Applications of Trigonometry: Triangles and Vectors

Transcription

1 7 Applitions of Trigonometry: Tringles nd Vetors Norfolk, Virgini Atlnti Oen Bermud Bermud In reent dedes, mny people hve ome to elieve tht n imginry re lled the Bermud Tringle, loted off the southestern Atlnti ost of the United Sttes, hs een the site of high inidene of losses of ships, smll Fort Luderdle, Florid Florid, Bermud, Puerto Rio Sn Jun, Puerto Rio Virgini, Bermud, Cu ots, nd irrft over the enturies. The U.S. Bord of Geogrphi Nmes does not reognize the Bermud Tringle s n offiil nme nd does not mintin n offiil file on the re. Assume for the moment, without judging the merits of the hypothesis, tht the Bermud Tringle either hs verties in Mimi (Florid), Sn Jun (Puerto Rio), nd Bermud, or it hs verties in Norfolk (Virgini), Bermud, nd Sntigo (Cu). In this hpter, you will develop formul tht determines the re of tringle from its perimeter nd side lengths. Whih Bermud Tringle hs lrger re: Mimi-Bermud-Puerto Rio or Norfolk-Bermud-Cu? You will lulte the nswer in this hpter.* Sntigo de Cu Atlnti Oen *Setion 7.3, Exerises 37 nd 38.

2 IN THIS CHAPTER, we disuss olique (nonright) tringles. We use the Lw of Sines nd the Lw of Cosines to solve olique tringles. Then on the sis of the Lw of Sines nd the Lw of Cosines nd with trigonometri identities, we develop formuls for lulting the re of n olique tringle. We lso define vetors nd use the Lw of Cosines nd the Lw of Sines to determine resulting veloity nd fore vetors. Finlly, we define dot produts (produt of two vetors), nd see how they re pplile to physil prolems suh s lulting work. APPLICATIONS OF TRIGONOMETRY: TRIANGLES AND VECTORS 7.1 Olique Tringles nd the Lw of Sines 7.2 The Lw of Cosines 7.3 The Are of Tringle 7.4 Vetors 7.5 The Dot Produt Solving Olique Tringles Solving Olique Tringles The Are of Tringle (SAS Cse) The Are of Tringle (SSS Cse) Vetors: Mgnitude nd Diretion Vetor Opertions Horizontl nd Vertil Components of Vetor Unit Vetors Resultnt Vetors Multiplying Two Vetors: The Dot Produt Angle Between Two Vetors Work LEARNING OBJECTIVES Solve olique tringles using the Lw of Sines. Solve olique tringles using the Lw of Cosines. Find res of olique tringles. Perform vetor opertions. Find the dot produt of two vetors. 371

3 SECTION 7.1 OBLIQUE TRIANGLES AND THE LAW OF SINES SKILLS OBJECTIVES Solve AAS or ASA tringle ses. Solve miguous SSA tringle ses. Solve pplition prolems involving olique tringles. CONCEPTUAL OBJECTIVES Understnd the derivtion of the Lw of Sines. Understnd tht the miguous se n yield no tringle, one tringle, or two tringles. Understnd why n AAA se nnot e solved. Solving Olique Tringles Thus fr, we hve disussed only right tringles. There re, however, two types of tringles, right nd olique. An olique tringle is ny tringle tht does not hve right ngle. An olique tringle will e either n ute tringle, hving three ute (less thn 90 ) ngles; or n otuse tringle, hving one otuse (etween 90 nd 180 ) ngle. TRIANGLES Olique tringles Right tringles Aute tringles Otuse tringles 180 It is ustomry to lel olique tringles the following wy: ngle (lph): opposite side. ngle (et): opposite side. ngle g (gmm): opposite side. Rememer tht the sum of the three ngles of ny tringle must equl 180. Rell in Setion 1.5 tht we solved right tringles. In this hpter, we solve olique tringles, whih mens we find the lengths of ll three sides nd the mesures of ll three ngles. Four Cses To solve n olique tringle, we need to know the length of one side nd one of the following three: two ngle mesures one ngle mesure nd nother side length the other two side lengths This requirement leds to four possile ses to onsider: 372

4 7.1 Olique Tringles nd the Lw of Sines 373 R EQUIRED INFORMATION TO SOLVE OBLIQUE TRIANGLES CASE WHAT S GIVEN EXAMPLES/NAMES Cse 1 One side nd two ngles AAS: Angle-Angle-Side ASA: Angle-Side-Angle Cse 2 Two sides nd the ngle opposite one of them SSA: Side-Side-Angle Cse 3 Two sides nd the ngle etween them SAS: Side-Angle-Side Cse 4 Three sides SSS: Side-Side-Side Notie tht there is no AAA se. This is euse two similr tringles n hve the sme ngle mesures ut different side lengths, so t lest one side must e known to determine unique tringle. Study Tip To solve tringles, t lest one side must e known.

