Right Triangle Trigonometry


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1 Right Tringle Trigonometry To the ncient Greeks, trigonometry ws the study of right tringles. Trigonometric functions (sine, cosine, tngent, cotngent, secnt, nd cosecnt) cn be defined s right tringle rtios (rtios of the lengths of sides of right tringle). Thousnds of yers lter, we still find pplictions of right tringle trigonometry tody in sports, surveying, nvigtion,* nd engineering. Jke Rjs/Photonic/Getty Imges, Inc. *Section.5, Emple 7 nd Eercises nd
2 IN THIS CHAPTER, we will review ngles, degree mesure, nd specil right tringles. We will discuss the properties of similr tringles. We will use the concept of similr right tringles to define the si trigonometric functions s rtios of the lengths of the sides of right tringles (right tringle trigonometry). RIGHT TRIANGLE TRIGONOMETRY. Angles, Degrees, nd Tringles. Similr Tringles.3 Definition of Trigonometric Functions: Right Tringle Rtios.4 Evluting Trigonometric Functions: Ectly nd with Clcultors.5 Solving Right Tringles Angles nd Degree Mesure Tringles Specil Right Tringles Finding Angle Mesures Using Geometry Clssifiction of Tringles Trigonometric Functions: Right Tringle Rtios Cofunctions Evluting Trigonometric Functions Ectly for Specil Angle Mesures: 30, 45, nd 60 Using Clcultors to Evlute (Approimte) Trigonometric Function Vlues Representing Prtil Degrees: DD or DMS Accurcy nd Significnt Digits Solving Right Tringle Given n Acute Angle Mesure nd Side Length Solving Right Tringle Given the Lengths of Two Sides LEARNING OBJECTIVES Understnd degree mesure. Lern the conditions tht mke two tringles similr. Define the si trigonometric functions s rtios of lengths of the sides of right tringles. Evlute trigonometric functions ectly nd with clcultors. Solve right tringles. 3
3 SECTION. ANGLES, DEGREES, AND TRIANGLES SKILLS OBJECTIVES Find the complement of n ngle. Find the supplement of n ngle. Use the Pythgoren theorem to solve right tringle. Solve nd tringles. CONCEPTUAL OBJECTIVES Understnd tht degrees re mesure of n ngle. Understnd tht the Pythgoren theorem pplies only to right tringles. Understnd specil reltionships between sides of nd tringles. Angles nd Degree Mesure A B Line AB A B Segment AB A B Ry AB Angle Verte The study of trigonometry relies hevily on the concept of ngles. Before we define ngles, let us review some bsic terminology. A line is the stright pth connecting two points (A nd B) nd etending beyond the points in both directions. The portion of the line between the two points (including the points) is clled line segment. A ry is the portion of the line tht strts t one point (A) nd etends to infinity (beyond B). Point A is clled the endpoint of the ry. In geometry, n ngle is formed when two rys shre the sme endpoint. The common endpoint is clled the verte. In trigonometry, we sy tht n ngle is formed when ry is rotted round its endpoint. The ry in its originl position is clled the initil ry or the initil side of n ngle. In the Crtesin plne (the rectngulr coordinte plne), we usully ssume the initil side of n ngle is the positive is nd the verte is locted t the origin. The ry fter it is rotted is clled the terminl ry or the terminl side of n ngle. Rottion in counterclockwise direction corresponds to positive ngle, wheres rottion in clockwise direction corresponds to negtive ngle. Study Tip Positive ngle: Counterclockwise Negtive ngle: Clockwise Terminl side Positive ngle Initil side Terminl side Initil side Negtive ngle Lengths, or distnces, cn be mesured in different units: feet, miles, nd meters re three common units. To compre ngles of different sizes, we need stndrd unit of mesure. One wy to mesure the size of n ngle is with degree mesure. We discuss degrees now, nd in Chpter 3, we discuss nother ngle mesure clled rdins. D EFINITION Degree Mesure of Angles An ngle formed by one complete counterclockwise rottion hs mesure 360 degrees, denoted 360. One complete revolution = 360º Therefore, counterclockwise 360 of rottion hs mesure degree. 4
4 . Angles, Degrees, nd Tringles 5 WORDS 360 represents complete rottion. 80 represents rottion represents rottion. MATH complete rottion complete rottion complete rottion The Greek letter u (thet) is the most common nme for n ngle in mthemtics. Other common nmes for ngles re (lph), b (bet), nd g (gmm). WORDS An ngle mesuring ectly 90 is clled right ngle. A right ngle is often represented by the djcent sides of rectngle, indicting tht the two rys re perpendiculr. MATH = 90º Right ngle: 4 rottion Study Tip Greek letters re often used to denote ngles in trigonometry. An ngle mesuring ectly 80 is clled stright ngle. = 80º Stright ngle: rottion An ngle mesuring greter thn 0, but less thn 90, is clled n cute ngle. Acute ngle 0º < < 90º An ngle mesuring greter thn 90, but less thn 80, is clled n obtuse ngle. Obtuse ngle 90º < < 80º If the sum of the mesures of two positive ngles is 90, the ngles re clled complementry. We sy tht is the complement of b (nd vice vers). Complementry ngles + = 90º If the sum of the mesures of two positive ngles is 80, the ngles re clled supplementry. We sy tht is the supplement of b (nd vice vers). Supplementry ngles + = 80º
5 6 CHAPTER Right Tringle Trigonometry Clssroom Emple... Find the complement of 3.. b. Find the supplement of 63.. c.* Find the complement of, ssuming tht d.* Find two supplementry ngles for which one is 5 less thn 4 times the other. Answer: b. 6.8 c. 90 d. 39, 4 EXAMPLE Finding Mesures of Complementry nd Supplementry Angles Find the mesure of ech ngle:. Find the complement of 50. b. Find the supplement of 0. c. Represent the complement of in terms of. d. Find two supplementry ngles such tht the first ngle is twice s lrge s the second ngle. Solution:. The sum of complementry ngles is 90. Solve for u. b. The sum of supplementry ngles is 80. Solve for u. c. Let b be the complement of. The sum of complementry ngles is 90. Solve for b. d. The sum of supplementry ngles is 80. Let b. Solve for. Substitute 60 into b. u u 40 u 0 80 u 70 b 90 b 90 b b 0 The ngles hve mesures 60 nd 0. Answer: The ngles hve mesures 45 nd 35. YOUR TURN Find two supplementry ngles such tht the first ngle is three times s lrge s the second ngle. Tringles Trigonometry originted s the study of tringles, with emphsis on clcultions involving the lengths of sides nd the mesures of ngles. Tringles re threesided closedplne figures. An importnt property of tringles is tht the sum of the mesures of the three ngles of ny tringle is 80. ANGLE SUM OF A TRIANGLE The sum of the mesures of the ngles of ny tringle is = 80º
6 . Angles, Degrees, nd Tringles 7 EXAMPLE Finding n Angle of Tringle If two ngles of tringle hve mesures 3 nd 68, wht is the mesure of the third ngle? Clssroom Emple.. If g b nd b 3, find ll three ngles of the tringle, b, nd g. Answer: 8, 54, 08 3º 68º Solution: The sum of the mesures of ll three ngles is Solve for. 80 YOUR TURN If two ngles of tringle hve mesures 6 nd 96, wht is the mesure of the third ngle? Answer: 68 In geometry, some tringles re clssified s equilterl, isosceles, nd right. An equilterl tringle hs three equl sides nd therefore hs three equl ngles (60 ). An isosceles tringle hs two equl sides (legs) nd therefore hs two equl ngles opposite those legs. The most importnt tringle tht we will discuss in this course is right tringle. A right tringle is tringle in which one of the ngles is right ngle (90 ). Since one ngle is 90, the other two ngles must be complementry (sum to 90 ) so tht the sum of ll three ngles is 80. The longest side of right tringle, clled the hypotenuse, is opposite the right ngle. The other two sides re clled the legs of the right tringle. Study Tip In this book when we sy equl ngles, this implies equl ngle mesures. Similrly, when we sy n ngle is, this implies tht the ngle s mesure is. Hypotenuse Right tringle: Leg Leg The Pythgoren theorem reltes the sides of right tringle. It is importnt to note tht length ( synonym of distnce) is lwys positive. PYTHAGOREAN THEOREM In ny right tringle, the squre of the length of the longest side (hypotenuse) is equl to the sum of the squres of the lengths of the other two sides (legs). b c c b
7 8 CHAPTER Right Tringle Trigonometry It is importnt to note tht the Pythgoren theorem pplies only to right tringles. It is lso importnt to note tht it does not mtter which leg is clled or b s long s the squre of the longest side is equl to the sum of the squres of the smller sides. Clssroom Emple..3 Suppose you hve 3foot ldder nd wnt to rech height of feet to clen out the gutters on your house. How fr from the bse of the house should the bse of the ldder be? Answer: 5 ft EXAMPLE 3 Using the Pythgoren Theorem to Find the Side of Right Tringle Suppose you hve 0foot ldder nd wnt to rech height of 8 feet to clen out the gutters on your house. How fr from the bse of the house should the bse of the ldder be? Solution: Lbel the unknown side s. Apply the Pythgoren theorem. Simplify. Solve for ft? ft Length must be positive. The ldder should be 6 feet 6 from the bse of the house long the ground. Answer: 5 ft YOUR TURN A steep rmp is being built for skteborders. The height, horizontl ground distnce, nd rmp length form right tringle. If the height is 9 feet nd the horizontl ground distnce is feet, wht is the length of the rmp? When solving right tringle ectly, simplifiction of rdicls is often necessry. For emple, if side length of tringle resulted in 7, the rdicl cnnot be simplified ny further. However, if side length of tringle resulted in 0, the rdicl would be simplified: EXAMPLE Using the Pythgoren Theorem with Rdicls Use the Pythgoren theorem to solve for the unknown side length in the given right tringle. Epress your nswer ectly in terms of simplified rdicls. Solution: Apply the Pythgoren theorem. Simplify known squres. Solve for. The side length is distnce tht is positive Simplify the rdicl } 7 3
8 . Angles, Degrees, nd Tringles 9 YOUR TURN Use the Pythgoren theorem to solve for the unknown side length in the given right tringle. Epress your nswer ectly in terms of simplified rdicls. 8 Answer: 43 4 Specil Right Tringles Right tringles whose sides re in the rtios of 345, 53, nd re emples of right tringles tht re specil becuse their side lengths re equl to whole numbers tht stisfy the Pythgoren theorem. A Pythgoren triple consists of three positive integers tht stisfy the Pythgoren theorem There re two other specil right tringles tht wrrnt ttention: tringle nd tringle. Although in trigonometry we focus more on the ngles thn on the side lengths, we re interested in specil reltionships between the lengths of the sides of these right tringles. We will strt with tringle. WORDS A tringle is n isosceles (two legs re equl) right tringle. MATH 45º 45º Apply the Pythgoren theorem. Simplify the left side of the eqution. Solve for the hypotenuse. The nd the hypotenuse re both lengths nd, therefore, must be positive. hypotenuse hypotenuse hypotenuse ƒ ƒ hypotenuse This shows tht the hypotenuse of is times the length of either leg. 45º 45º If we let, then the tringle will hve legs with length equl to nd the hypotenuse will hve length. Notice tht these lengths stisfy the Pythgoren theorem: A B, or. Lter when we discuss the unit circle pproch, we will let the hypotenuse hve length. The legs will then hve lengths nd : b b
9 0 CHAPTER Right Tringle Trigonometry Study Tip To solve tringle mens to find ll of the ngle mesures nd side lengths. EXAMPLE 5 Solving Tringle A house hs roof with 45 pitch (the ngle the roof mkes with the house). If the house is 60 feet wide, wht re the lengths of the sides of the roof tht form the ttic? Round to the nerest foot. 45º 45º 60 ft Clssroom Emple..5 A house hs roof with 45 pitch. If the house is 48 feet wide, wht re the lengths of the sides of the roof tht form the ttic? Round to the nerest foot. Answer: 4 34 ft Solution: Drw the tringle. Let represent the length of the unknown legs. 45º 45º 60 ft Recll tht the hypotenuse of tringle is times the length of either leg. 45º 45º Technology Tip 60 To clculte, press 6 0 nd ) ENTER Let the hypotenuse equl 60 feet. Solve for ft 4 ft 45º 45º 60 ft Round to the nerest foot. 4 ft Answer: ft YOUR TURN A house hs roof with 45 pitch. If the sides of the roof re 60 feet, how wide is the house? Round to the nerest foot.
10 . Angles, Degrees, nd Tringles Let us now determine the reltionship of the sides of tringle. We strt with n equilterl tringle (equl sides nd equl 60 ngles). WORDS Drw n equilterl tringle with sides. MATH 60º 60º 60º Drw line segment from one verte tht is perpendiculr to the opposite side; this line segment represents the height of the tringle h nd bisects the bse. There re now two identicl tringles. 30º 30º h 60º 60º Notice tht in ech tringle the hypotenuse is twice the length of the shortest leg, which is opposite the 30 ngle. 30º h 60º To find the length h, use the Pythgoren theorem. Solve for h. Both h nd re lengths nd must be positive. We see tht the hypotenuse of tringle is two times the length of the leg opposite the 30 ngle, the short side. The leg opposite the 60 ngle, the long leg, is 3 times the length of the leg opposite the 30 ngle, the short leg. h () h 4 h 3 h 3 3 ƒ ƒ h 3 30º 3 60º
11 CHAPTER Right Tringle Trigonometry If we let, then the tringle will hve legs with lengths nd 3 nd hypotenuse of length. These lengths stisfy the Pythgoren theorem: ( 3), or 4 4. Lter when we discuss the unit circle pproch, we will let the hypotenuse hve length. 3 The legs will then hve lengths nd. b 3 b EXAMPLE 6 Solving Tringle Before hurricne strikes it is wise to stke down trees for dditionl support during the storm. If the brnches llow for the rope to be tied 5 feet up the tree nd desired ngle between the rope nd the ground is 60, how much totl rope is needed? How fr from the bse of the tree should ech of the two stkes be hmmered? 30 60º 60º Solution: Recll the reltionship between the sides of tringle. Lbel ech side. In this cse, the leg opposite the 60 ngle is 5 feet. 3 = 5 ft 30º 60º Solve for. Find the length of the hypotenuse. The ropes should be stked pproimtely 8.7 feet from the bse of the tree, nd pproimtely (7) 34 totl feet of rope will be needed. 3 5 ft ft 3 hypotenuse 5 b 7 ft ft 7 ft 60º 60º 8.7 ft Answer: Ech rope should be stked pproimtely feet from the bse of the tree. Approimtely 46 totl feet of rope will be needed. YOUR TURN Rework Emple 6 with height (where the ropes re tied) of 0 feet. How fr from the bse of the tree should ech of the ropes be stked nd how much totl rope will be needed?
