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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)
Finding an Unknown Side Find the side labeled x. In Exercises 17 and 18 state your answer rounded to five decimal places. 19.Finding an Unknown Side Find the side labeled x. In Exercises 17 and 18 state your answer rounded to five decimal places. 20.21ETrigonometric Ratios Express x and y in terms of trigonometric ratios of . 22.23E24E25E26E27ETrigonometric Ratios Sketch a triangle that has acute angle , and find the other five trigonometric ratios of . 28. cot=5329E30E31E32EEvaluating an Expression Evaluate the expression without using a calculator. 33. (cos 30)2 (sin 30)234E35EEvaluating an Expression Evaluate the expression without using a calculator. 36. (sin3tan6+csc4)237E38E39E40E41E42E43E44EFinding an Unknown Side Find x rounded to one decimal place. 47.48E49E50ETrigonometric Ratios Express the length x in terms of the trigonometric ratios of .Trigonometric Ratios Express the lengths a, b, c, and d in the figure in terms of the trigonometric ratios of .Height of a Building The angle of elevation to the top of the Empire State Building in New York is found to be 11 from the ground at a distance of 1 mi from the base of the building. Using this information, find the height of the Empire State Building.Gateway Arch A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 35,000 ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is 22. (a) What is the distance between the plane and the arch? (b) What is the distance between a point on the ground directly below the plane and the arch?Deviation of a Laser Beam A laser beam is to be directed toward the center of the moon, but the beam strays 0.5 from its intended path. (a) How far has the beam diverged from its assigned target when it reaches the moon? (The distance from the earth to the moon is 240,000 mi.) (b) The radius of the moon is about 1000 mi. Will the beam strike the moon?Distance at Sea From the top of a 200-ft lighthouse, the angle of depression to a ship in the ocean is 23. How far is the ship from the base of the lighthouse?Leaning Ladder A 20-ft ladder leans against a building so that the angle between the ground and the ladder is 72. How high does the ladder reach on the building?58EElevation of a Kite A man is lying on the beach, flying a kite. He holds the end of the kite string at ground level and estimates the angle of elevation of the kite to be 50. If the string is 450 ft long, how high is the kite above the ground?Determining a Distance A woman standing on a hill sees a flagpole that she knows is 60 ft tall. The angle of depression to the bottom of the pole is 14, and the angle of elevation to the top of the pole is 18. Find her distance x from the pole.Height of a Tower A water tower is located 325 ft from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is 39 and that the angle of depression to the bottom of the tower is 25. How tall is the tower? How high is the window?62EDetermining a Distance If both cars in Exercise 62 are on one side of the plane and if the angle of depression to one car is 38 and that to the other car is 52, how far apart are the cars?Height of a Balloon A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 20 and 22. How high is the balloon?Height of a Mountain To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be 32. One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is 35. Estimate the height of the mountain.Height of Cloud Cover To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75 from the horizontal. An observer 600 m away measures the angle of elevation to the spot of light to be 45. Find the height h of the cloud cover.Distance to the Sun When the moon is exactly half full, the earth, moon, and sun form a right angle (see the figure). At that time the angle formed by the sun, earth, and moon is measured to be 89.85. If the distance from the earth to the moon is 240,000 mi, estimate the distance from the earth to the sun.Distance to the Moon To find the distance to the sun as in Exercise 67, we needed to know the distance to the moon. Here is a way to estimate that distance: When the moon is seen at its zenith at a point A on the earth, it is observed to be at the horizon from point B (see the following figure). Points A and B are 6155 mi apart, and the radius of the earth is 3960 mi. (a) Find the angle in degrees. (b) Estimate the distance from point A to the moon.Radius of the Earth In Exercise 80 of Section 6.1 a method was given for finding the radius of the earth. Here is a more modern method: From a satellite 600 mi above the earth it is observed that the angle formed by the vertical and the line of sight to the horizon is 60.276. Use this information to find the radius of the earth.Parallax To find the distance to nearby stars, the method of parallax is used. The idea is to find a triangle with the star at one vertex and with a base as large as possible. To do this, the star is observed at two different times exactly 6 months apart, and its apparent change in position is recorded. From these two observations E1SE2 can be calculated. (The times are