and angles.2ECPAt a point 65 feet from the base of a church, the angles of elevation to the bottom of the steeple and the top of the steeple are 35 and 43, respectively. Find the height of the steeple.From the time a small airplane is 100 feet high and 1600 ground feet from its landing runway, the plane descends in a straight line to the runway. Determine the angle of descent (in degrees) of the plane.5ECP6ECP7ECPFill in the blanks. A measures the acute angle that a path or line of sight makes with a fixed north-south line.A point that moves on a coordinate line is in simple when its distance d from the origin at time t is given by either d=asintord=acost.Fill in the blanks. The time for one complete cycle of a point in simple harmonic motion is its .The number of cycles per second of a point in simple harmonic motion is its .Solving a Right Triangle In Exercises 5-12, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. A=60,c=12Solving a Right Triangle In Exercises 5-12, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. B=25,b=4Solving a Right Triangle In Exercises 5-12, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. B=72.8,a=4.4Solving a Right Triangle In Exercises 5-12, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. A=8.4,a=40.59E10E11ESolving a Right Triangle In Exercises 5-12, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. b=1.32,c=9.45Finding an Altitude In Exercises 13-16, find the altitude of the isosceles triangle shown in the figure. Round your answers to two decimal places. =45,b=614EFinding an Altitude In Exercises 13-16, find the altitude of the isosceles triangle shown in the figure. Round your answers to two decimal places. =32,b=816E17E18E19E20EHeight At a point 50 feet from the base of a church, the angles of elevation to the bottom of the steeple and the top of the steeple are 35 and 48, respectively. Find the height of the steeple.Distance An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of depression to the ships are 4 and 6.5 (see figure). How far apart are the ships?Distance A passenger in an airplane at an altitude of 10 kilometers sees two towns directly to the east of the plane. The angles of depression to the towns are 28 and 55 (see figure). How far apart are the towns?Angle of Elevation The height of an outdoor basketball backboard is 1212 feet, and the backboard casts a shadow 17 feet long. (a) Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. (b) Use a trigonometric function to write an equation involving the unknown angle of elevation. (c) Find the angle of elevation.Angle of Elevation An engineer designs a 75-foot cellular telephone tower. Find the angle of elevation to the top of the tower at a point on level ground 50 feet from its base.Angle of Depression A cellular telephone tower that is 120 feet tall is placed on top of a mountain that is 1200 feet above sea level. What is the angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles away and 400 feet above sea level?Angle of Depression A Global Positioning System satellite orbits 12,500 miles above Earth’s surface (see figure). Find the angle of depression from the satellite to the horizon. Assume the radius of Earth is 4000 miles.Height You are holding one of the tethers attached to the top of a giant character balloon that is floating approximately 20 feet above ground level. You are standing approximately 100 feet ahead of the balloon (see figure). (a) Find an equation for the length l of the tether you are holding in terms of h, the height of the balloon from top to bottom. (b) Find an equation for the angle of elevation from you to the top of the balloon. (c) The angle of elevation to the top of the balloon is 35. Find the height h of the balloon.29EThe designers of a water park have sketched a preliminary drawing of a new slide (see figure). (a) Find the height h of the slide. (b) Find the angle of depression from the top of the slide to the end of the slide at the ground in terms of the horizontal distance d a rider travels. (c) Safety restrictions require the angle of depression to be no less than 25 and no more than 30. Find an interval for how far a rider travels horizontally.Speed Enforcement A police department has set up a speed enforcement zone on a straight length of highway. A patrol car is parked parallel to the zone, 200 feet from one end and 150 feet from the other end (see figure). (a) Find the length l of the zone and the measures of angles A and B (in degrees). (b) Find the minimum amount of time (in seconds) it takes for a vehicle to pass through the zone without exceeding the posted speed limit of 35 miles per hour.Airplane Ascent During takeoff, an airplane’s angle of ascent is 18 and its speed is 260 feet per second. (a) Find the plane’s altitude after 1 minute. (b) How long will it take for the plane to climb to an altitude of 10,000 feet?Air Navigation An airplane flying at 550 miles per hour has a bearing of 52. After flying for 1.5 hours, how far north and how far east will the plane have traveled from its point of departure?34E35E36E37E38ESurveying A surveyor wants to find the distance across a pond (see figure). The bearing from AtoB is N32W. The surveyor walks 50 meters from AtoC, and at the point C the bearing to B is N68W. (a) Find the bearing from AtoB. (b) Find the distance from AtoB.Location of a Fire tower A is 30 kilometers due west of fire tower B. A fire is spotted from the towers, and the bearings from A and B are N76E and N76W, respectively (see figure). Find the distance d of the fire from the line segment AB.41E42EGeometry Find the length of the sides of a regular pentagon inscribed in a circle of radius 25 inches.44ESimple Harmonic Motion In Exercises 45-48, find a model for simple harmonic motion satisfying the specified conditions. Displacementt=0AmplitudePeriod 04centimeters2secondsSimple Harmonic Motion In Exercises 45-48, find a model for simple harmonic