12 . Angles, Degrees, nd Tringles 3 SECTION. SUMMARY SMH In this section, you hve prcticed working with ngles nd tringles. One unit of mesure of ngles is degrees. An ngle mesuring ectly 90 is clled right ngle. The sum of the three ngles of ny tringle is lwys 80. Tringles tht contin right ngle re clled right tringles. With the Pythgoren theorem, you cn solve for one side of right tringle given the other two sides. The right tringles nd re specil becuse of the simple numericl reltions of their sides. 45º 45º 3 30º 60º SECTION. EXERCISES SKILLS In Eercises 8, specify the mesure of the ngle in degrees using the correct lgebric sign ( or ). 4. rottion counterclockwise. rottion counterclockwise 3. rottion clockwise 4. rottion clockwise rottion counterclockwise 6. rottion counterclockwise 7. rottion clockwise 8. rottion clockwise In Eercises 9 4, find () the complement nd (b) the supplement of ech of the given ngles In Eercises 5 8, find the mesure of ech ngle (3)º (5)º (6)º (4)º 7. Supplementry ngles with mesures 8 degrees nd 4 degrees 8. Complementry ngles with mesures 3 5 degrees nd 0 0 degrees In Eercises 9 4, refer to the tringle in the drwing. 9. If 7 nd b 33, find g. 0. If 0 nd b 45, find g.. If g b nd 4b, find ll three ngles.. If g b nd 3b, find ll three ngles. 3. If b 53.3 nd g 3.6, find. 4. If 05.6 nd g 3., find b. + + = 80º
13 4 CHAPTER Right Tringle Trigonometry In Eercises 5 34, refer to the right tringle in the drwing. Epress lengths ectly. 5. If 4 nd b 3, find c. 6. If 3 nd b 3, find c. 7. If 6 nd c 0, find b. 8. If b 7 nd c, find. 9. If 8 nd b 5, find c. 30. If 6 nd b 5, find c. 3. If 7 nd c, find b. 3. If b 5 nd c 9, find. 33. If b 7 nd c 5, find. 34. If 5 nd c 0, find b. b c In Eercises 35 40, refer to the tringle in the drwing. Epress lengths ectly. 35. If ech leg hs length 0 inches, how long is the hypotenuse? 36. If ech leg hs length 8 meters, how long is the hypotenuse? 45º 37. If the hypotenuse hs length centimeters, how long is ech leg? 38. If the hypotenuse hs length 0 feet, how long is ech leg? 45º 39. If ech leg hs length 4 inches, how long is the hypotenuse? 40. If the hypotenuse hs length 6 meters, how long is ech leg? In Eercises 4 46, refer to the tringle in the drwing. Epress lengths ectly. 4. If the shortest leg hs length 5 meters, wht re the lengths of the other leg nd the hypotenuse? 4. If the shortest leg hs length 9 feet, wht re the lengths of the other leg nd the hypotenuse? 43. If the longer leg hs length yrds, wht re the lengths of the other leg nd the hypotenuse? 44. If the longer leg hs length n units, wht re the lengths of the other leg nd the hypotenuse? 45. If the hypotenuse hs length 0 inches, wht re the lengths of the two legs? 46. If the hypotenuse hs length 8 centimeters, wht re the lengths of the two legs? 3 30º 60º APPLICATIONS 47. Clock. Wht is the mesure of the ngle (in degrees) tht the minute hnd trces in 0 minutes? 48. Clock. Wht is the mesure of the ngle (in degrees) tht the minute hnd trces in 5 minutes? 49. London Eye. The London Eye (similr to bicycle wheel) mkes one rottion in pproimtely 30 minutes. Wht is the mesure of the ngle (in degrees) tht crt (spoke) will rotte in minutes? 50. London Eye. The London Eye (similr to bicycle wheel) mkes one rottion in pproimtely 30 minutes. Wht is the mesure of the ngle (in degrees) tht crt (spoke) will rotte in 5 minutes? 5. Revolving Resturnt. If revolving resturnt cn rotte 70 in 45 minutes, how long does it tke for the resturnt to mke complete revolution? 5. Revolving Resturnt. If revolving resturnt cn rotte 7 in 9 minutes, how long does it tke for the resturnt to mke complete revolution? Dvid Bll/Inde Stock/Photolibrry
14 . Angles, Degrees, nd Tringles Field Tril. In Lbrdor retriever field tril, dog is judged by the strightness of the line it tkes to retrieve fllen bird. The competitors re required to go through the wter, not long the shore. If the judge wnts to clculte how fr dog will trvel long stright pth, she wlks the two legs of right tringle s shown in the drwing nd uses the Pythgoren theorem. How fr would this dog trvel (run nd swim) if it trveled long the hypotenuse? Round to the nerest foot. 80 ft 30 ft Pond 54. Field Tril. How fr would the dog in Eercise 53 run nd swim if it trveled long the hypotenuse if the judge wlks 5 feet long the shore nd then 00 feet out to the bird. Round to the nerest foot. 57. Tree Stke. A tree needs to be stked down before storm. If the ropes cn be tied on the tree trunk 7 feet bove the ground nd the stked rope should mke 60 ngle with the ground, how fr from the bse of the tree should ech rope be stked? Round to the nerest foot. 58. Tree Stke. Wht is the totl mount of rope (in feet) tht etends to the two stkes supporting the tree in Eercise 57? 59. Tree Stke. A tree needs to be stked down before storm. If the ropes cn be tied on the tree trunk 0 feet bove the ground nd the stked rope should mke 30 ngle with the ground, how fr from the bse of the tree should ech rope be stked? Round to the nerest foot. 60. Tree Stke. Wht is the totl mount of rope (in feet) tht etends to the four stkes supporting the tree in Eercise 59? Round to the nerest foot. 6. Prty Tent. Steve nd Peggy wnt to rent 40foot by 0foot tent for their bckyrd to host brbecue. The bse of the tent is supported 7 feet bove the ground by poles nd then roped stkes re used for support. The ropes mke 60 ngle with the ground. How lrge footprint in their yrd would they need for this tent (nd stked ropes)? In other words, wht re the dimensions of the rectngle formed by the stkes on the ground? Round to the nerest foot. 00 ft 5 ft Pond istockphoto 55. Christms Lights. A couple wnt to put up Christms lights long the roofline of their house. If the front of the house is 00 feet wide nd the roof hs 45 pitch, how mny liner feet of Christms lights should the couple buy? Round to the nerest foot. 45º 00 ft 56. Christms Lights. Repet Eercise 55 for house tht is 60 feet wide. Round to the nerest foot. 45º 6. Prty Tent. Ashley s prents re throwing grdution prty nd re renting 40foot by 80foot tent for their bckyrd. The bse of the tent is supported 7 feet bove the ground by poles nd then roped stkes re used for support. The ropes mke 45 ngle with the ground. How lrge footprint (see Eercise 6) in their yrd will they need for this tent (nd stked ropes)? Round to the nerest foot.
15 6 CHAPTER Right Tringle Trigonometry 63. Fence Corner. Ben is checking to see if his fence mesures 90 t the corner. To do so, he mesures out 0 feet long the fence nd plces stke mrking it point A. He then goes bck to the corner nd mesures out 5 feet long the fence in the other direction nd plces stke mrking it point B s shown in the figure below. Finlly, he mesures the distnce between points A nd B nd finds it to be 0 feet. Does his corner mesure 90? 64. Fence Corner. For nother corner of Ben s fence (see Eercise 63), he mesures from the corner up one side of the fence 8 feet nd mrks it with stke s point A. He then mesures digonlly cross the yrd 7 feet from point A to the fence going in the other direction s shown nd mrks it s point B. If this corner is 90, wht is the length of the fence from the corner to point B? B 5 ft 0 ft 7 ft A 8 ft 0 ft A B 65. Cr Engine. Bill s cr engine is sid to run t 700 RPM s (revolutions per minute) t idle. Through how mny degrees does his engine turn ech second? 66. Cr Engine. Upon ccelertion, Bill s cr engine turns 300,000 in 5 seconds. At wht RPM (revolutions per minute) is the cr engine turning? CATCH THE MISTAKE In Eercises 67 nd 68, eplin the mistke tht is mde. 67. In tringle, find the length of the side opposite the 60 ngle if the side opposite the 30 ngle is 0 inches. Solution: The length opposite the 60 ngle is twice the length opposite the 30 ngle. (0) 0 The side opposite the 60 ngle hs length 0 inches. This is incorrect. Wht mistke ws mde? 68. In tringle, find the length of the hypotenuse if ech leg hs length 5 centimeters. Solution: Use the Pythgoren theorem. Simplify. Solve for the hypotenuse. 5 5 hypotenuse 50 hypotenuse hypotenuse 5 The hypotenuse hs length 5 centimeters. This is incorrect. Wht mistke ws mde? CONCEPTUAL In Eercises 69 76, determine whether ech sttement is true or flse. 69. The Pythgoren theorem cn be pplied to ny equilterl tringle. 70. The Pythgoren theorem cn be pplied to ll isosceles tringles. 7. The two ngles opposite the legs of right tringle re complementry. 7. In tringle, the length of the side opposite the 60 ngle is twice the length of the side opposite the 30 ngle. 73. The two cute ngles in right tringle must be complementry ngles. 74. In tringle, the length of ech side is the sme. 75. Angle mesure in degrees cn be both positive nd negtive. 76. A right tringle cnnot contin n obtuse ngle.
16 . Similr Tringles 7 CHALLENGE 77. Wht is the mesure (in degrees) of the smller ngle the hour nd minute hnds form when the time is :0? 78. Wht is the mesure (in degrees) of the smller ngle the hour nd minute hnds form when the time is 9:0? In Eercises 79 8, use the figure below: A 8. If AC 30, AB 4, nd DC, find AD. 8. If AB 60, AD 6, nd DC 36, find AC. 83. Given squre with side length, drw the two digonls. The result is 4 specil tringles. Describe these tringles. Wht re the ngle mesures? 84. Solve for in the tringle below. 3 + B D C 79. If AB 3, AD 5, nd AC 58, find DC. 80. If AB 4, AD 5, nd AC 4, find DC. TECHNOLOGY 85. If the shortest leg of tringle hs length 6.68 feet, wht re the lengths of the other leg nd the hypotenuse? Round nswers to two deciml plces. 86. If the longer leg of tringle hs length centimeter, wht re the lengths of the other leg nd the hypotenuse? Round nswers to two deciml plces. SECTION. SIMILAR TRIANGLES SKILLS OBJECTIVES Find ngle mesures using geometry. Use similrity to determine the length of side of tringle. Solve ppliction problems using similr tringles. CONCEPTUAL OBJECTIVE Understnd tht two tringles re similr if they hve equl corresponding ngles. Understnd the difference between congruent nd similr tringles. Finding Angle Mesures Using Geometry We stted in Section. tht the sum of the mesures of three ngles of ny tringle is 80. We now review some ngle reltionships from geometry. Verticl ngles re ngles of equl mesure tht re opposite one nother nd shre the sme verte. In the figure on the right, ngles nd 3 re verticl ngles. Angles nd 4 re lso verticl ngles. A trnsversl is line tht intersects two other coplnr lines. If trnsversl intersects two prllel lines m nd n, or m n, the corresponding ngles hve equl mesure. Angles 3, 4, 5, nd 6 re clssified s interior ngles, wheres ngles,, 7, nd 8 re clssified s eterior ngles. 4 3 m n m n Trnsversl
17 8 CHAPTER Right Tringle Trigonometry In digrms, there re two trditionl wys to indicte ngles hving equl mesure: with the sme number of rcs, or with single rc nd the sme number of hsh mrks. In this tet, we will use the sme number of rcs. Properties of Angles NAME PICTURE RULE Verticl ngles Verticl ngles hve equl mesure nd 4 Alternte interior ngles m n 3 4 m Alternte interior ngles hve equl mesure. 3 6 nd n Alternte eterior ngles m n m Alternte eterior ngles hve equl mesure. 8 nd n Corresponding ngles m n 3 4 m Corresponding ngles hve equl mesure. 5 Interior ngles on the sme side of trnsversl m n n m = 80º = 80º 5 6 n Note tht the following re lso corresponding ngles: nd 6 3 nd 7 4 nd 8 Interior ngles on the sme side of the trnsversl hve mesures tht sum to 80 (they re supplementry).
18 . Similr Tringles 9 EXAMPLE Given tht 0 nd m n, find the mesure of ngle 7. Finding Angle Mesures m n = 0º m 7 =? n Solution: Corresponding ngles hve equl mesure, m n 5. = 0º 5 = 0º 5 nd 7 re supplementry ngles. m n 5 = 0º 7 Mesures of supplementry ngles sum to 80. Substitute Solve for Drw two prllel lines (solid) nd drw two trnsverse lines (dshed), which intersect t point on one of the prllel lines. The corresponding ngles nd 6 hve equl mesure. The verticl ngles nd 4 hve equl mesure. The corresponding ngles 3 nd 5 hve equl mesure. The sum of the mesures of ngles,, nd 3 is 80. Therefore, the sum of the mesures of ngles 4, 5, nd 6 is lso 80. So the sum of the mesures of three ngles of ny tringle is
19 0 CHAPTER Right Tringle Trigonometry Clssifiction of Tringles Angles with equl mesure re often lbeled with the sme number of rcs. Similrly, sides of equl length re often lbeled with the sme number of hsh mrks. NAME PICTURE RULE Equilterl tringle All sides re equl. Isosceles tringle Two sides re equl. Sclene tringle No sides re equl. The word similr in geometry mens identicl in shpe, lthough not necessrily the sme size. It is importnt to note tht two tringles cn hve the ect sme shpe (sme ngles) but be of different sizes. D EFINITION Similr Tringles Similr tringles re tringles with equl corresponding ngle mesures (equl ngles). The word congruent mens equl in ll corresponding mesures; therefore, congruent tringles hve ll corresponding ngles of equl mesure nd ll corresponding sides of equl mesure. Two tringles re congruent if they hve ectly the sme shpe nd size. In other words, if one tringle cn be picked up nd situted on top of nother tringle so tht the two tringles coincide, they re sid to be congruent.