5 374 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors In this setion, we will derive the Lw of Sines, whih will enle us to solve Cse 1 nd Cse 2 prolems. In the next setion, we will derive the Lw of Cosines, whih will enle us to solve Cse 3 nd Cse 4 prolems. The Lw of Sines Let us strt with two olique tringles, n ute tringle nd n otuse tringle. Aute tringle Otuse tringle The following disussion pplies to oth tringles. First, onstrut n ltitude h (perpendiulr) from the vertex t ngle g to the side (or its extension) opposite g. h Aute tringle h 180º Otuse tringle WORDS MATH Formulte sine rtios for the ute tringle. sin h nd sin h Formulte sine rtios for the otuse tringle. sin(180 ) h nd sin h For the otuse ngle, pply the differene identity for the sine funtion. sin(180 ) sin 180 os os 180 sin 0 os (1)sin sin Therefore, in oth tringles we find the sme two equtions. sin h nd sin h Solve for h in oth equtions. Sine h is equl to itself in eh tringle, equte the expressions for h. h sin nd sin sin h sin

6 7.1 Olique Tringles nd the Lw of Sines 375 Divide oth sides y. Divide out ommon ftors. sin sin sin sin In similr mnner, we n extend n ltitude (perpendiulr) from ngle, nd we will find tht sin g sin sin of the Lw of Sines: sin g.. Equting these two expressions leds us to the third rtio THE LAW OF SINES For tringle with side lengths,, nd nd opposite ngle mesures the following reltionship is true: sin sin sin g,, nd g, In other words, the rtio of the sine of n ngle in tringle to its opposite side is equl to the rtios of the sines of the other two ngles to their opposite sides. Tht is, the rtio of the sine of n ngle to its opposite side is onstnt in ny tringle. A few things to note efore we egin solving olique tringles: The ngles nd sides shre the sme progression of mgnitude: The longest side of tringle is opposite the lrgest ngle. The shortest side of tringle is opposite the smllest ngle. Drw the tringle nd lel the ngles nd sides. If two ngle mesures re known, strt y determining the third ngle mesure. Whenever possile, in suessive steps, lwys return to given vlues rther thn referring to lulted (pproximte) vlues. In this hpter, we will round nswers so tht the numer of signifint digits in the nswer is equl to the numer of signifint digits of the given informtion: We use the lest numer of signifint digits from the given informtion. Study Tip The longest side is opposite the lrgest ngle; the shortest side is opposite the smllest ngle. Study Tip When possile, use given vlues rther thn lulted (pproximted) vlues for etter ury.

7 376 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Clssroom Exmple Solve the tringle. 20º 14 in. 130º Answer: 30, 9.1 in., 6.3 in. Tehnology Tip Step 1: Set the lultor to degree mode y typing MODE. Step 2: Use the lultor to find 7 sin 37 sin 110. Cse 1: Two Angles nd One Side (AAS or ASA) EXAMPLE 1 Solve the tringle. STEP 2 Find. Use the Lw of Sines with the known side. Isolte. Using the Lw of Sines to Solve Tringle (AAS) Solution: This is n AAS (ngle-ngle-side) se euse two ngle mesures nd side length re given nd the = 33º = 7.0 m side is opposite one of the ngles. Bsed on the given informtion, nswers will e rounded to two signifint digits. = 110º STEP 1 Find. The sum of the mesures of the ngles in tringle is 180. g 180 Let 110 nd g Solve for. 37 Let 110, 37, nd 7 meters. Use lultor to pproximte. Round to two signifint digits. sin sin sin sin 7 sin 37 sin m 4.5 m Step 3: Use the lultor to find 7 sin 33. sin 110 STEP 3 Find. Use the Lw of Sines with the known side. Isolte. Let 110, g 33, nd 7 meters. Use lultor to pproximte. Round to two signifint digits. sin sin g sin g sin 7 sin 33 sin m 4.1 m Answer: g 32, 42 ft, nd 23 ft YOUR TURN Solve the tringle. Study Tip Notie in Step 3 tht we used, whih is given, s opposed to, whih hs een lulted (pproximted). = 30 ft = 105º = 43º