chosen so that E1SE2 is as large as possible, which guarantees that E1OS is 90.) The angle E1SO is called the parallax of the star. Alpha Centauri, the star nearest the earth, has a parallax of 0.000211. Estimate the distance to this star. (Take the distance from the earth to the sun to be 9.3 107 mi.)71EDISCUSS: Similar Triangles If two triangles are similar, what properties do they share? Explain how these properties make it possible to define the trigonometric ratios without regard to the size of the triangle.If the angle is in standard position and P(x, y) is a point on the terminal side of , and r is the distance from the origin to P, then sin=cos=tan=The sign of a trigonometric function of depends on the _____ in which the terminal side of the angle lies. In Quadrant II, sin is _______ (positive / negative). In Quadrant III, cos is _______ (positive / negative). In Quadrant IV, sin is _______(positive / negative).(a) If is in standard position, then the reference angle is the acute angle formed by the terminal side of and the _______. So the reference angle for = 100 is =, and that for = 190 is =. (b) If is any angle, the value of a trigonometric function of is the same, except possibly for sign, as the value of the trigonometric function of . So sin 100 = sin _____, and sin 190 = sin _____.The area A of a triangle with sides of lengths a and b and with included angle is given by the formula A=. So the area of the triangle with sides 4 and 7 and included angle = 30 is ______.5EReference Angle Find the reference angle for the given angle. 6. (a) 175 (b) 310 (c) 7307E8E9EReference Angle Find the reference angle for the given angle. 10. (a) 56 (b) 109 (c) 23711E12E13EValues of Trigonometric Functions Find the exact value of the trigonometric function. 14. sin 24015E16E17E18E19E20E21E22E23EValues of Trigonometric Functions Find the exact value of the trigonometric function. 24. cos 66025E26E27E28E29E30E31E32E33E34E35E36E37E38E39EQuadrant in Which an Angle Lies Find the quadrant in which lies from the information given. 40. csc 0 and cos 0Expressing One Trigonometric Function in Terms of Another Write the first trigonometric function in terms of the second for in the given quadrant. 41. tan , cos ; in Quadrant III42E43E44E45E46EValues of Trigonometric Functions Find the values of the trigonometric functions of from the information given. 47. sin=45, in Quadrant IV48EValues of Trigonometric Functions Find the values of the trigonometric functions of from the information given. 49. cos=712, sin 050EValues of Trigonometric Functions Find the values of the trigonometric functions of from the information given. 51. csc = 2, in Quadrant I52E53EValues of Trigonometric Functions Find the values of the trigonometric functions of from the information given. 54. tan = 4, sin 0Values of an Expression If = /3, find the value of each expression. 55. sin 2, 2 sin56E57EArea of a Triangle Find the area of the triangle with the given description. 58. A triangle with sides of length 10 and 22 and included angle 1059E60EFinding an Angle of a Triangle A triangle has an area of 16 in2, and two of the sides have lengths 5 in. and 7 in. Find the sine of the angle included by these two sides.Finding a Side of a Triangle An isosceles triangle has an area of 24 cm2, and the angle between the two equal sides is 5/6. Find the length of the two equal sides.Area of a Region Find the area of the shaded region in the figure. 63.Area of a Region Find the area of the shaded region in the figure. 64.Height of a Rocket A rocket fired straight up is tracked by an observer on the ground 1 mi away. (a) Show that when the angle of elevation is , the height of the rocket (in ft) is h = 5280 tan . (b) Complete the table to find the height of the rocket at the given angles of elevation.Rain Gutter A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle . (See the figure on the next page.) (a) Show that the cross-sectional area of the gutter is modeled by the function A()=100sin+100sincos (b) Graph the function A for 0 /2. (c) For what angle is the largest cross-sectional area achieved?Wooden Beam A rectangular beam is to be cut from a cylindrical log of diameter 20 cm. The figures show different ways this can be done. (a) Express the cross-sectional area of the beam as a function of the angle in the figures. (b) Graph the function you found in part (a). (c) Find the dimensions of the beam with largest cross-sectional area.Strength of a Beam The strength of a beam is proportional to the width and the square of the depth. A beam is cut from a log as in Exercise 67. Express the strength of the beam as a function of the angle in the figures.Throwing a Shot Put The range R and height H of a shot put thrown with an initial velocity of v0 ft/s at an angle are given by R=v02sin(2)gH=v02sin22g On the earth g = 32 ft/s2, and on the moon g = 5.2 ft/s2. Find the range and height of a shot put thrown under the given conditions. (a) On the earth with v0 = 12 ft/s and = /6 (b) On the moon with v0 = 12 ft/s and = /6Sledding The time in seconds that it takes for a sled to slide down a hillside inclined at an angle is t=d16sin where d is the length