20 . Similr Tringles D EFINITION Congruent Tringles Congruent tringles re tringles with equl corresponding ngle mesures (equl ngles) nd corresponding equl side lengths. Study Tip Similr tringles: Ect sme shpe Congruent tringles: Ect sme shpe nd size It is importnt to note tht ll congruent tringles re lso similr tringles, but not ll similr tringles re congruent tringles. Trigonometry (s you will see in Section.3) relies on the properties of similr tringles. Since similr tringles hve the sme shpe (equl corresponding ngles), the sides opposite the corresponding ngles must be proportionl. Given ny tringle (see figure in the mrgin), if we let correspond to tringle A nd 5 correspond to tringle B, then we would hve the following two tringles. Sides opposite 30 : Sides opposite 60 : Sides opposite 90 : Tringle B Tringle A 5 5 Tringle B 53 Tringle A 3 5 Tringle B 0 Tringle A 5 Notice tht the sides opposite the corresponding ngles re proportionl. This mens tht we will find tht ll three rtios of ech side of tringle B to its corresponding side of tringle A re equl. This proportionlity property holds for ll similr tringles. 3 30º A 60º º B º 3 30º 60º CONDITIONS FOR SIMILAR TRIANGLES One of the following must be verified in order for two tringles to be similr: Corresponding ngles must hve the sme mesure. or Corresponding sides must be proportionl (rtios must be equl) r b br c cr b b' c c' '
21 CHAPTER Right Tringle Trigonometry Let us now use the fct tht corresponding sides of similr tringles re proportionl to determine lengths of sides of similr tringles. Clssroom Emple..* Suppose the following two tringles ABC nd DEF re similr. Find the lengths of sides nd z (in terms of y). A 5y D Answer: 0y, z 30y z y B 3y E y C F EXAMPLE Finding Lengths of Sides in Similr Tringles Given tht the two tringles re similr, find the length of ech of the unknown sides (b nd c). Solution: Solve for b. The corresponding sides re proportionl. Multiply by 5. Solve for b. 8 b 5 b 4(5) b 0 b 8 c 5 6 Solve for c. The corresponding sides re proportionl. Multiply by 6. Solve for c. Check tht the rtios re equl: 8 c 6 c 4(6) c Answer: 5, b 8 YOUR TURN Given tht the two tringles re similr, find the length of ech of the unknown sides ( nd b). 4 7 b 9 5 Applictions Involving Similr Tringles As you hve seen, the common rtios ssocited with similr tringles re very useful. For emple, you cn quickly estimte the heights of flgpoles, trees, nd ny other tll objects by mesuring their shdows long the ground becuse similr tringles re formed. Surveyors rely on the properties of similr tringles to determine distnces tht re difficult to mesure.
22 . Similr Tringles 3 EXAMPLE 3 Clculting the Height of Tree Billy wnts to rent lift to trim his tll trees. However, he must decide which lift he needs: one tht will lift him 5 feet or more epensive lift tht will lift him 50 feet. His wife Jenine hmmered stke into the ground nd by mesuring found its shdow to be.75 feet long nd the tree s shdow to be 9 feet. (Assume both the stke nd tree re perpendiculr to the ground.) If the stke ws stnding 3 feet bove the ground, how tll is the tree? Which lift should Billy rent? Clssroom Emple..3 Re is trying to figure out the height of utility pole. He hs mesured its shdow to be 8 feet long while his 3foottll milbo hs shdow of.3 feet. How tll is the pole? Answer: pproimtely 8.5 ft ft (NOT TO SCALE) 3 ft 9 ft.75 ft Solution: Rys of sunlight re stright nd prllel to ech other. Therefore, the rys mke the sme ngle with the tree tht they do with the stke. Drw nd lbel the two similr tringles. (NOT TO SCALE) The rtios of corresponding sides of similr tringles re equl. Solve for. Simplify (9) The tree is pproimtely 33 feet tll. Billy should rent the more epensive lift to be sfe. YOUR TURN Billy s neighbor decides to do the sme thing. He borrows Jenine s stke nd mesures the shdows. If the shdow of his tree is 5 feet nd the shdow of the stke (3 feet bove the ground) is. feet, how tll is Billy s neighbor s tree? Round to the nerest foot. Answer: pproimtely 38 ft
23 4 CHAPTER Right Tringle Trigonometry SECTION. SUMMARY SMH Similr tringles (the sme shpe) hve corresponding ngles with equl mesure. Congruent tringles (the sme shpe nd size) re similr tringles tht lso hve equl corresponding side lengths. Similr tringles hve the property tht the rtios of their corresponding side lengths re equl. SECTION. EXERCISES SKILLS In Eercises, find the mesure of the indicted ngles, using the digrm on the right.. C 80, find B. 5. F 75, find B.. C 80, find E. 3. C 80, find F. 6. F 75, find A. 7. G 65, find B. 9. A (8) nd D (9 5), find the mesures of A nd D. 4. C 80, find G. 8. G 65, find A. m n A B C D m 0. B (9 7) nd F ( 7), find the mesures of B nd F.. A ( 4) nd G (9 ), find the mesures of A nd G.. C ( 3) nd E (30 5), find the mesures of C nd E. E F G H n In Eercises 3 8, mtch the corresponding tringle with the pproprite nme. 3. Equilterl tringle 4. Right tringle (nonisosceles) 5. Isosceles tringle (nonright) 6. Acute sclene tringle 7. Obtuse sclene tringle 8. Isosceles right tringle. b. c. d. e. f. In Eercises 9 6, clculte the specified lengths, given tht the two tringles re similr. 9. 4, c 6, d, f? 0., b 9, e 3, d?. 3. d 5, e.5, b 7.5,? d. m, f.5 m, c 6.5 km,? 5. b 4 5 in., c 5 in., f 3, e?. 4. e.4, f.6, c 3.9, b? e 0 cm, f 4 cm, c 35 m, b? 6. d 7 m, e m, b mm,? c d f APPLICATIONS b e 7. Height of Tree. The shdow of tree mesures 4 4 feet. 9. Height of Lighthouse. The Cpe Htters Lighthouse t At the sme time of dy, the shdow of 4foot pole the Outer Bnks of North Crolin is the tllest lighthouse mesures feet. How tll is the tree? in North Americ. If 5foot womn csts 5foot shdow nd the lighthouse csts 48foot shdow, 8. Height of Flgpole. The shdow of flgpole pproimtely how tll is the Cpe Htters Lighthouse? mesures 5 foot. At the sme time of dy, the shdow 3 of stke feet bove ground mesures 4 foot. How tll 30. Height of Mn. If 6foot mn csts foot shdow, is the flgpole? how long shdow will his 4foot son cst?
24 . Similr Tringles 5 For Eercises 3 nd 3, refer to the following: Although most people know tht list eists of the Seven Wonders of the Ancient World, only few cn nme them: the Gret Pyrmid of Giz, the Hnging Grdens of Bbylon, the Sttue of Zeus t Olympi, the Temple of Artemis t Ephesus, the Musoleum of Hlicrnssus, the Colossus of Rhodes, nd the Lighthouse of Alendri. 3. Seven Wonders. One of the Seven Wonders of the Ancient World ws lighthouse on the Islnd of Phros in Alendri, Egypt. It is the first lighthouse in recorded history nd ws built bout 80 BC. It survived for 500 yers until it ws completely destroyed by n erthquke in the fourteenth century. On sunny dy, if metertll mn csts shdow pproimtely 5 centimeters (0.05 meters) long, nd the lighthouse pproimtely 3meter shdow, how tll ws this fntstic structure? Mry Evns Picture Librry/Almy In Eercises 33 nd 34, use the drwing below: Islnd In home remodeling project, your rchitect gives you plns tht hve n indicted distnce of r4s, which mesures 4 inch with ruler on the blueprint. 33. Mesurement. How long is the pntry in the kitchen if it mesures 6 inch with ruler? 34. Mesurement. How wide is the islnd if it mesures 7 6 inch with ruler? 35. Cmping. The front of Brin s tent is tringle tht mesures 5 feet high nd 4 feet wide t the bse. The tent hs one zipper down the middle tht is perpendiculr to the bse of the tent nd one zipper tht is prllel to the bse. With the tent unzipped, the opening nd the outline of the tent re equilterl tringles s shown below. If the verticl zipper mesures 3.5 feet, how wide is the opening t the bse? Pntry 3. Seven Wonders. Only one of the gret Seven Wonders of the Ancient World is still stnding the Gret Pyrmid of Giz. Ech of the bse sides long the ground mesures 30 meters. If meter child csts 90centimeter shdow t the sme time the shdow of the pyrmid etends 6 meters long the ground (beyond the bse), pproimtely how tll is the Gret Pyrmid of Giz? 36. Cmping. Zch sees tree growing t n ngle with the ground. If the ngle between the tree nd the ground is 60, how fr is it from the ground to the tree 5 feet from the bse of the tree? 37. Design. Rit is designing logo for her compny s letterhed. As prt of her design, she is trying to include two similr right tringles within circle s shown. If the verticl leg of the lrge tringle mesures.6 inches, while the hypotenuse mesures. inches, wht is the length of the verticl leg of the smller tringle given tht its hypotenuse mesures. inches? Lonely Plnet Imges/Getty Imges, Inc. 38. Design. Rit s supervisor sks tht the tringles in Rit s design (shown in Eercise 37) be isosceles right tringles. If Rit decides to mke the hypotenuse of the lrger tringle inches long, how long re its legs?
25 6 CHAPTER Right Tringle Trigonometry For Eercises 39 4, refer to the following: A collimtor is device used in rdition tretment tht nrrows bems or wves, cusing the wves to be more ligned in specific direction. The use of collimtor fcilittes the focusing of rdition to tret n ffected region of tissue beneth the skin. In the figure, d s is the distnce from the rdition source to the skin, d t is the distnce from the outer lyer of skin to the trgeted tissue, f s is the field size on the skin (dimeter of the circulr treted skin) nd f d is the trgeted field size t depth d t (the dimeter of the trgeted tissue t the specified depth beneth the skin surfce). Rdition source Collimtor 39. Helth/Medicine. Rdition tretment is pplied to field size of 8 centimeters t depth.5 centimeters below the skin surfce. If the tretment hed is positioned 80 centimeters from the skin, find the trgeted field size to the nerest millimeter. 40. Helth/Medicine. Rdition tretment is pplied to field size of 4 centimeters lying t depth of 3.5 centimeters below the skin surfce. If the field size on the skin is required to be 3.8 centimeters, find the distnce from the skin tht the rdition source must be locted to the nerest millimeter. 4. Helth/Medicine. Rdition tretment is pplied to field size on the skin of 3.75 centimeters to rech n ffected region of tissue with field size of 4 centimeters t some depth below the skin. If the tretment hed is positioned 60 centimeters from the skin surfce, find the desired depth below the skin to the trget re to the nerest millimeter. f s f d d s dt 4. Helth/Medicine. Rdition tretment is pplied to field size on the skin of 4.5 centimeters to rech n ffected re lying 4.5 centimeters below the skin surfce. If the tretment hed is positioned 60 centimeters from the skin surfce, find the field size of the trgeted re to the nerest millimeter. CATCH THE MISTAKE In Eercises 43 nd 44, eplin the mistke tht is mde. 43. In the similr tringles shown, if A 8, B 5, nd D 3, find E. 44. In the similr tringles shown in Eercise 43, if A 8, B 5, nd D 3, find F. A C D F B E Solution: Set up rtio of corresponding sides. Substitute A 8, B 5, nd D 3. Cross multiply. Solve for E. This is incorrect. Wht mistke ws mde? A E B D 8 E 5 3 5E 8(3) E 4 5 Solution: Set up rtio of similr tringles. Substitute A 8, B 5, nd D 3. A B D F F Cross multiply. 8F 5(3) Solve for F. F 5 8 This is incorrect. Wht mistke ws mde?
26 . Similr Tringles 7 CONCEPTUAL In Eercises 45 5, determine whether ech sttement is true or flse. 45. Two similr tringles must hve equl corresponding ngles. 49. Alternte interior ngles re supplementry. 46. All congruent tringles re similr, but not ll similr tringles re congruent. 47. Two ngles in tringle cnnot hve mesures 8 nd Two equilterl tringles re similr but do not hve to be congruent. 50. Verticl ngles re congruent. 5. All isosceles tringles re similr to ech other. 5. Corresponding sides of similr tringles re congruent. CHALLENGE 53. Find. 55. Eplin why tringles nd re similr tringles nd why tringles 3 nd 4 re similr tringles Set up the similr tringle rtios for. tringles nd b. tringles 3 nd Use the rtios in Eercise 56 to derive the Lens lw. For Eercises 58 60, use the figure below: 54. Find nd y. BF is prllel to CG, CD is congruent to CG, nd the mesure of ngle A y A 5 8 B E F 0 C G For Eercises 55 57, refer to the following: D The Lens lw reltes three quntities: the distnce from the object to the lens, D o ; the distnce from the lens to the imge, D i ; nd the focl length of the lens, f. D o D i f 58. Nme ll similr tringles in the figure. 59. If AB mesures centimeters nd BC mesures 3 centimeters, find the mesure of EF nd EG. 60. Wht is the mesure of DG? D o Imge H o D o f f 3 f D i f 4 H i Object D i
27 SECTION.3 DEFINITION OF TRIGONOMETRIC FUNCTIONS: RIGHT TRIANGLE RATIOS SKILLS OBJECTIVES Clculte trigonometric rtios of generl ngles. Epress trigonometric function vlues in terms of their cofunctions. CONCEPTUAL OBJECTIVES Understnd tht right tringle rtios re bsed on the properties of similr tringles. Lern the trigonometric functions s rtios of lengths of sides of right tringle. Trigonometric Functions: Right Tringle Rtios The word trigonometry stems from the Greek words trigonon, which mens tringle, nd metrein, which mens to mesure. Trigonometry begn s brnch of geometry nd ws utilized etensively by erly Greek mthemticins to determine unknown distnces. The trigonometric functions were first defined s rtios of side lengths in right tringle. This is the wy we will define them in this section. Since the two ngles besides the right ngle in right tringle hve to be cute, second kind of definition ws needed to etend the domin of trigonometric functions to noncute ngles in the Crtesin plne (Sections. nd.3). Strting in the eighteenth century, broder definitions of the trigonometric functions cme into use, under which the functions re ssocited with points long the unit circle (Section 3.4). In Section., we reviewed the fct tht tringles with equl corresponding ngles lso hve proportionl sides nd re clled similr tringles. The concept of similr tringles (one of the bsic insights in trigonometry) llows us to determine the length of side of one tringle if we know the length of one side of tht tringle nd the length of certin sides of similr tringle. Since the two right tringles to the right hve equl ngles, they re similr tringles, nd the following rtios hold true: r b br c cr c ' c' b b' WORDS Strt with the first rtio. Cross multiply. Divide both sides by bbr. Simplify. Similrly, it cn be shown tht MATH b br c cr r b br br rb br rb bbr bbr r b br nd c r cr. 8
28 .3 Definition of Trigonometric Functions: Right Tringle Rtios 9 Notice tht even though the sizes of the tringles re different, since the corresponding ngles re equl, the rtio of the lengths of the two legs of the lrge tringle is equl to the r rtio of the lengths of the legs of the smll tringle, or b br. Similrly, the rtios of the lengths of leg nd the hypotenuse of the lrge tringle nd the corresponding leg nd b br r hypotenuse of the smll tringle re lso equl; tht is, nd. c cr c cr For ny right tringle, there re si possible rtios of the length of the sides tht cn be clculted for ech cute ngle u: b c c b c b c b c b These rtios re referred to s trigonometric rtios or trigonometric functions, since they depend on u, nd ech is given nme: FUNCTION NAME ABBREVIATION WORDS MATH Sine sin The sine of u sin u Cosine cos The cosine of u cos u Tngent tn The tngent of u tn u Cosecnt csc The cosecnt of u csc u Secnt sec The secnt of u sec u Cotngent cot The cotngent of u cot u Sine, cosine, tngent, cotngent, secnt, nd cosecnt re nmes given to specific rtios of lengths of sides of right tringle. These re the si trigonometric functions. D EFINITION Trigonometric Functions Let u be n cute ngle in right tringle; then sin u b c csc u c b cos u c sec u c tn u b cot u b c b In this right tringle, we sy tht: Side c is the hypotenuse. Side b is the side (leg) opposite ngle u. Side is the side (leg) djcent to ngle u.