8 7.1 Olique Tringles nd the Lw of Sines 377 EXAMPLE 2 Solve the tringle. Using the Lw of Sines to Solve Tringle (ASA) Clssroom Exmple Solve the tringle. 4 ft 86.1º STEP 1 Find g. The sum of the mesures of the ngles in tringle is 180. Let 80 nd 32. Solve for g. STEP 2 Find. Use the Lw of Sines with the known side. Isolte. = 80º = 17 mi Let 32, g 68, nd 17 miles. Use lultor to pproximte. = 32º Solution: This is n ASA (ngle-side-ngle) se, euse the mesure of two ngles nd side length re given nd the side is not opposite one of the ngles. Bsed on the given informtion, nswers will e rounded to two signifint digits. g g 180 sin g 68 sin g sin sin g 17 sin 32 sin º Answer: 53.4, 5 ft, 3 ft Note: Rounding yields tringle, ut this is not right tringle. Study Tip Whenever possile, use given vlues s opposed to lulted (pproximte) vlues. Tehnology Tip Step 2: Use the lultor to find 17 sin 32. sin 68 Round to two signifint digits. STEP 3 Find. Use the Lw of Sines with the known side. Isolte. Let 80, g 68, nd 17 miles. Use lultor to pproximte. Round to two signifint digits. sin 9.7 mi sin g sin sin g 17 sin 80 sin mi Step 3: Use the lultor to find 17 sin 80. sin 68 YOUR TURN Solve the tringle. = 60º = 12 in. Answer: 35, 21 in., nd 18 in. = 85º

9 378 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Cse 2 (Amiguous Cse): Two Sides nd One Angle (SSA) h If we re given two side lengths nd the mesure of n ngle opposite one of the sides, we ll tht Cse 2, SSA (side-side-ngle). This se is lled the miguous se, euse the given informtion y itself n represent one tringle, two tringles, or no tringle t ll. If the ngle given is ute, then the possiilities re zero, one, or two tringles. If the ngle given is otuse, then the possiilities re zero or one tringle. The possiilities ome from the ft tht sin k, where 0 k 1, hs two solutions for : one in qudrnt I (ute ngle) nd one in qudrnt II (otuse ngle). In the figure on the left, note tht h sin y the definition of the sine rtio. When the Given Angle Is Aute Study Tip h sin CONDITION PICTURE NUMBER OF TRIANGLES 0 h sin 1 h sin 1 No Tringle h Right Tringle = h 0 1 h 0 sin 1 Aute Tringle h Otuse Tringle 2 0 sin 1 h Aute Tringle γ h α β 1 *Note: is the unknown ngle opposite the known side. Study Tip Notie if the ngle given is otuse, the side opposite tht ngle must e longer thn the other given side (longest side opposite the lrgest ngle). When the Given Angle Is Otuse CONDITION PICTURE NUMBER OF TRIANGLES sin 1 0 sin 1 h h 0 1

10 7.1 Olique Tringles nd the Lw of Sines 379 EXAMPLE 3 Solving the Amiguous Cse (SSA) One Tringle Solve the tringle 23 feet, 11 feet, nd 122. Solution: This is n miguous se euse two side lengths nd the mesure of n ngle opposite one of those sides re given. Sine the given ngle is otuse, we know tht the other ngles in the tringle re ute. So we know tht the Lw of Sines will give the orret vlues for the other ngles nd only one tringle will exist sine Bsed on the given informtion, nswers will e rounded to two signifint digits. STEP 1 Find. Use the Lw of Sines. Isolte sin. = 30 ft sin = 23 ft = 11 ft sin sin sin = 122º = 24º Clssroom Exmple Solve the tringle. 70º 5.0 m 4.6 m Answer: 60, 50, 4.1 m STEP 2 Let 23 feet, 11 feet, nd 122. Use lultor to evlute sin. Solve for using the inverse sine funtion. Round the nswer to two signifint digits. Find g. The mesures of the ngles in tringle sum to 180. Let 122 nd 24. Solve for g. STEP 3 Find. Use the Lw of Sines. Isolte. Let 23 feet, 122, nd g 34. sin sin sin 1 ( ) g 180 sin 11 sin g 180 sin g sin g sin 23 sin 34 sin 122 g 34 Tehnology Tip Step 1: Use lultor to find the vlue of. Step 3: Use the lultor to find 23 sin 34. sin 122 Use lultor to evlute nd round to two signifint digits. 15 ft YOUR TURN Solve the tringle 133, 48 millimeters, nd 17 millimeters. Answer: 32, g 15, nd 35 mm

11 380 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors EXAMPLE 4 Solving the Amiguous Cse (SSA) Two Tringles Solve the tringle 8.1 meters, 8.3 meters, nd 72. Solution: This is n miguous se euse two side lengths nd the mesure of n ngle opposite one of those sides re given. Sine the given ngle is ute nd, we might expet two tringles. Bsed on the given informtion, nswers will e rounded to two signifint digits. STEP 1 Find. Use the Lw of Sines. Isolte sin. Let 8.1 meters, 8.3 meters, nd 72. sin sin sin sin sin 8.3 sin Use lultor to evlute sin. Solve for using the inverse sine funtion. Note tht n e ute or otuse. This is the qudrnt I ( is ute) solution. The qudrnt II ( is otuse) solution is sin sin 1 ( ) STEP 2 Find g (two vlues). The mesures of the ngles in tringle sum to 180. Let 72 nd Solve for g 1. Let 72 nd Solve for g 2. STEP 3 Find. Use the Lw of Sines. Isolte. Let 8.1 meters, 72, nd g Use lultor to pproximte 1. Let 8.1 meters, 72, nd g 2 5. Use lultor to pproximte 2. g g g g g 2 5 sin sin g sin 8.1 sin 31 1 sin m 2 sin g 8.1 sin 5 sin m