of the slope in feet. Find the time it takes to slide down a 2000-ft slope inclined at 30.Beehives In a beehive each cell is a regular hexagonal prism, as shown in the figure. The amount of wax W in the cell depends on the apex angle and is given by W=3.020.38cot+0.65csc Bees instinctively choose so as to use the least amount of wax possible. (a) Use a graphing device to graph W as a function of for 0 . (b) For what value of does W have its minimum value? [Note: Biologists have discovered that bees rarely deviate from this value by more than a degree or two.]Turning a Comer A steel pipe is being carried down a hallway that is 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. (a) Show that the length of the pipe in the figure is modeled by the function L()=9csc+6sec (b) Graph the function L for 0 /2. (c) Find the minimum value of the function L. (d) Explain why the value of L you found in part (c) is the length of the longest pipe that can be carried around the corner.Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation of a rainbow is always the same. It can be shown that = 4 2. Where sin=ksin and = 59.4 and k = 1.33 is the index of refraction of water. Use the given information to find the angle of elevation of a rainbow. [Hint: Find sin , then use the SIN1 key on your calculator to find .] (For a mathematical explanation of rainbows see Calculus Early Transcendentals, 7th Edition, by James Stewart, page 282.)74EDISCUSS DISCOVER: Vites Trigonometric Diagram In the 16th century the French mathematician Franois Vite (see page 50) published the following remarkable diagram. Each of the six trigonometric functions of is equal to the length of a line segment in the figure. For instance, sin = | PR|, since from OPR we see that sin=opphyp=PROR=PR1=PR For each of the live other trigonometric functions, find a line segment in the figure whose length equals the value of the function at . [Note: The radius of the circle is 1, the center is O, segment QS is tangent to the circle at R, and SOQ is a right angle.]PROVE: Pythagorean Identities To prove the following Pythagorean identities, start with the first Pythagorean identity, sin2 + cos2 = 1, which was proved in the text, and then divide both sides by an appropriate trigonometric function of . (a) tan2 + 1 = sec2 (b) 1 + cot2 = csc2DISCUSS DISCOVER: Degrees and Radians What is the smallest positive real number x with the property that the sine of x degrees is equal to the sine of x radians ?For a function to have an inverse, it must be _______________. To define the inverse sine function, we restrict the _______________ of the sine function to the interval _______________.The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges. (a) The function sin1 has domain _______ and range _______. (b) The function cos1 has domain _______ and range _______. (c) The function tan1 has domain _______and range _______.In the triangle shown we can find the angle as follows. (a) =sin1 (b) =cos1 (c) =tan1To find sin(cos1513), we let =cos1(513) and complete the right triangle at the top of the next column. We find that sin(cos1513)=.5E6EEvaluating Inverse Trigonometric Functions Find the exact value of each expression, if it is defined. Express your answer in radians. 7. (a) sin1(22) (b) cos1(22) (c) tan1(1)8E9E10E11E12EEvaluating Inverse Trigonometric Functions Use a calculator to find an approximate value (in radians) of each expression rounded to five decimal places, if it is defined. 13. tan1 314E15E16EFinding Angles in Right Triangles Find the angle in degrees, rounded to one decimal place. 17.18E19EFinding Angles in Right Triangles Find the angle in degrees, rounded to one decimal place. 20.21EFinding Angles in Right Triangles Find the angle in degrees, rounded to one decimal place. 22.Basic Trigonometric Equations Find all angles between 0 and 180 satisfying the given equation. Round your answer to one decimal place. 23. sin=23Basic Trigonometric Equations Find all angles between 0 and 180 satisfying the given equation. Round your answer to one decimal place. 24. cos=3425E26E27E28E29EValue of an Expression Find the exact value of the expression. 30. cos(tan143)Value of an Expression Find the exact value of the expression. 31. sec(sin11213)32E33E34EAlgebraic Expressions Rewrite the expression as an algebraic expression in x. 35. cos(sin1 x)36E37E38ELeaning Ladder A 20-ft ladder is leaning against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle of elevation of the ladder? How high does the ladder reach on the building?40EHeight of the Space Shuttle An observer views the space shuttle from a distance of 2 mi from the launch pad. (a) Express the height of the space shuttle as a function of the angle of elevation . (b) Express the angle of elevation as a function of the height h of the space shuttle.Height of a Pole A 50-ft pole casts a shadow as shown in the figure. (a) Express the angle of elevation of the sun as a function of the length s of the shadow. (b) Find the angle of elevation of the sun when the shadow is 20 ft long.Height of a Balloon A 680-ft rope anchors a hot-air balloon as shown in the figure. (a) Express the angle as