29 30 CHAPTER Right Tringle Trigonometry b Also notice tht since sin u b sin u nd cos u then. cos u c b c tn u c, c The three min trigonometric functions should be lerned in terms of the rtios. sin opposite hypotenuse cos djcent hypotenuse tn opposite djcent Study Tip You only need to memorize the three min trigonometric rtios: sin u, cos u, nd tn u. The remining three cn lwys be clculted s reciprocls of the min three for n cute ngle u by remembering the reciprocl identities. Study Tip Trigonometric functions re functions of specified ngle. Alwys specify the ngle. Sin lone mens nothing. Sin u specifies the ngle dependency. The sme is true for the other five trigonometric functions. Study Tip SOHCAHTOA is n crostic tht is often used to remember the right tringle definitions of sine, cosine, nd tngent. SOH: sin u opposite hypotenuse CAH: cos u djcent hypotenuse TOA: tn u opposite djcent The remining three trigonometric functions cn be derived from sin u, cos u, nd tn u using the reciprocl identities. Recll tht the reciprocl of is for 0. RECIPROCAL D EFINITION IDENTITIES csc u sin u sec u cos u Let u be n cute ngle in right tringle; then sin u cos u nd their reciprocls opposite hypotenuse djcent hypotenuse tn u opposite djcent csc u sin u sec u cos u cot u tn u It is importnt to note tht the reciprocl identities only hold for vlues of u tht do not mke the denomintor equl to zero (i.e., when sin u, cos u, or tn u re not equl to 0). Using this terminology, we hve n lterntive definition tht is esier to remember. Trigonometric Functions (Alternte Form) c Hypotenuse Adjcent b Opposite cot u tn u Notice tht the tngent function cn lso be written s quotient of the sine function nd the cosine function: nd similrly, cot u cos u sin u. tn u sin u cos u opposite hypotenuse djcent hypotenuse opposite djcent These reltionships re clled the quotient identities.
30 .3 Definition of Trigonometric Functions: Right Tringle Rtios 3 EXAMPLE Finding Trigonometric Function Vlues of Generl Angle in Right Tringle For the given right tringle, clculte the vlues of sin u, tn u, nd csc u. Clssroom Emple.3.* Compute the si trigonometric functions: sin u, cos u, tn u, cot u, sec u, nd csc u for the following tringle: 4 3! Solution: STEP Find the length of the hypotenuse. Apply the Pythgoren theorem. Lengths of sides cn be only positive Answer: sin u cos u 3 3 tn u cot u 4 sec u 3 csc u 3 4 STEP Lbel the sides of the tringle: with numbers representing lengths. s hypotenuse or s opposite or djcent with respect to u. 4 Opposite 5 Hypotenuse STEP 3 Set up the trigonometric functions s rtios. Sine is Opposite over Hypotenuse (SOH). Tngent is Opposite over Adjcent (TOA). Cosecnt is the reciprocl of sine. 3 Adjcent sin u opposite hypotenuse 4 5 tn u opposite djcent 4 3 csc u sin u YOUR TURN For the tringle in Emple, clculte the vlues of nd cot u. cos u, sec u, Answer: cos u 3 5, sec u 5 3, nd cot u 3 4
31 3 CHAPTER Right Tringle Trigonometry Clssroom Emple.3. Compute the si trigonometric functions: sin u, cos u, tn u, cot u, sec u, nd csc u for the following tringle: EXAMPLE Finding Trigonometric Function Vlues for Generl Angle in Right Tringle For the given right tringle, clculte cos u, tn u, nd sec u Solution: STEP Find the length of the unknown leg Answer: sin u cos u tn u sec u 6 5 Answer: cot u 5 csc u 6 Study Tip 6 5 Becuse we rtionlize epressions contining rdicls in the denomintor, sometimes reciprocl rtios my not lwys look like reciprocl frctions: cos u ; sec u sin u , csc u 65 nd cot u 4 7, 7 STEP STEP 3 Apply the Pythgoren theorem. Lengths of sides cn be only positive. Lbel the sides of the tringle: with numbers representing lengths. s hypotenuse or s opposite or djcent with respect to u. Set up the trigonometric functions s rtios. Cosine is Adjcent over Hypotenuse (CAH). Tngent is Opposite over Adjcent (TOA). Secnt is the reciprocl of cosine. Adjcent 4 cos u 65 4 Epressions tht contin rdicl in the denomintor like cn be rtionlized by 65 multiplying both the numertor nd the denomintor by the rdicl, b (65) djcent hypotenuse 4 65 tn u opposite djcent 7 4 sec u cos u 4 In this emple, the cosine function cn now be written s cos u YOUR TURN For the tringle in Emple, clculte sin u, csc u, nd cot u. 65 Hypotenuse Opposite 7
32 .3 Definition of Trigonometric Functions: Right Tringle Rtios 33 Cofunctions Notice the co in cosine, cosecnt, nd cotngent functions. These cofunctions re bsed on the reltionship of complementry ngles. Let us look t right tringle with lbeled sides nd ngles. c b opposite of b sin b hypotenuse b c djcent to cos hypotenuse b sin b cos c Recll tht the sum of the mesures of the three ngles in tringle is 80. In right tringle, one ngle is 90 ; therefore, the two cute ngles re complementry ngles (the mesures sum to 90 ). You cn see in the tringle bove tht b nd re complementry ngles. In other words, the sine of n ngle is the sme s the cosine of the complement of tht ngle. This is true for ll trigonometric cofunction pirs. COFUNCTION THEOREM A trigonometric function of n ngle is lwys equl to the cofunction of the complement of the ngle. If b 90, then sin b cos sec b csc tn b cot COFUNCTION IDENTITIES sin u cos(90 u) tn u cot(90 u) sec u csc(90 u) cos u sin(90 u) cot u tn(90 u) csc u sec(90 u) c 90º b
33 34 CHAPTER Right Tringle Trigonometry Clssroom Emple.3.3 Write the functions in terms of their cofunctions.. cos(88.4 ) b.* cot( ) c.* sec(90 ) Answer:. sin(.6 ) b. tn(88 ) c. csc() EXAMPLE 3 Writing Trigonometric Function Vlues in Terms of Their Cofunctions Write ech function or function vlue in terms of its cofunction.. sin 30 b. tn c. csc 40 Solution (): Cosine is the cofunction of sine. Substitute u 30. Simplify. Solution (b): Cotngent is the cofunction of tngent. Substitute u. sin u cos(90 u) sin 30 cos(90 30 ) sin 30 cos 60 tn u cot(90 u) tn cot(90 ) Solution (c): Secnt is the cofunction of cosecnt. Substitute u 40. Simplify. csc u sec(90 u) csc 40 sec(90 40 ) csc 40 sec 50 Answer:. sin 45 b. sec(90 y) YOUR TURN Write ech function or function vlue in terms of its cofunction.. cos 45 b. csc y SECTION.3 SUMMARY SMH In this section, we hve defined trigonometric functions s rtios of the lengths of sides of right tringles. This pproch is clled right tringle trigonometry. This is the first of three definitions of trigonometric functions (others will follow in Chpters nd 3). We now cn find trigonometric functions of n cute ngle by tking rtios of the three sides of right tringle: djcent, opposite, nd hypotenuse. sin u opposite hypotenuse cos u djcent hypotenuse tn u opposite djcent It is importnt to remember tht djcent nd opposite re with respect to one of the cute ngles we re considering. We lerned tht trigonometric functions of n ngle re equl to the cofunctions of the complement to the ngle. SECTION.3 EXERCISES SKILLS In Eercises 6, refer to the tringle in the drwing to find the indicted trigonometric function vlues.. sin u. cos u 3. csc u 4. sec u 5. tn u 6. cot u 0 8 6
34 .3 Definition of Trigonometric Functions: Right Tringle Rtios 35 In Eercises 7, refer to the tringle in the drwing to find the indicted trigonometric function vlues. Rtionlize ny denomintors contining rdicls tht you encounter in the nswers. 7. cos u 8. sin u 9. sec u 0. csc u. tn u. cot u In Eercises 3 8, refer to the tringle in the drwing to find the indicted trigonometric function vlues. Rtionlize ny denomintors contining rdicls tht you encounter in the nswers. 3. sin u 4. cos u 5. tn u 6. csc u 7. sec u 8. cot u 5 In Eercises 9 4, use the cofunction identities to fill in the blnks. 9. sin 60 cos 0. sin 45 cos. cos sin. cot A tn 3. csc 30 sec 4. sec B csc 34 In Eercises 5 30, write the trigonometric function vlues in terms of its cofunction. 5. sin( y) 6. sin(60 ) 7. cos(0 A) 8. cos(a B) 9. cot(45 ) 30. sec(30 u) APPLICATIONS For Eercises 3 nd 3, consider the following scenrio: A mn lives in house tht borders psture. He decides to go to the grocery store to get some milk. He is trying to decide whether to drive long the rods in his cr or tke his ll terrin vehicle (ATV) cross the psture. His cr drives fster thn the ATV, but the distnce the ATV would trvel is less thn the distnce he would trvel in his cr. 3. Shortcut. If sin u 3 nd cos u nd if he drove his cr long the streets, it would be 4 miles round trip. How fr would he hve to go on his ATV round trip? Round your nswer to the nerest mile. 3. Shortcut. If tn u nd if he drove his cr long the streets, it would be 00 yrds round trip. How fr would he hve to go on his ATV round trip? Round your nswer to the nerest yrd. 33. Roofing. Bob is told tht the pitch on the roof of his grge is 5, mening tht for every 5 feet the roof increses verticlly, it increses feet horizontlly. If u is defined s the ngle t the corner of the roof formed by the pitch of the roof nd horizontl line, wht is sin u? 34. Roofing. Bob s roof hs 5 pitch while his neighbor s roof hs 7 pitch. With u defined s the ngle formed t the corner of the roof by the pitch of the roof nd horizontl line, whose roof hs lrger vlue for cos u? Eplin. 35. Automotive Design. The ngle u formed by cr s windshield y nd dshbord is such tht tn u 3. Wht is the mesure of ngle u? Dshbord Windshield
35 36 CHAPTER Right Tringle Trigonometry 36. Automotive Design. Consider the informtion relted to the interior of the cr given in Eercise 35. If the verticl distnce y between the top of the windshield nd the horizontl plne of the dshbord is 3 feet, wht is the length of the windshield? 37. Bookshelves. Jun is building bookcse. To give ech shelf etr strength, he is plnning to dd tringleshped brce on ech end. If A B ech brce is right tringle s shown below, such tht 3 cos A 5, find sin B. 4. Forensic Science. If drop of blood found t crime scene hs width of 6 millimeters nd length of millimeters, find the ngle u tht represents the directionlity. 4. Forensic Science. If drop of blood found t crime scene hs width of millimeters nd length of millimeters, find the ngle u tht represents the directionlity. For Eercises 43 nd 44, refer to the following: The monthly profits of PizzRi re function of sles, tht is, p(s). A finncil nlysis hs determined tht the sles s in thousnds of dollrs of PizzRi re lso relted to monthly profits p in thousnds of dollrs by the reltionship tn u p s for 0 s 55 nd 0 p Bookshelves. If the height of the brce shown in Eercise 37 is 8 inches, find the width of the brce. 39. Hiking. Rj nd Ariel re plnning to hike to the top of hill. If u is the ngle formed by the hill nd the ground s shown below, such tht sec u.75, find sin u. Profits (p) p(s) s p Sles (s) 40. Hiking. Hving recently hd knee surgery, Lil is under doctor s orders not to hike nything steeper thn 45 incline. If u is the ngle formed by the hill nd the ground such tht tn u., is the hill too steep for Lil to hike? Eplin. For Eercises 4 nd 4, refer to the following: When trveling through ir, sphericl drop of blood with dimeter d mintins its sphericl shpe until hitting flt surfce. The direction of trvel of the drop of blood dicttes the directionlity of the blood spltter on the surfce. For this reson, the dimeter of the blood drop is equl to the width of the blood spltter on the surfce. The ngle t which sphericl drop of blood is deposited on surfce, clled ngle of impct, is relted to the width w nd the length l of the spltter by sin u w. l Top view Side view Bsed on sles nd profits, it cn be determined tht the domin for ngle u is 0 u 40 The rtio tn u represents the slope of the hypotenuse of the right tringle formed by sles s nd profit p (see figure bove). The ngle u cn be interpreted s mesure of the reltive size of s to p. The lrger the ngle u is, the greter profit is reltive to sles, nd conversely, the smller the ngle u is, the smller profit is reltive to sles. 43. Business. If PizzRi s monthly sles re $5,000 nd monthly profits re $0,000, find. tn u b. cot u 44. Business. If PizzRi s monthly sles re s nd monthly profits re p:. Determine the hypotenuse in terms of s nd p. b. Determine formul for cos u in terms of s nd p. Blood drop Direction of trvel Drop Blood spltter w l
36 .3 Definition of Trigonometric Functions: Right Tringle Rtios 37 CATCH THE MISTAKE In Eercises 45 48, eplin the mistke tht is mde. 45. Clculte sin y. Solution: Write the sine rtio. The opposite side is 4; the hypotenuse side is 5. This is incorrect. Wht mistke ws mde? 46. Clculte tn. Solution: Write the tngent rtio. The djcent side is 3; the opposite side is 4. 4 y 3 5 sin y opposite hypotenuse sin y 4 5 tn djcent opposite tn Clculte sec. Solution: Write the sine rtio. The opposite side is 4; the sin 4 hypotenuse side is 5. 5 Write secnt s the reciprocl sec of sine. sin Simplify. sec 4/5 5 4 This is incorrect. Wht mistke ws mde? 48. Clculte csc y. Solution: Write the cosine rtio. cos y djcent hypotenuse The djcent side is 4; the cos y 4 hypotenuse side is 5. 5 Write cosecnt s the reciprocl of cosine. sin opposite hypotenuse csc y cos y Simplify. csc y 4/5 5 4 This is incorrect. Wht mistke ws mde? This is incorrect. Wht mistke ws mde? CONCEPTUAL In Eercises 49 5, determine whether ech sttement is true or flse. 49. sin 45 cos sin 60 cos sec 60 csc cot 45 tn 45 In Eercises 53 58, use the specil tringles ( nd ) shown below: 53. Clculte sin 30 nd cos Clculte sin 45, cos 45, nd tn Clculte sin 60 nd cos Clculte tn 30 nd tn Clculte sec 45 nd csc Clculte tn 60 nd cot º 3 30º 45º 60º CHALLENGE 59. Wht vlues cn sin u nd cos u tke on? 60. Wht vlues cn tn u nd cot u tke on? 6. Wht vlues cn sec u nd csc u tke on? 6. As u increses from 0 to 90, how does sin u chnge? 63. As u increses from 0 to 90, how does the cofunction of sin u chnge? 64. As u increses from 0 to 90, how does csc u sin u chnge?