12 7.1 Olique Tringles nd the Lw of Sines 381 STEP 4 Drw nd lel the two tringles. = 31º = = 5º = 8.3 m = 8.1 m = 8.3 m = 8.1 m = 72º = 4.9 m = 77º = 72º = 0.83 m = 103º Notie tht when there re two solutions for the SSA se, one of the tringles will e otuse. EXAMPLE 5 Solving the Amiguous Cse (SSA) No Tringle Solve the tringle 107, 6, nd 8. Solution: This is n miguous se euse two side lengths nd the mesure of n ngle opposite one of those sides re given. Sine the given ngle is otuse nd, there is no tringle. (Notie 107 is the lrgest ngle, whih implies tht side is the longest side.) Notie wht would hppen in our lultions. Clssroom Exmple Solve the tringle g 140, 7.1, nd 6. Answer: no tringle Use the Lw of Sines. sin sin Isolte sin. Let 107, 6, nd 8. Use lultor to pproximte sin. sin sin sin 8 sin sin Sine the rnge of the sine funtion is [1, 1], there is no ngle suh tht sin Therefore, there is no tringle with the given mesurements. Applitions The solution of olique tringles hs pplitions in stronomy, surveying, irrft design, piloting, nd mny other res.

13 382 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors EXAMPLE 6 How Fr Over Is the Tower of Pis Lening? Clssroom Exmple A 15-foot pole is lening towrd the south mking n ngle of 95 with the ground. An 18-foot guy wire runs from the ground on the north side of the pole to the top of the pole. How fr from the pole is the guy wire tthed to the ground? Answer: 8.7 ft The Tower of Pis ws originlly uilt 56 meters tll. Beuse of poor soil in the foundtion, it strted to len. At distne 44 meters from the se of the tower, the ngle of elevtion is 55. How muh is the Tower of Pis lening wy from the vertil position? Solution: 56 m? 44 m 55º Coris Digitl Stok 55, 44 meters, nd 56 meters is the given informtion, so this is n SSA prolem. There is only one tringle euse. = 56 m = 55º STEP 1 STEP 2 Find g. Use the Lw of Sines. Isolte sin g. Let 55, 44 meters, nd 56 meters. Evlute the right side using lultor. Solve for g using the inverse sine funtion. Round to two signifint digits. Find. The mesures of ngles in tringle sum to 180. Let 55 nd g 40. Solve for. = 44 m sin sin g sin g sin 44 sin 55 sin g 56 sin g g sin 1 ( ) g 40 g The Tower of Pis mkes n ngle of 85 with the ground. It is lening t n ngle of 5.

14 7.1 Olique Tringles nd the Lw of Sines 383 SECTION 7.1 SUMMARY In this setion, we solved olique tringles. When given three mesurements of tringle (t lest one side), we lssify the tringle ording to the dt (sides nd ngles). Four ses rise: one side nd two ngles (AAS or ASA) two sides nd the ngle opposite one of the sides (SSA) two sides nd the ngle etween sides (SAS) three sides (SSS) The Lw of Sines sin sin sin g n e used to solve the first two ses (AAS or ASA nd SSA). It is importnt to note tht the SSA se is lled the miguous se euse ny one of three results is possile: no tringle, one tringle, or two tringles. SECTION 7.1 EXERCISES SKILLS In Exerises 1 6, lssify eh tringle s AAS, ASA, SAS, SSA, SAS, or SSS on the sis of the given informtion. 1.,, nd 2. 3.,, nd 4.,, nd g,, nd g + + = 180º 5.,, nd 6., g, nd In Exerises 7 20, solve the given tringles , 60, 10 m , g 72, 200 m , g 60, 25 in. g 100, 40, 16 ft + + = 180º , g 47.6, 211 yd , 30, 12 m , g 57, 100 yd , 10, 12 m , g 33.6, 26 in. 45, g 75, 9 in. 80, g 30, 3 ft g 46, 37, 10 yd , g 13.4, 57 m 20. 5, 15, 2.3 mm In Exerises 21 40, two sides nd n ngle re given. Determine whether tringle (or two) exists, nd if so, solve the tringle(s) , 5, , 20, , 12, g , 80, , 14, , 26, , 330, g , 9, , 17, , 27.2, g , 12, g 14.5, 10, , 18, , 18, g 20, 6.2, , 5, , 5, , 9, g , 1.5, g , 10, 80