a function of the height h of the balloon. (b) Find the angle if the balloon is 500 ft high.View from a Satellite The figures on the next page indicate that the higher the orbit of a satellite, the more of the earth the satellite can see. Let , s, and h be as in the figure, and assume that the earth is a sphere of radius 3960 mi. (a) Express the angle as a function of h. (b) Express the distance s as a function of . (c) Express the distance s as a function of h. [Hint: Find the composition of the functions in parts (a) and (b).] (d) If the satellite is 100 mi above the earth, what is the distance s that it can see? (e) How high does the satellite have to be to see both Los Angeles and New York, 2450 mi apart?Surfing the Perfect Wave For a wave to be surfable, it cant break all at once. Robert Gura and Tony Bowen have shown that a wave has a surfable shoulder if it hits the shoreline at an angle given by =sin1(12n+1tan) where is the angle at which the beach slopes down and where n = 0, 1, 2, . (a) For = 10, find when n = 3. (b) For = 15, find when n = 2, 3, and 4. Explain why the formula does not give a value for when n = 0 or 1.46EIn triangle ABC with sides a, b, and c the Law of Sines states that ==The four cases in which we can solve a triangle are ASASSASASSSS (a) In which of these cases can we use the Law of Sines to solve the triangle? (b) Which of the cases listed can lead to more than one solution (the ambiguous case)?3E4E5E6E7E8E9ESolving a Triangle Solve the triangle using the Law of Sines. 10.11E12ESolving a Triangle Sketch each triangle, and then solve the triangle using the Law of Sines. 13. A = 50, B = 68, c = 23014E15E16E17E18ESolving a Triangle Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. 19. a = 28, b = 15, A = 11020E21E22E23ESolving a Triangle Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. 24. a = 75, b = 100, A = 3025E26E27ESolving a Triangle Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. 28. b = 73, c = 82, B = 58Finding Angles For the triangle shown, find (a) BCD and (b) DCA.Finding a Side For the triangle shown, find the length AD.Tracking a Satellite The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 50 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0 and 84.2, respectively. (a) How far is the satellite from station A? (b) How high is the satellite above the ground?Flight of a Plane A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 5 mi apart, to be 32 and 48, as shown in the figure. (a) Find the distance of the plane from point A. (b) Find the elevation of the plane.33EDistance Across a Lake Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such that CAB = 48.6. He also measures CA as 312 ft and CB as 527 ft. Find the distance between A and B.The Leaning Tower of Pisa The bell lower of the cathedral in Pisa, Italy, leans 5.6 from the vertical. A tourist stands 105 m from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be 29.2. Find the length of the tower to the nearest meter.Radio Antenna A short-wave radio antenna is supported by two guy wires, 165 ft and 180 ft long. Each wire is attached to the top of the antenna and anchored to the ground at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of 67 with the ground. How far apart are the anchor points?Height of a Tree A tree on a hillside casts a shadow 215 ft down the hill. If the angle of inclination of the hillside is 22 to the horizontal and the angle of elevation of the sun is 52, find the height of the tree.Length of a Guy Wire A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58. A guy wire is to be attached to the top of the tower and to the ground, 100 m downhill from the base of the tower. The angle in the figure is determined to be 12. Find the length of cable required for the guy wire.Calculating a Distance Observers at P and Q are located on the side of a hill that is inclined 32 to the horizontal, as shown. The observer at P determines the angle of elevation to a hot-air balloon to be 62. At the same instant the observer at Q measures the angle of elevation to the balloon to be 71. If P is 60 m down the hill from Q, find the distance from Q to the balloon.Calculating an Angle A water tower 30 m tall is located at the top of a hill. From a distance of 120 m down the hill it is observed that the angle formed between the top and base of the tower is 8. Find the angle of inclination of the hill.Distances to Venus The elongation of a planet is the angle formed by the planet, earth, and sun (see the figure). It is known that the distance from the sun to Venus is 0.723 AU (see Exercise 71 in Section 6.2). At a certain time the elongation of Venus is found to be 39.4. Find the possible distances from the earth to Venus at that time in astronomical units (AU).Soap Bubbles When two bubbles cling together in midair, their common surface is part of a sphere whose center D lies on the line passing through the centers of the bubbles (see the figure). Also, ACB and ACD each have measure 60. (a) Show that the radius r of the common face is given by r=abab [Hint: Use the Law of Sines together with the fact that an angle and its supplement 180 have the same sine.] (b) Find the radius of the common face if the radii of the bubbles are 4 cm and 3 cm. (c) What shape does the common face take if the two bubbles have equal radii?43E44E45EFor triangle ABC with sides a, b, and c the Law of Cosines states c2=In which of the following cases must the Law of Cosines be used to solve a triangle? ASASSSSASSSA3EFinding an Angle or Side Use the Law of Cosines to determine the indicated side x or angle . 