37 38 CHAPTER Right Tringle Trigonometry TECHNOLOGY 65. Clculte sec 70 the following two wys:. Find cos 70 to three deciml plces nd then divide by tht number. Write tht number to five deciml plces. b. First find cos 70 nd then find its reciprocl. Round the result to five deciml plces. 66. Clculte csc 40 the following two wys:. Find sin 40, write tht down (round to three deciml plces), nd then divide by tht number. Write this lst result to five deciml plces. b. First find sin 40 nd then find its reciprocl. Round the result to five deciml plces. 67. Clculte cot(54.9 ) the following two wys:. Find tn(54.9 ) to three deciml plces nd then divide by tht number. Write tht number to five deciml plces. b. First find tn(54.9 ) nd then find its reciprocl. Round the result to five deciml plces. 68. Clculte sec(8.6 ) the following two wys:. Find cos(8.6 ) to three deciml plces nd then divide by tht number. Write tht number to five deciml plces. b. First find cos(8.6 ) nd then find its reciprocl. Round the result to five deciml plces. SECTION.4 EVALUATING TRIGONOMETRIC FUNCTIONS: EXACTLY AND WITH CALCULATORS SKILLS OBJECTIVES Evlute trigonometric function vlues ectly for specil ngles. Evlute (pproimte) trigonometric function vlues using clcultor. Represent prtil degrees in either deciml degrees (DD) or degreesminutesseconds (DMS). CONCEPTUAL OBJECTIVE Understnd the difference between ect nd pproimte vlues for trigonometric functions. Evluting Trigonometric Functions Ectly for Specil Angle Mesures: 30, 45, nd 60 We now turn our ttention to evluting trigonometric functions for known ngles. In this section, we will distinguish between evluting trigonometric function ectly nd pproimting the vlue of trigonometric function with clcultor. Throughout this tet instructions will specify which is desired (ect or pproimte) nd the proper nottion, or, will be used. There re three specil cute ngles tht re very importnt in trigonometry: 30, 45, nd 60. In Section., we discussed two importnt tringles: nd Recll the reltionships between the lengths of the sides of these two right tringles. 30º 45º 3 45º 60º
38 .4 Evluting Trigonometric Functions: Ectly nd with Clcultors 39 We cn combine these reltionships with the trigonometric rtios developed in Section.3 to evlute the trigonometric functions for the specil ngle mesures of 30, 45, nd 60. EXAMPLE Evluting the Trigonometric Functions Ectly for 30 Evlute the si trigonometric functions for n ngle tht mesures 30. Solution: Lbel the sides (opposite, djcent, nd hypotenuse) of the right tringle with respect to the 30 ngle. 30º 3 Adjcent Hypotenuse Opposite Use the right tringle rtio definitions of the sine, cosine, nd tngent functions. sin 30 opposite hypotenuse cos 30 djcent hypotenuse tn 30 opposite djcent 3 Use the reciprocl identities to obtin vlues for the vlues of the cosecnt, secnt, nd cotngent functions. csc 30 sin sec 30 cos cot 30 tn Study Tip cos 30 3 is ect, wheres if we evlute with clcultor, we get n pproimtion: cos The si trigonometric functions evluted for n ngle mesuring 30 re sin 30 cos 30 3 tn csc 30 sec cot 30 3 Answer: sin 60 3 csc cos 60 sec 60 YOUR TURN Evlute the si trigonometric functions for n ngle tht mesures 60. tn 60 3 cot
39 40 CHAPTER Right Tringle Trigonometry In compring our nswers in Emple nd the Your Turn, we see tht the following cofunction reltionships re true: sin 30 cos 60 sin 60 cos 30 sec 30 csc 60 sec 60 csc 30 which is epected, since 30 nd 60 re complementry ngles. tn 30 cot 60 tn 60 cot 30 EXAMPLE Evluting the Trigonometric Functions Ectly for 45 Clssroom Emple.4.* Assuming tringle is nd the hypotenuse hs mesure y, find the length of ech leg. Answer: y Evlute the si trigonometric functions for n ngle tht mesures 45. Solution: Lbel the sides of the right tringle with respect to one of the 45 ngles. Opposite Hypotenuse Adjcent 45º Use the right tringle rtio definitions of the sine, cosine, nd tngent functions. Study Tip sin 45 is ect, wheres if we evlute with clcultor, we get n pproimtion: sin sin 45 cos 45 opposite hypotenuse djcent hypotenuse tn 45 opposite djcent Use the reciprocl identities to obtin the vlues of the cosecnt, secnt, nd cotngent functions. csc 45 sin 45 sec 45 cos 45 cot 45 tn 45 The si trigonometric functions evluted for n ngle mesuring 45 re sin 45 csc 45 cos 45 sec 45 tn 45 cot 45 We see tht the following cofunction reltionships re indeed true: sin 45 cos 45 sec 45 csc 45 tn 45 cot 45 which is epected, since 45 nd 45 re complementry ngles.
40 .4 Evluting Trigonometric Functions: Ectly nd with Clcultors 4 The trigonometric function vlues for the three specil ngle mesures (30, 45, nd 60 ) re summrized in the following tble: Trigonometric Function Vlues for Specil Angles (30, 45, nd 60) SIN COS TAN COT SEC CSC Study Tip If you memorize the vlues for sine nd cosine for the ngles given in the tble, then the other trigonometric function vlues in the tble cn be found using the quotient nd reciprocl identities. It is importnt to lern the specil vlues in red for the sine nd cosine functions. All other vlues in the tble cn be found through reciprocls or quotients of these two functions. Remember tht the tngent function is the rtio of the sine to cosine functions. sin u opposite hypotenuse cos u djcent hypotenuse opposite tn u sin u cos u hypotenuse opposite djcent djcent hypotenuse Study Tip SOHCAHTOA: Sine is Opposite over Hypotenuse Cosine is Adjcent over Hypotenuse Tngent is Opposite over Adjcent Using Clcultors to Evlute (Approimte) Trigonometric Function Vlues We will now turn our ttention to using clcultors to evlute trigonometric functions, which sometimes results in n pproimtion. Scientific nd grphing clcultors hve buttons for the sine (sin), cosine (cos), nd tngent (tn) functions. Let us strt with wht we lredy know nd confirm it with our clcultors. EXAMPLE 3 Evluting Trigonometric Functions with Clcultor Use clcultor to find the vlues of. sin 75 b. tn 67 c. sec 5 d. cos 30 Round your nswers to four deciml plces. Solution:. sin b. tn c. sec cos 5 d. cos Note: We know cos YOUR TURN Use clcultor to find the vlues of. cos b. tn 8 c. csc 37 d. sin 45 Round your nswers to four deciml plces. Clssroom Emple.4.3 Use clcultor to find the vlues:. cot 4 b. csc(6.76 ) Round nswers to four deciml plces. Answer:..06 b Study Tip Mke sure your clcultor is set in degrees (DEG) mode. Answer: b c..666 d
41 4 CHAPTER Right Tringle Trigonometry Representing Prtil Degrees: DD or DMS Since one revolution is equl to 360, n ngle of seems very smll. However, there re times when we wnt to brek down degree even further. For emple, if we re off even onethousndth of degree in pointing our ntenn towrd geosttionry stellite, we won t receive signl. Tht is becuse the distnce the signl trvels (35,000 kilometers) is so much lrger thn the size of the stellite (5 meters). Stellite Less thn /000 of degree 35,000 km There re two trditionl wys of representing prt of degree: degreesminutesseconds (DMS) nd deciml degrees (DD). Most clcultors llow you to enter ngles in either DD or DMS formt, nd some even mke the conversion between them. However, we will illustrte mnul conversion technique. The degreesminutesseconds wy of breking down degrees is similr to how we brek down time in hoursminutesseconds. There re 60 minutes in n hour nd 60 seconds in minute, which results in n hour being broken down into 3,600 seconds. Similrly, we cn think of degrees like hours. We cn divide into 60 equl prts. Ech prt is clled minute nd is denoted r. One minute is therefore of degree. We cn then brek down ech minute into 60 seconds. Therefore, second, denoted s, is of minute or of degree r 60 b or 60r s or 60s r 60 br 3600 b The following emples represent how DMS epressions re stted: 60 WORDS degrees, 5 minutes 49 degrees, minutes, 7 seconds MATH 5r 49 r 7s We dd nd subtrct frctionl vlues in DMS form similr to how we dd nd subtrct time. For emple, if Crol nd Donn both run the Boston Mrthon nd Crol crosses the finish line in 5 hours, 4 minutes, nd 0 seconds nd Donn crosses the finish line in 6 hours, 55 minutes, nd seconds, how much time elpsed between the two women crossing the finish line? The nswer is hour, 3 minutes, nd seconds. We combine hours with hours, minutes with minutes, nd seconds with seconds.