15 384 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors APPLICATIONS For Exerises 41 nd 42, refer to the following: NASA Kennedy Spe Center 195 ft 1º 10º On the lunh pd t Kennedy Spe Center, the stronuts hve n espe sket tht n hold four stronuts. The sket slides down wire tht is tthed 195 feet ove the se of the lunh pd. The ngle of inlintion mesured from where the sket would touh the ground to the se of the lunh pd is 1, nd the ngle of inlintion from tht sme point to where the wire is tthed is NASA. How long is the wire? Find length. 42. NASA. How fr from the lunh pd does the sket touh the ground? Find length. 43. Hot-Air Blloon. A hot-ir lloon is sighted t the sme time y two friends who re 1.0 mile prt on the sme side of the lloon. The ngles of elevtion of the lloon from the two friends re 20.5 nd How high is the lloon? 20.5º 25.5º 1.0 mi 45. Roket Trking. A trking sttion hs two telesopes tht re 1.0 mile prt. Eh telesope n lok onto roket fter it is lunhed nd reord its ngle of elevtion to the roket. If the ngles of elevtion from telesopes A nd B re 30 nd 80, respetively, then how fr is the roket from telesope A? 80º B 30º 46. Roket Trking. Given the informtion in Exerise 45, how fr is the roket from telesope B? 47. Distne Aross River. An engineer wnts to onstrut ridge ross fst-moving river. Using stright-line segment etween two points tht re 100 feet prt long his side of the river, he mesures the ngles formed when sighting the point C on the other side where he wnts to hve the ridge end. If the ngles formed t points A nd B re 65 nd 15, respetively, how fr is it from point A to point C on the other side of the river? Round to the nerest foot. C 1.0 mi A 44. Hot-Air Blloon. A hot-ir lloon is sighted t the sme time y two friends who re 2 miles prt on the sme side of the lloon. The ngles of elevtion of the lloon from the two friends re 10 nd 15. How high is the lloon? A 65º 15º 100 ft B

16 7.1 Olique Tringles nd the Lw of Sines Distne Aross River. Given the dt in Exerise 47, how fr is it from point B to the point on the other side of the river? 49. Sumrine. A sumrine trvels N45 E 15 miles. It then turns to ourse of 120 mesured lokwise off of due north nd trvels until it is due est of its originl position. How fr is it from where it strted? 50. Sumrine. A sumrine trvels N45 W 20 miles. It then turns to ourse of 200 mesured lokwise off of due north nd trvels until it is due west of its originl position. How fr is it from where it strted? 51. Hiking. A photogrpher prks his r long stright ountry rod (tht runs due north nd south) to hike into the woods to tke some pitures. Initilly, he trvels 850 yrds on ourse of 130 mesured lokwise from due north. He hnges diretion only to rrive k t the rod fter wlking 700 yrds. How fr from his strting point is he when he finds the rod? Hint: There re two possiilities: one leding to n ute tringle nd one leding to n otuse tringle. 53. Crpentry. A woodworker fshions hir suh tht the legs ome down t n ngle to the floor s shown in the figure. If the legs re 28 inhes long, how fr prt re they long the floor? 75º 54. Crpentry. A woodworker fshions hir suh tht the legs ome down t n ngle to the floor s shown in the figure. If the legs re 30 inhes long, how fr prt re they long the floor? 35º yd B 700 yd A 70º 33º yd For Exerises 55 nd 56, refer to the following: To quntify the torque (rottionl fore) of the elow joint of humn rm (see the figure to the right), it is neessry to identify ngles A, B, nd C s well s lengths,, nd. Mesurements performed on n rm determine tht the mesure of ngle C is 95, the mesure of ngle A is 82, nd the length of the musle is 23 entimeters. B 700 yd A B 52. Hiking. Suppose the hiker in Exerise 51 initilly trvels 800 yrds on ourse of 135 mesured lokwise from due north. He hnges diretion only to rrive k t the rod fter wlking 700 yrds. How fr from his strting point is he when he finds the rod? Hint: There re two possiilities: one leding to n ute tringle nd one leding to n otuse tringle. Joint A C Musle 55. Helth/Mediine. Find the length of the forerm from the elow joint to the musle tthment. 56. Helth/Mediine. Find the length of the upper rm from the musle tthment to the elow joint.