motion satisfying the specified conditions. Displacementt=0AmplitudePeriod 03meters6seconds47E48ETuning Fork A point on the end of a tuning fork moves in simple harmonic motion described by d=asint. Find given that the tuning fork for middle C has a frequency of 262 vibrations per second.50E51E52E53ESimple Harmonic Motion In Exercises 51-54, for the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of d when t=5, and (d) the least positive value of t for which d=0. Use a graphing utility to verify your results. d=164sin792tOscillation of a Spring A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by y=14cos16t,t0, where y is measured in feet and t is the time in seconds. (a) Graph the function. (b) What is the period of the oscillations? (c) Determine the first time the weight passes the point of equilibrium y=0.Hours of Daylight The numbers of hours H of daylight in Denver, Colorado, on the 15th of each month starting with January are :9.68,10.72,11.92,13.25,14.35,14.97,14.72,13.73,12.47,11.18,10.00, and 9.37. A model for the data is Ht=12.13+2.77sint61.60 where t represents the month, with t=1 corresponding to January. (Source: United States Navy) (a) Use a graphing utility to graph the data and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem?Sales The table shows the average sales S (in millions of dollars) of an outerwear manufacturer for each month t, where t=1 corresponds to January. (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data? (c) What is the period of the model? Do you think it is reasonable given the context? Explain. (d) Interpret the meaning of the model’s amplitude in the context of the problem.HOW DO YOU SEE IT? The graph below shows the displacement of an object in simple harmonic motion. (a) What is the amplitude? (b) What is the period? (c) Is the equation of the simple harmonic motion of the form d=asintord=acost?True or False? In Exercises 59 and 60, determine whether the statement is true or false. Justify your answer. The Leaning Tower of Pisa is not vertical, but when you know the angle of elevation to the top of the tower as you stand d feet away from it, its height h can be found using the formula h=dtan.60EUsing Radian or Degree Measure In Exercises 1-4, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine two coterminal angles (one positive and one negative). 154Using Radian or Degree Measure In Exercises 1-4, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine two coterminal angles (one positive and one negative). 43Using Radian or Degree Measure In Exercises 1-4, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine two coterminal angles (one positive and one negative). 1104RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15REPhonograph Phonograph records are vinyl discs that rotate on a turntable. A typical record album is 12 inches in diameter and plays at 3313 revolutions per minute (a) Find the angular speed of a record album. (b) Find the linear speed (in inches per minute) of the outer edge of a record album.17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RERailroad Grade A train travels 3.5 kilometers on a straight track with a grade of 1.2 (see figure). What is the vertical rise of the train in that distance?42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RESketching the Graph of a Sine or Cosine Function In Exercises 63-68, sketch the graph of the function. (Include two full periods.) fx=cos3x65RESketching the Graph of a Sine or Cosine Function In Exercises 63-68, sketch the graph of the function. (Include two full periods.) y=4cosx67RE68RE69REMeteorology The times S of sunset (Greenwich Mean Time) at 40 north latitude on the 15th of each month starting with January are :16:59,17:35,18:06,18:38,19:08,19:30,19:28,18:57,18:10,17:21,16:44,and16:36. A model (in which minutes have been converted to the decimal parts of an hour) for the data is St=18.101.41sint6+1.55 where t represents the month, with t=1 corresponding to January. (a) Use a graphing utility to graph the data and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem?71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE81RECalculators and Inverse Trigonometric Functions In Exercises 81-84, use a calculator to approximate the value of the expression, if possible. Round your result to two decimal places. arccos0.372Calculators and Inverse Trigonometric Functions In Exercises 81-84, use a calculator to approximate the value of the expression, if possible. Round your result to two decimal places. arccot15.5Calculators and Inverse Trigonometric Functions In Exercises 81-84, use a calculator to approximate the value of the expression, if possible. Round your result to two decimal places. arccsc4.0385RE86RE87REEvaluating a Composition of Functions In Exercises 87-90, find the exact value of the expression. tanarccos35Evaluating a Composition of Functions In Exercises 87-90, find the exact value of the expression. secarctan12590RE91REWriting an Expression In Exercises 91 and 92, write an algebraic expression that is equivalent to the given expression. secarcsinx193RE94REAir Navigation From city A to city B, a plane flies 650 miles at a bearing of 48. From city B to city C, the plane flies 810 miles at a bearing of 115. Find the distance from city A to city C and the bearing from city A to city C.96RETrue or False? In Exercises 97 and 98, determine whether the statement is true or false. Justify your answer. The equation y=sin does not represent y as a function of because sin30=sin150.98RE99RE100REWriting When graphing the sine and cosine functions, determining the amplitude is part of the analysis. Explain why this is not true for the other four trigonometric functions.Graphical Reasoning Use the formulas for the area of a circular sector and arc length given in Section 1.1. (a) For =0.8, write the area and arc length as functions of r. What is the domain of each function? Use a graphing utility to graph the functions. Use the graphs to determine which function changes more rapidly as r increases. Explain. (b) For r=10 centimeters, write the area and arc length as functions of . What is the domain of each function? Use the graphing utility to graph the functions.Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. Consider an angle that measures 54 radians. (a) Sketch the angle in standard position. (b) Determine two coterminal angles (one positive and one negative). (c) Convert the radian measure to degree measure.2TTake this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. A water sprinkler sprays water on a lawn over a distance of 25 feet and rotates through an angle of 130. Find the area of the lawn watered by the sprinkler.Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. Given that is an acute angle and tan=32, find the exact values of the other five trigonometric functions of .Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. Find the exact values of the six trigonometric functions of the angle shown in the figure.6T7T8T9TIn Exercises 9 and 10, find the exact values of the remaining five trigonometric functions of satisfying the given conditions. sec=2920,sin011TIn Exercises 11-13, sketch the graph of the function. (Include two full periods.) ft=cost+21In Exercises 11-13, sketch the graph of the function. (Include two full periods.) fx=12tan2xIn Exercises 14 and 15, use a graphing utility to graph the function. If the function is periodic, find its period. If not, describe the behavior of the function as x increases without bound. y=sin2x+2cosxIn Exercises 14 and 15, use a graphing utility to graph the function. If the function is periodic, find its period. If not, describe the behavior of the function as x increases without bound. y=6xcos0.25x16T17TSketch the graph of the function fx=2arcsin12x.An airplane is 90 miles south and 110 miles east of an airport. What bearing should the pilot take to fly directly to the airport?A ball on a spring starts at its lowest point of 6 inches below equilibrium, bounces to its maximum height of 6 inches above equilibrium, and returns to its lowest point in a total of 2 seconds. Write an equation for the simple harmonic motion of the ball.1PS2PS3PSPeriodic Function The function f is periodic, with period c.So,ft+c=ft. Determine whether each statement is true or false. Explain. aft2c=ftbft+12c=f12t cf12t+c=f12tdf12t+4c=f12tSurveying A surveyor in a helicopter is determining the width of an island, as shown in the figure. (a) What is the shortest distance d the helicopter must travel to land on the island? (b) What is the horizontal distance x the helicopter must travel before it is directly over the nearer end of the island? (c) Find the width w of the island. Explain how you found your answer.Similar Triangles and Trigonometric Functions Use the figure below. (a) Explain why ABC,ADE, and AFG are similar triangles. (b) What does similarity imply about the ratios BCAB,DEAD,andFGAF? (c) Does the value of sinA depend on which triangle from part a is used to calculate it? Does the value of sinA change when you use a different right triangle similar to the three given triangles? (d) Do your conclusions from part c apply to the other five trigonometric functions? Explain.7PS8PSBlood Pressure The pressure P (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by P=10020cos8t3 where t is the time (in seconds). (a) Use a graphing utility to graph the model. (b) What is the period of the model? What does it represent in the context of the problem? (c) What is the amplitude of the model? What does it represent in the context of the problem? (d) If one cycle of this model is equivalent to one heartbeat, what is the pulse of the patient? (e) A physician wants the patient’s pulse rate to be 64 beats per minute or less. What should the period be? What should the coefficient of t be?Biorhythms A popular theory that attempts to explain the ups and downs of everyday life states that each person has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by the sine functions below, where t is the number of days since birth. Physical (23 days): P=sin2t23,t0 Emotional (28 days): E=sin2t28,t0 Intellectual (33 days): I=sin2t33,t0 Consider a person who was born on July 20,1995. (a) Use a graphing utility to graph the three models in the same viewing window for 7300t7380. (b) Describe the person’s biorhythms during the month of September 2015. (c) Calculate the person’s three energy levels on September 22,2015.11PS12PSRefraction When you stand in shallow water and look at an object below the surface of the water, the object will look farther away from you than it really is. This is because when light rays pass between air and water, the water refracts, or bends, the light rays. The index of refraction for water is 1.333. This is the ratio of the sine of 1 and the sine of 2 (see figure). (a) While standing in water that is 2 feet deep, you look at a rock at angle 1=60 (measured from a line perpendicular to the surface of the water). Find 2. (b) Find the distances x and y. (c) Find the distance d between where the rock is and where it appears to be. (d) What happens to d as you move closer to the rock? Explain.14PSUse the conditions tanx=13 and cosx0 to find the values of all six trigonometric functions.2ECP3ECP4ECP5ECPPerform the addition and simplify: 11+sin+11sin.Rewrite cos21sin so that it not in fractional form.8ECPFill in the blank to complete the trigonometric identity. sinucosu=2E3E4E5E6EUsing Identities to Evaluate a Function In Exercises 7-12, use the given conditions to find the values of all six trigonometric functions. secx=52,tanx08E9E10E11E12E13E14E15E16EMatching Trigonometric Expressions In Exercises 13–18, match the trigonometric expression with its simplified form. (a)cscx(b)1(c)1(d)sinxtanx(e)sec2x(f)secx sec2x1sin2x18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E