42 .4 Evluting Trigonometric Functions: Ectly nd with Clcultors 43 EXAMPLE 4 Adding Degrees in DMS Form Add 7 5r 7s nd 35 7r 5s. Solution: Technology Tip Align degrees with degrees, minutes with minutes, nd seconds with seconds r 7r 7s 5s Add degrees, minutes, nd seconds, respectively. Note tht 69 seconds is equl to minute, 9 seconds nd simplify. 6 6 r 3r 69s r 9s 9s YOUR TURN Add 35 r4s nd 7 5r30s. Answer: 4 7rs EXAMPLE 5 Subtrct 5 8r from 90. Subtrcting Degrees in DMS Form Technology Tip Solution: Align degrees with degrees nd minutes with minutes r 8r Borrow from 90 nd write it s 60 minutes. Subtrct degrees nd minutes, respectively r 5 8r 74 3r Clssroom Emple.4.5 Subtrct 7 48r from 80. Answer: 5 r YOUR TURN Subtrct 3 8r from 90. Answer: 66 5r An lternte wy of representing prts of degrees is with deciml degrees. For emple, 33.4 nd 9.75 re mesures of ngles in deciml degrees. To convert from DD to DMS, multiply the deciml prt once by 60 to get the minutes nd if necessry multiply the resulting deciml prt by 60 to get the seconds. A similr twostge reverse process is necessry for the opposite conversion of DMS to DD. EXAMPLE 6 Converting from DegreesMinutesSeconds to Deciml Degrees Convert 7 39rs to deciml degrees. Round to the nerest thousndth. Solution: Write the number of minutes in deciml form, where r ( 60) b 0.65 Write the number of seconds in deciml form, where. s ( ) 3600 b Write the epression s sum. 7 39rs Add nd round to the nerest thousndth. 7 39rs YOUR TURN Convert 6 8r 5s to deciml degrees. Round to the nerest thousndth. Technology Tip Clssroom Emple.4.6 Convert 8 9r8s to deciml degrees. Answer: 8.34 Answer: 6.38
43 44 CHAPTER Right Tringle Trigonometry Technology Tip To convert from deciml degrees to DMS, press DMS t the end nd APPS 4 ENTER Answer: 35 5r 34s Clssroom Emple.4.8 Evlute the following trigonometric functions for the specified ngle mesures. Round your nswers to four deciml plces.. cos(36 40r) b. cot(3.069 ) Answer: b Technology Tip. b. If the TI clcultor is in degree mode nd ngles re entered without, then degrees will be used. EXAMPLE 7 Converting from Deciml Degrees to DegreesMinutesSeconds Convert to degrees, minutes, nd seconds. Round to the nerest second. Solution: Write s sum. To find the number of minutes, multiply the deciml prt by 60, since 60. Simplify. Write s sum. To find the number of seconds, multiply the deciml by 60, since 60. Simplify. Round to the nerest second. YOUR TURN Convert to degrees, minutes, nd seconds. Round to the nerest second. In this tet, we will primrily use deciml degrees for ngle mesure, nd we will lso use deciml pproimtions for trigonometric function vlues. A common question tht rises is how to round the decimls. There is difference between specifying tht number be rounded to prticulr deciml plce nd specifying rounding to certin number of significnt digits. More discussion on tht topic will follow in the net section when solving right tringles. For now, we will round ngle mesures to the nerest minute in DMS or the nerest hundredth in DD, nd we will round trigonometric function vlues to four deciml plces for convenience. EXAMPLE 8 Evluting Trigonometric Functions with Clcultors Evlute the following trigonometric functions for the specified ngle mesurements. Round your nswers to four deciml plces.. sin(8 9r) b. sec(9.54 ) Solution (): Write 8 9r in deciml degrees b r r 0.8r r 0.8r 60 b r 6.8s r 7s 8 9r b Use clcultor to evlute the sine function. sin(8.5 ) Round to four deciml plces Solution (b): Write secnt s the reciprocl of cosine. sec(9.54 ) cos(9.54 ) Use clcultor to evlute the epression. sec(9.54 ) Round to four deciml plces. sec(9.54 ).49
44 .4 Evluting Trigonometric Functions: Ectly nd with Clcultors 45 SECTION.4 SUMMARY SMH In this section, you hve lerned the ect vlues of the sine, cosine, nd tngent functions for the specil ngle mesures: 30, 45, nd 60. The vlues for ech of the other trigonometric functions cn be determined through reciprocl properties. Clcultors cn be used to pproimte trigonometric vlues of ny cute ngle. Degrees cn be broken down into smller prts using one of two systems: deciml degrees nd degreesminutesseconds. SIN COS TAN SECTION.4 EXERCISES SKILLS In Eercises 6, lbel ech trigonometric function vlue with its correct vlue in (), (b), nd (c). 3. b. c.. sin 30. sin cos cos sin cos 45 sin For Eercises 7 9, use the results in Eercises 6 nd the trigonometric quotient identity, tn, to clculte the cos following vlues. 7. tn tn tn 60 For Eercises 0 8, use the results in Eercises 9 nd the reciprocl identities, csc sec nd cot, sin, cos, tn to clculte the following vlues. 0. csc 30. sec 30. cot csc sec cot csc sec cot 45 In Eercises 9 30, use clcultor to evlute the trigonometric functions for the indicted vlues. Round your nswers to four deciml plces. 9. sin sin(7.8 ). cos 8. cos(.9 ) 3. tn tn(43. ) 5. sec 8 6. sec csc csc 5 9. cot cot 9 In Eercises 3 38, perform the indicted opertions using the following ngles: A 5 7r 9s B 63 8r 35s C 6 r 30s 3. A B 3. A C 33. B C 34. B A 35. B C 36. C A A B In Eercises 39 46, convert from degreesminutesseconds to deciml degrees. Round to the nerest hundredth if only minutes re given nd to the nerest thousndth if seconds re given r r r r r 5s r 30s r s r 9s
45 46 CHAPTER Right Tringle Trigonometry In Eercises 47 54, convert from deciml degrees to degreesminutesseconds. In Eercises 47 50, round to the nerest minute. In Eercises 5 54, round to the nerest second In Eercises 55 60, use clcultor to evlute the trigonometric functions for the indicted vlues. Round your nswers to four deciml plces. 55. sin(0 5r) 56. cos(75 3r) 57. tn( 5r) 58. sec(68 r) 59. csc(8 5r 35s) 60. sec(50 0r 9s) APPLICATIONS For Eercises 6 nd 6, refer to the following: Xry crystllogrphy is method of determining the rrngement of toms within crystl. This method hs reveled the structure nd functioning of mny biologicl molecules including vitmins, drugs, proteins, nd nucleic cids (including DNA). The structure of crystl cn be determined eperimentlly using Brgg s lw: n i sin(u i ) n r sin(u r ) nl d sin u where l is the wvelength of ry (mesured in ngstroms), d is the distnce between tomic plnes (mesured in ngstroms), u is the ngle of reflection (in degrees), nd n is the order of Brgg reflection ( positive integer). 6. Physics/Life Sciences. A diffrctometer ws used to mke diffrction pttern for protein crystl from which it ws determined eperimentlly tht rys of wvelength.54 ngstroms produced n ngle of reflection of 45 corresponding to Brgg reflection of order. Find the distnce between tomic plnes for the protein crystl to the nerest hundredth of n ngstrom. 6. Physics/Life Sciences. A diffrctometer ws used to mke diffrction pttern for slt crystl from which it ws determined eperimentlly tht rys of wvelength.67 ngstroms produced n ngle of reflection of 7.3 corresponding to Brgg reflection of order 4. Find the distnce between tomic plnes for the slt crystl to the nerest hundredth of n ngstrom. For Eercises 63 66, refer to the following: Hve you ever noticed tht if you put stick in the wter, it looks bent? We know the stick didn t bend. Insted, the light rys bend, which mde the imge pper to bend. Light rys propgting from one medium (like ir) to nother medium (like wter) eperience refrction, or bending, with respect to the surfce. Light bends ccording to Snell s lw, which sttes: n i is the refrctive inde of the medium the light is leving, the incident medium. u i is the incident ngle between the light ry nd the norml (perpendiculr) to the interfce between mediums. n r is the refrctive inde of the medium the light is entering. u r is the refrctive ngle between the light ry nd the norml (perpendiculr) to the interfce between mediums. Light ry Surfce Incident ngle Refrctive ngle Clculte the inde of refrction n r of the indicted refrctive medium given the following ssumptions. Round nswers to three deciml plces. The incident medium is ir. Air hs n inde of refrction of n i.00. The incidence ngle is u i Optics. Dimond, u r 64. Optics. Emerld, u r Optics. Wter, u r 66. Optics. Plstic, u r 0 i º r º n i n r
46 .4 Evluting Trigonometric Functions: Ectly nd with Clcultors Sprinkler. A wter sprinkler is plced in the corner of the yrd. If it dissemintes wter through n ngle of 85 8r, how much of the corner is it missing? 68. Sprinkler. Sprinklers re set longside circle drive. If ech sprinkler dissemintes wter through n ngle of 36 9r, wht ngle is covered by both sprinklers? Obstcle Course. If the ldder in Eercise 69 is plced closer to the wll, the formul for distnce trveled becomes d 5. Approimte the distnce trveled by the csc u 4 prticipnts if u HotAir Blloon. A hotir blloon is tethered by ropes on two sides tht form 45 ngle with the ground. If the height of the blloon cn be determined by multiplying the length of one tether by sin 45, find the ect height of the blloon when 00foot ropes re used. 7. HotAir Blloon. A hotir blloon is tethered by ropes on two sides tht form 60 ngle with the ground. If the height of the blloon cn be determined by multiplying the length of one tether by sin 60, find the ect height of the blloon when 00foot ropes re used. 73. Stircse. The pitch of stircse is given s 40 8r7s. Write the pitch in deciml degrees. 69. Obstcle Course. As prt of n obstcle course, prticipnts re required to scend to the top of ldder plced ginst building nd then use rope to climb the rest of the wy to the roof. The distnce trveled cn be clculted using the formul d 5 sin u 43, where u is the ngle the ldder mkes with the ground nd d is the distnce trveled, mesured in feet. Find the ect distnce trveled by the prticipnts if u Stircse. The height, mesured in feet, of certin stircse is given by the formul h 5 tn u, where u is the pitch of the stircse. Wht is the height of stircse with pitch of 39 8r37s? CATCH THE MISTAKE In Eercises 75 nd 76, eplin the mistke tht is mde. 75. sec sec(36 5r) Solution: Solution: Write secnt s the reciprocl sec u Convert 36 5r to cos u of cosine. deciml degrees. Substitute u 60. sec 60 cos 60 Use the reciprocl Recll cos sec 60 identity, sec u 3 cos u. Approimte with Simplify. Rtionlize the denomintor. sec 60 clcultor. 3 sec This is incorrect. Wht mistke ws mde? r 36.5 sec(36.5 ) cos(36.5 ) sec(36.5 ).400 This is incorrect. Wht mistke ws mde?
47 48 CHAPTER Right Tringle Trigonometry CONCEPTUAL In Eercises 77 80, determine whether ech sttement is true or flse. 77. cos 30 sec 30 b 79. When pproimting vlues of sin 0 nd cos 0 with clcultor, it is importnt for your clcultor to be in degree mode. 78. sin tn(0 50r) cot(70 0r) For Eercises 8 84, refer to the following: Thus fr in this tet, we hve only discussed trigonometric vlues of cute ngles, 0 u 90. Wht bout when u is pproimtely 0 or 90? We will formlly consider these cses in the net chpter, but for now, drw nd lbel right tringle tht hs one ngle very close to 0 so tht the opposite side is very smll compred to the djcent side. Then the hypotenuse nd the djcent side re very close in size. Use trigonometric rtios nd the ssumption tht is much lrger thn b to pproimte the vlues without using clcultor. b 8. sin 0 8. cos cos sin 90 CHALLENGE sec 45 tn Find the ect vlue of. cos 30 (sin 45 )(cot 30 ) 86. Find the ect vlue of. csc Find the ect vlue of cos 75 (csc 45 )(cos 30 ), 6 given tht sin Find the ect vlue of cot 75 (cos 45 sin 30 ), given tht tn 5 3. tn(5 0r 5s) sec(46 4r 6s) 89. Clculte. csc(3 7r 3s) 90. Clculte cos(33 38rs) cot(36 58r 6s). csc(50 33rs) TECHNOLOGY In Eercises 9 nd 9, perform the indicted opertions. Which gives you more ccurte vlue? 9. Clculte sec 70 the following two wys:. Write down cos 70 (round to three deciml plces), nd then divide by tht number. Write the number to five deciml plces. b. First find cos 70 nd then find its reciprocl. Round the result to five deciml plces. 9. Clculte csc 40 the following two wys:. Find sin 40 (round to three deciml plces), nd then divide by tht number. Write this lst result to five deciml plces. b. First find sin 40 nd then find its reciprocl. Round the result to five deciml plces. In Eercises 93 nd 94, illustrte clcultor procedures for converting between DMS nd DD. 93. Convert 3 4r5s to deciml degrees. Round to three deciml plces. 94. Convert to degreesminutesseconds. Round to three deciml plces.
48 SECTION.5 SOLVING RIGHT TRIANGLES SKILLS OBJECTIVES Identify the number of significnt digits to epress the lengths of sides nd mesures of ngles when solving right tringles. Solve right tringles given the mesure of n cute ngle nd the length of side. Solve right tringles given two side lengths. CONCEPTUAL OBJECTIVES Recognize the importnce of significnt digits in solving right tringles. Understnd tht the trigonometric inverse keys on clcultor cn be used to pproimte the mesure of n ngle given its trigonometric function vlue. To solve tringle mens to find the mesure of the three ngles nd three side lengths of the tringle. In this section, we will discuss only right tringles. Therefore, we know one ngle hs mesure of 90. We will know some informtion (the lengths of two sides or the length of side nd the mesure of n cute ngle) nd we will determine the mesures of the unknown ngles nd the lengths of the unknown sides. However, before we strt solving right tringles, we must first discuss ccurcy nd significnt digits. Accurcy nd Significnt Digits If we re upgrding our flooring nd quickly mesure room s 0 feet by feet nd we wnt to clculte the digonl length of the room, we use the Pythgoren theorem. WORDS Apply the Pythgoren theorem. Simplify. Use the squre root property. The length of the digonl must be positive. Approimte the rdicl with clcultor. MATH 0 d d 44 d 44 d 44 d Would you sy tht the 0foot by foot room hs digonl of feet? No, becuse the known room mesurements were given only with n ccurcy of foot nd the digonl bove is clculted to eight deciml plces. Your results re no more ccurte thn the lest ccurte mesure in your clcultion. In this emple we round to the nerest foot, nd hence we sy tht the digonl of the 0foot by foot room is bout 6 feet. Significnt digits re used to determine the precision of mesurement. D EFINITION Significnt Digits The number of significnt digits in number is found by counting ll of the digits from left to right strting with the first nonzero digit. The reson for the question mrk net to 8000 is tht we don t know. If 8000 is result from rounding to the nerest thousnd, then it hs one significnt digit. If 8000 is the result of rounding to the nerest ten, then it hs three significnt digits, nd if there re ectly 8000 people surveyed, then 8000 is n ect vlue nd it hs four significnt digits. In this tet we will ssume tht integers hve the gretest number of significnt digits. Therefore, 8000 hs four significnt digits nd cn be epressed in scientific nottion s d =? ft ft 0 ft Study Tip The lest ccurte mesure in your clcultion determines the ccurcy of your result. NUMBER SIGNIFICANT DIGITS ? 49
49 50 CHAPTER Right Tringle Trigonometry In solving right tringles, we first determine which of the given mesurements hs the lest number of significnt digits so we cn round our finl nswers to the sme number of significnt digits. EXAMPLE Identifying the Lest Number of Significnt Digits Determine the number of significnt digits corresponding to the given informtion in the following tringle: mesure of n cute ngle nd side length. In solving this right tringle, wht number of significnt digits should be used to epress the remining ngle mesure nd side lengths? c b 7.3 ft 45º Solution: Determine the significnt digits corresponding to b 45. Two significnt digits Determine the significnt digits corresponding to b 7.3 feet. Three significnt digits In solving this tringle, the remining side lengths of nd c nd the mesure of should be epressed to two significnt digits. Solving Right Tringle Given n Acute Angle Mesure nd Side Length When solving right tringle, we lredy know tht one ngle hs mesure 90. Let us now consider the cse when the mesure of n cute ngle nd side length re given. Since the mesure of one of the cute ngles is given, the remining cute ngle cn be found using the fct tht the sum of three ngles in tringle is 80. Right tringle trigonometry is then used to find the remining side lengths. Clssroom Emple.5. Solve the tringle. 7º c b 4 m Answer: 7. m, c 6 m, b 63 EXAMPLE Solving Right Tringle Given n Angle nd Side Solve the right tringle find, b, nd. Solution: STEP Determine ccurcy. Since the given quntities (5 feet nd 56 ) both re epressed to two significnt digits, we will round finl clculted vlues to two significnt digits. STEP Solve for A. Two cute ngles in right tringle re complementry. 5 ft = 56º Solve for. 34 b
50 .5 Solving Right Tringles 5 STEP 3 Solve for. The cosine of n ngle is equl to the djcent side over the hypotenuse. cos 56 5 Study Tip Mke sure your clcultor is in degrees mode. Solve for. 5 cos 56 Evlute the right side of the epression using clcultor. Round to two significnt digits. STEP 4 Solve for b. Notice tht there re two wys to solve for b: trigonometric functions or the Pythgoren theorem. Although it is tempting to use the Pythgoren theorem, it is better to use the given informtion with trigonometric functions thn to use vlue tht hs lredy been rounded, which could mke results less ccurte. The sine of n ngle is equl to the opposite side over the hypotenuse ft sin 56 b 5 Technology Tip Using TI clcultor to find 5 cos 56 nd 5 sin 56, press 5 COS 5 6 ) ENTER 5 SIN 5 6 ) ENTER Solve for b. Evlute the right side of the epression using clcultor. Round b to two significnt digits. STEP 5 Check the solution. Angles nd sides re rounded to two significnt digits. b 5 sin 56 b b ft 34º Using scientific clcultor, press 5 COS ( 56 ) ENTER 5 SIN ( 56 ) ENTER 5 ft ft Check the trigonometric vlues of the specific ngles by clculting the trigonometric rtios. 56º 8.4 ft sin 34? cos 34? tn 34? YOUR TURN Given the tringle below, solve the right tringle find, b, nd u. 33 in. b Answer: u 53, 6 in., nd b 0 in. 37º
51 5 CHAPTER Right Tringle Trigonometry Solving Right Tringle Given the Lengths of Two Sides When solving right tringle, we lredy know tht one ngle hs mesure 90. Let us now consider the cse when the lengths of two sides re given. In this cse, the third side cn be found using the Pythgoren theorem. If we cn determine the mesure of one of the cute ngles, then we cn find the mesure of the third cute ngle using the fct tht the sum of the three ngle mesures in tringle is 80. How do we find the mesure of one of the cute ngles? Since we know the side lengths, we cn use right tringle rtios to determine the trigonometric function (sine, cosine, or tngent) vlues nd then sk ourselves: Wht ngle corresponds to tht vlue? Sometimes, we my know the nswer ectly. For emple, if we determine tht sin u, then we know tht the cute ngle u is 30 becuse sin 30. Other times we my not know the corresponding ngle, such s sin u Clcultors hve three keys (sin, cos, nd tn ) tht help us determine the unknown ngle. For emple, clcultor cn be used to ssist us in finding wht ngle u corresponds to sin u sin (0.95) At first glnce, these three keys might pper to yield the reciprocl; however, the superscript corresponds to n inverse function. We will lern more bout inverse trigonometric functions in Chpter 6, but for now we will use these three clcultor keys to help us solve right tringles. EXAMPLE 3 Using Clcultor to Determine n Acute Angle Mesure Use clcultor to find u. Round nswers to the nerest degree.. b. cos u tn u.75 Solution (): Use clcultor to evlute the inverse cosine function. Round to the nerest degree. u cos (0.8734) u 9 Solution (b): Use clcultor to evlute the inverse tngent function. Round to the nerest degree. u tn (.75) u 70 Answer: 5 YOUR TURN Use clcultor to find u, given sin u Round the nswer to the nerest degree.