17 386 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors CATCH THE MISTAKE In Exerises 57 nd 58, explin the mistke tht is mde. 57. Solve the tringle 120, 7, nd 9. Solution: sin Use the Lw of Sines to find. sin sin 120 Let 120, 7, nd 9. sin 7 9 Solve for sin. sin Solve for. 42 Sum the ngle mesures to g 180 Solve for g. g 18 Use the Lw of Sines to find. Let 120, 7, nd g 18 Solve for. sin sin sin g 120, 42, g 18, 7, 9, nd 2.5. This is inorret. The longest side is not opposite the longest ngle. There is no tringle tht mkes the originl mesurements work. Wht mistke ws mde? 2.5 sin Solve the tringle 40, 7, nd 9. Solution: sin Use the Lw of Sines to find. sin Let 40, 7, nd 9. Solve for sin. Solve for. Find g. Use the Lw of Sines to find. Let 40, 7, nd g 84. Solve for. sin g 180 sin sin , 56, g 84, 7, 9, nd 11. This is inorret. Wht mistke ws mde? sin 9 sin g 84 sin g 11 sin 84 CONCEPTUAL In Exerises 59 62, determine whether eh sttement is true or flse. 59. The Lw of Sines pplies only to right tringles. 61. An ute tringle is n olique tringle. 60. If you re given two sides nd ny ngle, there is unique solution for the tringle. 62. An otuse tringle is n olique tringle. CHALLENGE 63. Consider tringle in whih 30. If nd re given, determine the vlues of in reltion to wherey there re two tringles possile. 64. Consider tringle in whih 60. If nd re given, determine the vlues of in reltion to wherey there re two tringles possile. 65. Mollweide s Identity. For ny tringle, the following identity is true. It is often used to hek the solution of tringle sine ll six piees of informtion (three sides nd three ngles) re involved. Derive the identity using the Lw of Sines. ( ) sin 1 2 g os 1 ( )d The Lw of Tngents. Use the Lw of Sines nd trigonometri identities to show tht for ny tringle, the following is true: tn 1 ( )d 2 tn 1 ( )d Let 30, 45, nd 22. Use the Lw of Sines long with sum nd differene identity to find the ext vlue of. 68. Let 120, 45, nd 26. Use the Lw of Sines long with sum nd differene identity to find the ext vlue of.

18 7.2 The Lw of Cosines 387 TECHNOLOGY For Exerises 69 72, let A, B, nd C e the lengths of the three sides with X, Y, nd Z s the orresponding ngles. Write progrm using lultor to solve the given tringle. Go to this ook s wesite ( for instrutions or help on how to write progrm in grphing lultor. 69. A 10, Y 45, nd Z B 42.8, X 31.6, nd Y A 22, B 17, nd X B 16.5, C 9.8, nd Z 79.2 B Z X A C Y SECTION 7.2 THE LAW OF COSINES SKILLS OBJECTIVES Solve SAS tringle ses. Solve SSS tringle ses. Solve pplition prolems involving olique tringles. CONCEPTUAL OBJECTIVES Understnd the derivtion of the Lw of Cosines. Develop strtegy for whih ngles (lrger or smller) nd whih method (the Lw of Sines or the Lw of Cosines) to selet when solving olique tringles. Solving Olique Tringles In the previous setion (Setion 7.1), we lerned tht to solve olique tringles mens to find ll three side lengths nd ll three ngle mesures. At lest one side length must e known. Two dditionl piees of informtion re needed to solve tringle (omintions of side lengths nd/or ngle mesures). We found tht there re four ses: Cse 1: AAS or ASA (the mesure of two ngles nd side length re given) Cse 2: SSA (two side lengths nd the mesure of n ngle opposite one of the sides re given) Cse 3: SAS (two side lengths nd the mesure of the ngle etween them re given) Cse 4: SSS (three side lengths re given) We used the Lw of Sines to solve Cse 1 nd Cse 2 tringles. Now, we use the Lw of Cosines to solve Cse 3 nd Cse 4 tringles. WORDS Strt with n ute tringle. (The sme n e shown for n otuse tringle.) MATH

19 388 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Drop perpendiulr line segment with height h from g to the opposite side. The result is two tringles within the lrger tringle. h Sine the segment of length h is shred, set h 2 h 2 for the two tringles. Multiply out the squred inomil on the right. Eliminte the prentheses. Add x 2 to oth sides. Isolte Use the Pythgoren theorem Tringle 1: for oth right tringles. Tringle 2: Solve for h 2. Tringle 1: h 2 2 x 2 Tringle 2: h 2 2 ( x) 2 2 x 2 2 ( x) 2 2 x 2 2 ( 2 2x x 2 ) 2 x x x x x Notie tht os x Let x os os. Note: If we insted drop the perpendiulr line segment with length h from the ngle or the ngle to the opposite side, we n derive the other two forms of the Lw of Cosines. x x x 2 h 2 2 ( x) 2 h os nd os g THE LAW OF COSINES For tringle with sides,, nd, nd opposite ngles,, nd g, the following is true: os os os g Study Tip The Pythgoren theorem is speil se of the Lw of Cosines. It is importnt to note tht the Lw of Cosines n e used to find unknown side lengths or ngle mesures. As long s three of the four vriles in ny of the equtions re known, the fourth n e lulted. Notie tht in the speil se of right tringle (sy, 90 ), os 90 one of the omponents of the Lw of Cosines redues to the Pythgoren theorem: { hyp { leg { leg 0 The Pythgoren theorem n thus e regrded s speil se of the Lw of Cosines.