4.5E6E7E8E9E10E11E12E13E14ESolving a Triangle Solve triangle ABC. 15. a = 20, b = 25, c = 2216ESolving a Triangle Solve triangle ABC. 17. b = 125, c = 162, B = 40Solving a Triangle Solve triangle ABC. 18. a = 65, c = 50, C = 5219ESolving a Triangle Solve triangle ABC. 20. a = 73.5, B = 61, C = 8321E22E23E24E25E26E27E28E29E30E31EHerons Formula Find the area of the triangle whose sides have the given lengths. 32. a = 11, b = 100, c = 10133E34E35E36EArea of a Region Three circles of radii 4, 5, and 6 cm are mutually tangent. Find the shaded area enclosed between the circles.Finding a Length In the figure, triangle ABC is a right triangle, CQ = 6, and BQ = 4. Also, AQC = 30 and CQB = 45. Find the length of AQ. [Hint: First use the Law of Cosines to find expressions for a2, b2, and c2.]Surveying To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information.Geometry A parallelogram has sides of lengths 3 and 5, and one angle is 50. Find the lengths of the diagonals.Calculating Distance Two straight roads diverge at an angle of 65. Two cars leave the intersection at 2:00 p.m., one traveling at 50 mi/h and the other at 30 mi/h. How far apart are the cars at 2:30 p.m.?Calculating Distance A car travels along a straight road, heading east for 1 h, then traveling for 30 min on another road that leads northeast. If the car has maintained a constant speed of 40 mi/h, how far is it from its starting position?Dead Reckoning A pilot flies in a straight path for 1 h 30 min. She then makes a course correction, heading 10 to the right of her original course, and flies 2 h in the new direction. If she maintains a constant speed of 625 mi/h, how far is she from her starting position?Navigation Two boats leave the same port at the same time. One travels at a speed of 30 mi/h in the direction N 50E, and the other travels at a speed of 26 mi/h in a direction S 70 E (see the figure). How far apart are the two boats after 1 h?45ENavigation Airport B is 300 mi from airport A at a bearing N 50 E (see the figure). A pilot wishing to fly from A to B mistakenly flies due east at 200 mi/h for 30 min, when he notices his error. (a) How far is the pilot from his destination at the time he notices the error? (b) What bearing should he head his plane to arrive at airport B?Triangular Field A triangular field has sides of lengths 22, 36, and 44 yd. Find the largest angle.Towing a Barge Two tugboats that are 120 ft apart pull a barge, as shown. If the length of one cable is 212 ft and the length of the other is 230 ft, find the angle formed by the two cables.Flying Kites A boy is flying two kites at the same time. He has 380 ft of line out to one kite and 420 ft to the other. He estimates the angle between the two lines to be 30. Approximate the distance between the kites.Securing a Tower A 125-ft tower is located on the side of a mountain that is inclined 32 to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 ft downhill from the base of the tower. Find the shortest length of wire needed.Cable Car A steep mountain is inclined 74 to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 800 ft from the base to the top of the mountain, as shown. Find the shortest length of cable needed.CN Tower The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft above the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is 43. She also observes that the angle between the vertical and the line of sight to one of the landmarks is 62 and that to the other landmark is 54. Find the distance between the two landmarks.Land Value Land in downtown Columbia is valued at 20 a square foot. What is the value of a triangular lot with sides of lengths 112, 148, and 190 ft?54EPROVE: Projection Laws Prove that in triangle ABC a=bcosC+ccosBb=ccosA+acosCc=acosB+bcosA These are called the Projection Laws. [Hint: To get the first equation, add the second and third equations in the Law of Cosines and solve for a.](a) How is the degree measure of an angle defined? (b) How is the radian measure of an angle defined? (c) How do you convert from degrees to radians? Convert 45 to radians. (d) How do you convert from radians to degrees? Convert 2 rad to degrees.2RCC3RCC4RCC5RCC6RCC7RCC(a) Describe the steps we use to find the value of a trigonometric function of an angle . (b) Find sin 5/6.9RCC10RCC11RCC12RCC13RCC1RE2RE3RE4RE5REA central angle in a circle of radius 2.5 cm is subtended by an arc of length 7 cm. Find the measure of in degrees and radians.7REA circular arc of length 13 m subtends a central angle of 130. Find the radius of the circle.9RENew York and Los Angeles are 2450 mi apart. Find the angle that the arc between these two cities subtends at the center of the earth. (The radius of the earth is 3960 mi.)11RE12RE