52 .5 Solving Right Tringles 53 EXAMPLE 4 Solve the right tringle find,, nd b. Solution: Solving Right Tringle Given Two Sides STEP Determine ccurcy. The given sides hve four significnt digits; therefore, round finl clculted vlues to four significnt digits. 37. cm 9.67 cm STEP Solve for A. The cosine of n ngle is equl to the 9.67 cm djcent side over the hypotenuse. cos 37. cm Evlute the right side with clcultor. cos Write the ngle in terms of the inverse cosine function. cos ( ) Use clcultor to evlute the inverse cosine function Round to four significnt digits STEP 3 Solve for B. The two cute ngles in right tringle re complementry. b 90 Substitute b 90 Solve for b. b 3.9 The nswer is lredy rounded to four significnt digits. STEP 4 Solve for. Use the Pythgoren theorem since the lengths of two sides re given. Substitute given vlues for b nd c. Solve for. Round to four significnt digits. STEP 5 Check the solution. Angles re rounded to the nerest hundredth degree, nd sides re rounded to four significnt digits of ccurcy. Check the trigonometric vlues of the specific ngles by clculting the trigonometric rtios. sin(3.9 )? b c cm 3.9º sin(58.09 )? cm 3.59 cm 58.09º 9.67 cm Technology Tip Clssroom Emple.5.4 Solve the tringle..83 ft Answer: 9.0 ft, u 78.3, b.7 Study Tip To find length of side in right tringle, use sine, cosine, nd tngent functions when n ngle mesure is given. To find the mesure of n ngle in right tringle, given the proper rtio, use inverse sine, inverse cosine, nd inverse tngent functions when the lengths of the sides re known ft YOUR TURN Solve the right tringle find,, nd b. 3.5 mi 7. mi Answer: 6.0 mi, 43.0, nd b 47.0
53 54 CHAPTER Right Tringle Trigonometry Applictions In mny pplictions of solving right tringles, you re given the length of side nd the mesure of n cute ngle nd sked to find one of the other sides. Two common emples involve n observer (or point of reference) locted on the horizontl nd n object tht is either bove or below the horizontl. If the object is bove the horizontl, then the ngle mde is clled the ngle of elevtion, nd if the object is below the horizontl, then the ngle mde is clled the ngle of depression. For emple, if rce cr driver is looking stright hed (in horizontl line of sight), then looking up is elevtion nd looking down is depression. Angle of elevtion Horizontl Angle of depression John Hrrelson/Stringer/ Getty Imges If the ngle is physicl one (like sktebord rmp), then the pproprite nme is the ngle of inclintion. Angle of inclintion EXAMPLE 5 Angle of Depression (NASCAR) In this picture, cr 9 is behind the leder cr. If the ngle of depression is 8 from the cr 9 s driver s eyes to the bottom of the 3foot high bck end of cr (side opposite the ngle of depression), how fr prt re their bumpers? Assume tht the horizontl distnce from the cr 9 s driver s eyes to the front of his cr is 5 feet. Wlter G Arce/Icon SMI/Corbis Imges
54 .5 Solving Right Tringles 55 Solution: Drw n pproprite right tringle nd lbel the known quntities. Becuse the sides of interest re djcent nd opposite to the known ngle, identify the tngent rtio. Solve for. Evlute the right side. Round to the nerest foot. Subtrct 5 feet from. 3 ft tn 8 3 ft 3 ft tn ft 9 ft ft Their bumpers re pproimtely 4 feet prt. Suppose NASA wnts to tlk with stronuts on the Interntionl Spce Sttion (ISS), which is trveling t speed of 7,700 mph, 400 kilometers (50 miles) bove the surfce of the Erth. If the ntenns t the ground sttion in Houston hve pointing error of even minute, tht is r, they will miss the chnce to tlk with the stronuts. 60 b Pointing Error EXAMPLE 6 Assume tht the ISS (which is 08 meters long nd 73 meters wide) is in 400kilometer low Erth orbit. If the communictions ntenns hve minute pointing error, how mny meters off will the communictions link be? Solution: Drw right tringle tht depicts this scenrio. Becuse the sides of interest re djcent nd opposite to the known ngle, identify the tngent rtio. Solve for. Convert r to deciml degrees. Convert the eqution for in terms of deciml degrees. 400 km (NOT TO SCALE) tn(r) ISS 400 km (400 km) tn(r) 8º r 60 b (400 km) tn( ) Technology Tip 3 To clculte = press tn 8, Clssroom Emple.5.5 A circus specttor is sitting in the blechers so tht her line of sight is t the sfety net, which is 0 feet bove the ground. The trpeze pltform is 80 feet wy from her nd rises 40 feet bove the ground (30 feet bove the net). Find the ngle her eyes mke while wtching the performers lunch off the trpeze pltform. Answer: Clssroom Emple.5.6 Assume tht the ISS (which is 08 meters long nd 73 meters wide) is in 400 kilometer low Erth orbit.. If the communictions ntenns hve 0.00 pointing error, how mny meters off will the communictions link be? b.* Eperiment to find the ngle pointing error such tht the communictions link is off by only meter. Answer:. 7 m b Technology Tip Evlute the epression on the right km 6 m 400 kilometers is ccurte to three significnt digits. So we epress the nswer to three significnt digits. The pointing error cuses the signl to be off by pproimtely 6 meters. Since the ISS is only 08 meters long, it is epected tht the signl will be missed by the stronut crew.
55 56 CHAPTER Right Tringle Trigonometry In nvigtion, the word bering mens the direction vessel is pointed. Heding is the direction the vessel is ctully trveling. Heding nd bering re only synonyms when there is no wind on lnd. Direction is often given s bering, which is the mesure of n cute ngle with respect to the northsouth verticl line. The plne hs bering N 0 E mens tht the plne is pointed 0 to the est of due north. N N 0º E W 0º E S Clssroom Emple.5.7 A jet tkes off bering N 3 W nd flies 6 miles, nd then mkes right turn ( 90 ) nd flies 0 miles frther. If the control tower opertor wnted to locte the plne, wht bering would he use? Round to the nerest degree. Answer: N 4 E EXAMPLE 7 Bering (Nvigtion) A jet tkes off bering N 8 E nd flies 5 miles nd then mkes left (90 ) turn nd flies miles frther. If the control tower opertor wnted to locte the plne, wht bering would she use? Round to the nerest degree. Solution: Drw picture tht represents this scenrio. Identify the tngent rtio. tn u 5 mi N 5 mi 8º Use the inverse tngent function to solve for u. Subtrct 8 from u to find the bering, b. Round to the nerest degree. u tn b b b N 39 W SECTION.5 SUMMARY In this section, we hve solved right tringles. When either side length nd n cute ngle mesure re given or two side lengths re given, it is possible to solve the right tringle (find ll unknown side lengths nd ngle mesures). The lest ccurte number used in your clcultions determines the pproprite number of significnt digits for your results.
56 .5 Solving Right Tringles 57 SECTION.5 EXERCISES SKILLS In Eercises 4, determine the number of significnt digits corresponding to ech of the given ngle mesures nd side lengths b km 4. b 0. mi In Eercises 5 0, use clcultor to find the mesure of ngle. Round nswer to the nerest degree. 5. sin u sin u cos u cos u tn u tn u In Eercises 30, refer to the right tringle digrm nd the given informtion to find the indicted mesure. Write your nswers for ngle mesures in deciml degrees.. b 35, c 7 in. ; find.. b 35, c 7 in. ; find b , c ft; find , c ft; find b , b 4.7 mi; find. 6. b 69.3, 0.75 mi; find b. 7. b 5, km; find c. 8. b 75, b 6 km; find c , 5.37 cm; find c , b 6.79 cm; find c.. 9 mm, c 38 mm; find.. 89 mm, c 99 mm; find b. 3. b.3 m, c 4.9 m; find. 4. b 7.8 m, c 3 m; find b 5. 7r, b 0.8 yd; find. 6. b 7 r, 7.0 yd; find b. 7. b 5 0r, 0. km; find c. 8. b 65 30r, b 8.6 km; find c r0s,,5 km; find c r50s, b 7,986 km; find c. c b In Eercises 3 48, refer to the right tringle digrm nd the given informtion to solve the right tringle. Write your nswers for ngle mesures in deciml degrees nd c ft nd c 37 ft nd b.6 cm nd b 0.0 m nd c 5 in nd c 5.38 in. 37. b 7 nd c 9.7 mm 38. b 45 nd c 7.8 mm nd mi 40. b 47. nd 9.75 mi 4. 45, b 0. km 4. b 85.5, b 4.3 ft r nd b 734 ft r nd 75 ft ft nd b 8.7 ft ft nd c 48.7 ft ,36 km nd c 4,766 km 48. b 0.45 mm nd c mm c b APPLICATIONS 5 ft (NOT TO SCALE) º 49. Golf. If the flgpole tht golfer ims t on green mesures 5 feet from the ground to the top of the flg nd golfer mesures ngle from top to bottom of the pole, how fr (in horizontl distnce) is the golfer from the flg? Round to the nerest foot. 50. Golf. If the flgpole tht golfer ims t on green mesures 5 feet from the ground to the top of the flg nd golfer mesures 3 ngle from top to bottom of the pole, how fr (in horizontl distnce) is the golfer from the flg? Round to the nerest foot.
57 RESCUE 58 CHAPTER Right Tringle Trigonometry Eercises 5 nd 5 illustrte midir refueling scenrio tht militry ircrft often enct. Assume the elevtion ngle tht the hose mkes with the plne being fueled is 36. In Eercises 57 nd 58, refer to the illustrtion below tht shows serch nd rescue helicopter with 30 field of view with serch light. Hose 5. Midir Refueling. If the hose is 50 feet long, wht should be the ltitude difference between the two plnes? Round to the nerest foot. 5. Midir Refueling. If the smllest cceptble ltitude difference between the two plnes is 00 feet, how long should the hose be? Round to the nerest foot. Eercises re bsed on the ide of glide slope (the ngle the flight pth mkes with the ground). Precision Approch Pth Indictor (PAPI) lights re used s visul pproch slope id for pilots lnding ircrft. Typicl glide pth for commercil jet irliners is 3. The spce shuttle hs n outer glide pproch of 8 0. PAPI lights re typiclly configured s row of four lights. All four lights re on, but in different combintions of red or white. If ll four lights re white, then the ngle of descent is too high; if ll four lights re red, then the ngle of descent is too low; nd if there re two white nd two red, then the pproch is perfect. PAPI Runwy = 36º Ground 53. Glide Pth of Commercil Jet Airliner. If commercil jetliner is 5000 feet (bout mile ground distnce) from the runwy, wht should be the ltitude of the plne to chieve red/ white PAPI lights? (Assume tht this corresponds to 3 glide pth.) 54. Glide Pth of Commercil Jet Airliner. If commercil jetliner is t n ltitude of 450 feet when it is 500 feet from the runwy (pproimtely mile ground distnce), wht is the glide slope ngle? Will the pilot see white lights, red lights, or both? 55. Glide Pth of the Spce Shuttle Orbiter. If the pilot of the spce shuttle orbiter is t n ltitude of 3000 feet when she is 5,500 feet (pproimtely 3 miles ground distnce) from the shuttle lnding fcility, wht is her glide slope ngle (round to the nerest degree)? Is she too high or too low? 56. Glide Pth of the Spce Shuttle Orbiter. If the sme pilot in Eercise 55 rises the nose of the gliding shuttle so tht she drops only 500 feet by the time she is 7800 feet from the shuttle lnding strip (ground distnce), wht is her glide ngle t tht time (round to the nerest degree)? Is she within the specs (8 0 ) to lnd the shuttle? b 3º Altitude 30º 57. Serch nd Rescue. If the serch nd rescue helicopter is flying t n ltitude of 50 feet bove se level, wht is the dimeter of the circle illuminted on the surfce of the wter? 58. Serch nd Rescue. If the serch nd rescue helicopter is flying t n ltitude of 500 feet bove se level, wht is the dimeter of the circle illuminted on the surfce of the wter? For Eercises 59 6, refer to the following: Geosttionry orbits re useful becuse they cuse stellite to pper sttionry with respect to fied point on the rotting Erth. As result, n ntenn (dish TV) cn point in fied direction nd mintin link with the stellite. The stellite orbits in the direction of the Erth s rottion t n ltitude of pproimtely 35,000 kilometers. 59. Dish TV. If your dish TV ntenn hs 35,000 km pointing error of s ( second), how long would the stellite hve to be to mintin link? Round your nswer to the nerest meter. 60. Dish TV. If your dish TV ntenn hs s pointing error of (hlf second), how long would the stellite hve to be to mintin link? Round your nswer to the nerest meter. 6. Dish TV. If the stellite in geosttionry orbit (t 35,000 kilometers) ws only 0 meters long, bout how ccurtely pointed would the dish hve to be? Give the nswer in degrees to two significnt digits. 6. Dish TV. If the stellite in geosttionry orbit (t 35,000 kilometers) ws only 30 meters long, bout how ccurtely pointed would the dish hve to be? Give the nswer in degrees to two significnt digits.