20 7.2 The Lw of Cosines 389 Cse 3: Solving Olique Tringles (SAS) We n now solve SAS tringle prolems, where the ngle etween two known sides is given. We strt y using the Lw of Cosines to solve for the length of the side opposite the given ngle. We then n use either the Lw of Sines or the Lw of Cosines to find the mesure of seond ngle. Using the Lw of Cosines to Solve Tringle (SAS) EXAMPLE 1 Solve the tringle 13, 6.0, nd 20. Solution: Two side lengths nd the mesure of the ngle etween them re given (SAS). Notie tht the Lw of Sines nnot e used euse it requires knowledge of t lest one ngle mesure nd the length of the side opposite tht ngle. STEP 1 Find. STEP 2 Use the Lw of Cosines tht involves. Let 13, 6.0, nd 20. Evlute the right side using lultor. Solve for. Round to two signifint digits; n e only positive. Find the ute ngle g os (13)(6)os = 13 = 20º = 6.0 Clssroom Exmple Solve the tringle. 15 m 17º 6.8 m Answer: 150, 13, 8.7 m Tehnology Tip Step 1: Use lultor to find the vlue of. Step 2: Use lultor to find the mesure of g. STEP 3 Use the Lw of Sines to find the smller ngle g. Isolte sin g. Let 7.6, 6.0, nd 20. Use the inverse sine funtion. Evlute the right side with lultor. Round to the nerest degree. Find. The three ngle mesures must sum to 180. Solve for. sin g sin sin g sin 6 sin 20 sin g 7.6 g sin 1 6 sin g g YOUR TURN Solve the tringle 4.2, 1.8, nd 35. Answer: 2.9, g 21, nd 124

21 390 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Study Tip Although the Lw of Sines n sometimes led to the miguous se, the Lw of Cosines never leds to the miguous se. Notie the steps we took in solving SAS tringle: 1. Find the length of the side opposite the given ngle using the Lw of Cosines. 2. Solve for the smller ngle (whih hs to e ute) using the Lw of Sines. 3. Solve for the lrger ngle using properties of tringles. You my e thinking, would it mtter if we solved for efore solving for g? Yes, it does mtter in this prolem, you nnot solve for y the Lw of Sines efore finding g. The Lw of Sines should e used only on the smller ngle (opposite the shortest side). If we hd tried to use the Lw of Sines with the otuse ngle, the inverse sine would hve resulted in 36. Sine the sine funtion is positive in qudrnts I nd II, we would not know if tht ngle ws 36 or its supplementry ngle 144. Notie tht ; therefore, the ngles opposite those sides must hve the sme reltionship g. We hoose the smller ngle first to void the miguity with the Lw of Sines. Alterntively, if we wnted to solve for the otuse ngle first, we ould hve used the Lw of Cosines to solve for. Clssroom Exmple Two irplnes tke off from the sme irport. One trvels 100 miles t n ngle of 50 lokwise from due north efore lnding, while the other flies for 80 miles t n ngle of 185 lokwise from due north efore lnding. After lnding, how fr prt re the plnes? Answer: pproximtely 166 mi EXAMPLE 2 Using the Lw of Cosines in n Applition (SAS) In n AKC (Amerin Kennel Clu)-sntioned field tril, judge sets up mrk (ird) tht requires the dog to swim ross ody of wter (the dogs re judged on how losely they dhere to the stright line to the ird, not the time it tkes to retrieve the ird). The judge is trying to lulte how fr the dog would hve to swim to this mrk, so she wlks off the two legs ross the lnd nd mesures the ngle s shown in the figure. How fr will the dog swim from the strting line to the ird? 176 yd 152 yd 117º Pond Solution: Lel the tringle. = 152 yd = 117º = 176 yd Pond Tehnology Tip Use lultor to find the vlue of. Use the Lw of Cosines. Let 176, 152, nd g 117. Use lultor to pproximte the right side. Solve for nd round to the nerest yrd (three signifint digits) os g (176)(152)os yd