58 .5 Solving Right Tringles Angle of Elevtion (Trffic). A person driving in sedn is driving too close to the bck of n 8 wheeler on n interstte highwy. He decides to bck off until he cn see the entire truck (to the top). If the height of the triler is 5 feet nd the sedn driver s ngle of elevtion (to the top of the triler from the horizontl line with the bottom of the triler) is roughly 30, how fr is he sitting from the end of the triler? 5 ft Angle of Depression (Oper). The blcony sets t the oper house hve n ngle of depression of 55 to center stge. If the horizontl (ground) distnce to the center of the stge is 50 feet, how fr re the ptrons in the blcony to the singer t center stge? 50 ft 55º 65. Angle of Inclintion (Skiing). The ngle of inclintion of mountin with triple blck dimond ski pths is 65. If skier t the top of the mountin is t n elevtion of 4000 feet, how long is the ski run from the top to the bse of the mountin? Foto World/The Imge Bnk/ Getty Imges, Inc. 66. Bering (Nvigtion). If plne tkes off bering N 33 W nd flies 6 miles nd then mkes right turn (90 ) nd flies 0 miles frther, wht bering will the trffic controller use to locte the plne? 67. Bering (Nvigtion). If plne tkes off bering N35 E nd flies 3 miles nd then mkes left turn (90 ) nd flies 8 miles frther, wht bering will the trffic controller use to locte the plne? 68. Bering (Nvigtion). If plne tkes off bering N48 W nd flies 6 miles nd then mkes right turn (90 ) nd flies 7 miles frther, wht bering will the trffic controller use to locte the plne? For Eercises 69 nd 70, refer to the following: With the dvent of new technology, tennis rcquets cn now be constructed to permit plyer to serve t speeds in ecess of 0 mph. One of the most effective serves in tennis is power serve tht is hit t top speed directly t the top left corner of the right service court (or top right corner of the left service court). When ttempting this serve, plyer will toss the bll rther high into the ir, bring the rcquet bck, nd then mke contct with the bll t the precise moment when the position of the bll in the ir coincides with the top of the netted prt of the rcquet when the plyer s rm is fully stretched over his or her hed. 69. Tennis. Assume tht the plyer is serving into the right service court nd stnds just inches to the right of the center line behind the bseline. If, t the moment the rcquet strikes the bll, both re 7 inches from the ground nd the serve ctully hits the top left corner of the right service court, determine the ngle t which the bll meets the ground in the right service court. Round to the nerest degree. 70. Tennis. Assume tht the plyer is serving into the right service court nd stnds just inches to the right of the center line behind the bseline. If the bll hits the top left corner of the right service court t n ngle of 44, t wht height bove the ground must the bll be struck? ft 36 ft Doubles 7 ft Singles Center mrk 8 ft ft ft 8 ft ft Side line Post 4 ft 6 in. Alley line Right service court Left service court 3 ft 6 in. Center service line 3 ft 6 in. Right service court Left service court Net Fore court Service line Bck court Bseline Bck screen 4 ft Side screen 78 ft
59 60 CHAPTER Right Tringle Trigonometry For Eercises 7 nd 7, refer to the following: The structure of molecules is criticl to the study of mterils science nd orgnic chemistry, nd hs countless pplictions to vriety of interesting phenomen. Trigonometry plys criticl role in determining bonding ngles of molecules. For instnce, the structure of the (FeCl 4 Br ) 3 ion (dibromtetetrchlorideferrte III) is shown in the figure below. 7. Chemistry. Determine the ngle u [i.e., the ngle between the is contining the picl bromide tom (Br) nd the segment connecting Br to Cl]. Br 73. Nvigtion. A plne tkes off heded due north. With the wind, the irplne ctully trvels on heding of N 5 E. After trveling for 00 miles, how fr north is the plne from its strting position? 74. Nvigtion. A bot must cross 50foot river. While the bot is pointed due est, perpendiculr to the river, the current cuses it to lnd 5 feet down river. Wht is the heding of the bot? For Eercises 75 nd 76, refer to the following: A cnl constructed by wterusers ssocition cn be pproimted by n isosceles tringle (see the figure below). When the cnl ws originlly constructed, the depth of the cnl ws 5.0 feet nd the ngle defining the shpe of the cnl ws 60. Cl.354 Cl Width Depth.9 Fe Cl Cl Br 7. Chemistry. Now, suppose one of the chlorides (Cl) is removed. The resulting structure is trigonl in nture, resulting in the figure below. Does the ngle u chnge? If so, wht is its new vlue? Br 75. Environmentl Science. If the width of the wter surfce tody is 4.0 feet, find the depth of the wter running through the cnl. 76. Environmentl Science. One yer lter survey is performed to mesure the effects of erosion on the cnl. It is determined tht when the wter depth is 4.0 feet, the width of the wter surfce is 5.0 feet. Find the ngle u defining the shpe of the cnl to the nerest degree. Hs erosion ffected the shpe of the cnl? Eplin. For Eercises 77 nd 78, refer to the following: After breking femur, ptient is plced in trction. The end of femur of length l is lifted to n elevtion forming n ngle u with the horizontl (ngle of elevtion)..354 Cl 0º Fe 0º.097 Cl Length Elevtion Br Cl 77. Helth/Medicine. A femur 8 inches long is plced into trction, forming n ngle of 5 with the horizontl. Find the height of elevtion t the end of the femur. 78. Helth/Medicine. A femur 8 inches long is plced in trction with n elevtion of 6. inches. Wht is the ngle of elevtion of the femur?
60 .5 Solving Right Tringles 6 CATCH THE MISTAKE For Eercises 79 nd 80, refer to the right tringle digrm below nd eplin the mistke tht is mde. 79. If b 800 feet nd 0 feet, find b. Solution: Represent tngent s the opposite side over the tn b b djcent side. Substitute b 800 feet nd 0 feet. c Use clcultor to evlute b. b tn This is incorrect. Wht mistke ws mde? b tn b If b 56 nd c 5 feet, find b nd then find. Solution: Write sine s the opposite side sin 56 b over the hypotenuse. 5 Solve for b. b 5 sin 56 Use clcultor to pproimte b. b.4356 Round the nswer to two significnt digits. b ft Use the Pythgoren theorem to find. b c Substitute b feet nd c 5 feet. 5 Solve for. 9 Round the nswer to two significnt digits. 9.0 ft Compre this with the results from Emple. Why did we get different vlue for here? CONCEPTUAL In Eercises 8 88, determine whether ech sttement is true or flse. 8. If you re given the lengths of two sides of right tringle, 85. The number 0.03 hs five significnt digits. you cn solve the right tringle. 8. If you re given the length of one side nd the mesure of one cute ngle of right tringle, you cn solve the right tringle. 83. If you re given the mesures of the two cute ngles of right tringle, you cn solve the right tringle. 84. If you re given the length of the hypotenuse of right tringle nd the mesures of the ngle opposite the hypotenuse, you cn solve the right tringle. 86. If the mesurement 700 feet hs been rounded to the nerest whole number, it hs three significnt digits. 87. If you re given the length of one side nd the mesure of one cute ngle, you cn solve the right tringle using either the sine or cosine function. 88. If you re given the lengths of two sides of right tringle, you will need to use n inverse trigonometric function to find the third side length.
61 6 CHAPTER Right Tringle Trigonometry CHALLENGE 89. Use the informtion in the picture below to determine the height of the mountin. 9. From the top of foot ldder, the ngle of depression to the fr side of sidewlk is 45, while the ngle of depression to the ner side of the sidewlk is 65. How wide is the sidewlk? 9. Find the perimeter of tringle A. 0 y 5º 38º 900 ft 5 48º A 90. Two friends who re engineers t Kennedy Spce Center (KSC) wtch the shuttle lunch. Crolyn is t the Vehicle Assembly Building (VAB) 3 miles from the lunch pd nd Jckie is cross the Bnn River, which is 8 miles from the lunch pd. They cll ech other t liftoff, nd fter 0 seconds they ech estimte the ngle of elevtion with respect to the ground. Crolyn thinks the ngle of elevtion is pproimtely 40 nd Jckie thinks the ngle of elevtion is pproimtely 5. Approimtely how high is the shuttle fter 0 seconds? (Averge their estimtes nd round to the nerest mile.) Jckie Crolyn 93. Solve for. º An electric line is strung from 0foot pole to point foot up on the side of house. If the pole is 50 feet from the house, wht ngle does the electric line mke with the pole? 35º 5º 40º 5 mi 3 mi Imge Bnk/Getty Imges, Inc.; Photonic/Getty Imges TECHNOLOGY 95. Use clcultor to find sin (sin 40 ). 96. Use clcultor to find cos (cos 7 ). 97. Use clcultor to find cos(cos 0.8). 98. Use clcultor to find sin(sin 0.3). 99. Bsed on the result from Eercise 95, wht would sin (sin u) be for n cute ngle u? 00. Bsed on the result from Eercise 96, wht would be for n cute ngle u? cos (cos u)
62 CHAPTER INQUIRYBASED LEARNING PROJECT Dr. Prkinson hs cquired two 30foot sections of fence from her neighbor Mr. Wilson. She hs decided to build tringulr corrl for her nimls. She plns to use brn wll s the third side. (The brn wll is 00 feet long see the digrm below.) As her contrctor, you re epected to mimize the corrl re. You hve decided to pproch this from trigonometric viewpoint. Hence, you wnt to find the ngle u tht mimizes the re. To do this, follow the steps below. (As side note: This problem is n emple of optimiztion nd you will revisit this type of problem in preclculus nd/or clculus courses. The outline below is designed to give you n understnding of how to set up nd solve this type of problem. Relize tht this tringle is n isosceles tringle two equl side lengths nd tht its perpendiculr bisector cn be used to find the height of the tringle.) Brn wll: 00 ft Write out the generl formul for the re of tringle.. To get n understnding of wht hppens to the re of the tringle s the ngle u chnges, you will clculte the following dimensions using right tringle trigonometry. Do these clcultions on sheet of scrtch pper. Since none of the tringles formed below re ctully right tringles, you will need to construct right tringle (using perpendiculr bisector) long with using the sine nd cosine reltionships to help you identify the bse nd height. (Use two decimls.) 0º 40º 60º 80º b(u) = bse h(u) = height A(u) = re You don t need to do every emple in the chrt by hnd. However, do s mny s you need to see wht ptterns emerge for clculting ech bse, height, nd re. When you see the pttern, you will hopefully then be ble to write function for ech piece of informtion. You cn then use the tble in your grphing clcultor to list ll the nswers. However, you re encourged to do t lest two by hnd before jumping to the function writing. Also be sure to check tht the results you get on your tble gree with the numbers you get by hnd. 3. Write the bse b(u) of the tringle s function of u. (Show how you rrived t this nswer.) Describe wht hppens to the bse vlues s the u vlues increse. 63
63 4. Write the height h(u) of the tringle s function of u. (Show how you rrived t this nswer.) Describe wht hppens to the height vlues s the u vlues increse. 5. Write the re A(u) s function of u using your results from 3 nd 4. Describe wht hppens to your re vlues s u increses. 6. Specify the domin for the re within the contet of this problem. (Vocbulry reminder: The domin for this problem is the set of vlues of u tht mkes sense from physicl stndpoint; tht is, you wouldn t build corrl using u = 0.) 7. Use grphing utility to grph your re function on its domin. 8. Estimte the mimum re nd the u tht this corresponds to. 9. Wht cn you conclude bout the shpe of the tringle tht yields the mimum re in this emple? 64
64 MODELING OUR WORLD The Intergovernmentl Pnel on Climte Chnge (IPCC) clims tht crbon dioide (CO ) production from incresed industril ctivity (such s fossil fuel burning nd other humn ctivities) hs incresed the CO concentrtions in the tmosphere. Becuse it is greenhouse gs, elevted CO levels will increse globl men (verge) temperture. In this section, we will emine the incresing rte of crbon emissions on Erth. In 955, there were (globlly) billion tons of crbon emitted per yer. In 005, the crbon emission hd more thn tripled, reching pproimtely 7 billion tons of crbon emitted per yer. Currently, we re on the pth to doubling our current crbon emissions in the net 50 yers. Tons of Crbon Emitted / Yer (in billions) The Stbiliztion Tringle Historicl emissions Currently projected Pth = Rmp Stbiliztion tringle Interim Flt pth gol Yer Two Princeton professors* (Stephen Pcl nd Rob Socolow) introduced the Climte Crbon Wedge concept. A wedge is strtegy to reduce crbon emissions tht grow.0 GtC/yr (gigtons of crbon per yer) over 50yer time period. Totl = 5 GtC/yr 50 yr GtC/yr. Consider eight scenrios (stying on pth of one of the seven wedges). A check is done t the 0yer mrk. Wht totl GtC per yer would we hve to mesure to correspond to the following projected pths?. Flt pth (no increse) over 50 yers (005 to 055) b. Increse of GtC over 50 yers (005 to 055) c. Increse of GtC over 50 yers (005 to 055) d. Increse of 3 GtC over 50 yers (005 to 055) *S. Pcl nd R. Socolow, Stbiliztion Wedges: Solving the Climte Problem for the Net 50 Yers with Current Technologies, Science 305 (004):
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