22 7.2 The Lw of Cosines 391 Cse 4: Solving Olique Tringles (SSS) We now solve olique tringles when ll three side lengths re given (the SSS se). In this se, strt y finding the mesure of the lrgest ngle (opposite the longest side) using the Lw of Cosines. Then use the Lw of Sines to find either of the remining two ngle mesures. Finlly, find the third ngle mesure using the ft tht the three ngles in tringle lwys sum to 180. Clssroom Exmple Solve the tringle. EXAMPLE 3 Using the Lw of Cosines to Solve Tringle (SSS) Solve the tringle 8, 6, nd 7. Solution: 3 3 STEP 1 STEP 2 Identify the lrgest ngle, whih is. Use the Lw of Cosines tht involves. Let 8, 6, nd 7. Simplify nd isolte os. Apply the inverse osine funtion. Approximte with lultor. Round to the nerest degree. Find either of the remining ngle mesures. To solve for ute ngle : Use the Lw of Sines. Isolte sin. Let 8, 6, nd 76. Use the inverse sine funtion os (6)(7) os os (6)(7) os 1 (0.25) 76 sin sin sin sin 6 sin 76 sin 8 sin 1 6 sin Answer: 66, 48, nd g 66 Tehnology Tip Step 1: Use lultor to find the vlue of. Step 2: Use the lultor to find. STEP 3 Approximte with lultor. Find the mesure of the third ngle g. The sum of the ngle mesures is 180. Solve for g g 180 g 57 YOUR TURN Solve the tringle 5, 7, nd 8. Answer: 38, 60, nd g 82 In the next exmple, insted of immeditely sustituting vlues into the Lw of Cosines eqution, we will solve for the ngle mesure in generl, nd then sustitute in vlues.

23 392 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Virgini = 894 nm Cu = 850 nm Bermud = 810 nm EXAMPLE 4 Using the Lw of Cosines in n Applition (SSS) In reent dedes, mny people hve ome to elieve tht n imginry re lled the Bermud Tringle, loted off the southestern Atlnti ost of the United Sttes, hs een the site of high inidene of losses of ships, smll ots, nd irrft over the enturies. Assume for the moment, without judging the merits of the hypothesis, tht the Bermud Tringle hs verties in Norfolk (Virgini), Bermud, nd Sntigo (Cu). Find the ngles of the Bermud Tringle given the following distnes: LOCATION LOCATION DISTANCE (NAUTICAL MILES) Norfolk Bermud 850 Bermud Sntigo 810 Norfolk Sntigo 894 Ignore the urvture of the Erth in your lultions. Round nswers to the nerest degree. Tehnology Tip Step 1: Use lultor to find the vlue of. Solution: STEP 1 Find (the lrgest ngle). Use the Lw of Cosines. Isolte os os os Step 2: Use the lultor to find. Use the inverse osine funtion to solve for. os Let os , 894, nd (810)(850) Use lultor to pproximte, nd round to the nerest degree. 65 Insted of using the pproximted vlue of to lulte, you n use ANS vlue of. to retrieve the tul STEP 2 Find. Use the Lw of Sines. Isolte sin. Use the inverse sine funtion to solve for. sin sin sin sin sin 1 sin Let 810, 894, nd 65. Use lultor to pproximte, nd round to nerest degree. sin sin STEP 3 Find g. The ngle mesures must sum to g 180 Solve for g. g 60

24 7.2 The Lw of Cosines 393 SECTION 7.2 SUMMARY We n solve ny tringle given three piees of informtion (mesurements), s long s one of the piees is side length. Depending on the informtion given, we either use the Lw of Sines sin sin sin g or we use omintion of the Lw of Cosines os os os g nd the Lw of Sines. The tle summrizes the strtegies for solving olique tringles. OBLIQUE TRIANGLE WHAT IS KNOWN PROCEDURE FOR SOLVING AAS or ASA Two ngles nd side Step 1: Find the remining ngle mesure using g 180. Step 2: Find the remining sides using the Lw of Sines. SSA Two sides nd n ngle This is the miguous se, so there is either no tringle, one tringle, opposite one of the sides or two tringles. If the given ngle is otuse, then there is either one or no tringles. If the given ngle is ute, then there is no tringle, one tringle, or two tringles. Step 1: Use the Lw of Sines to find one of the ngle mesures. Step 2: Find the remining ngle mesure using g 180. Step 3: Find the remining side using the Lw of Sines. If two tringles exist, then the ngle mesure found in Step 1 n e either ute or otuse, nd Steps 2 nd 3 must e performed for eh tringle. SAS Two sides nd n ngle Step 1: Find the third side length using the Lw of Cosines. etween the sides Step 2: Find the smller ngle mesure using the Lw of Sines. Step 3: Find the remining ngle mesure using g 180. SSS Three sides Step 1: Find the lrgest ngle mesure using the Lw of Cosines, whih voids the miguity with the Lw of Sines. Step 2: Find either remining ngle mesure using the Lw of Sines. Step 3: Find the lst remining ngle mesure using g 180. SECTION 7.2 EXERCISES SKILLS In Exerises 1 8, for eh of the given tringles, the ngle sum identity, A B G 180, will e used in solving the tringle. Lel the prolem s S if only the Lw of Sines is needed to solve the tringle. Lel the prolem s C if the Lw of Cosines is needed to solve the tringle. 1.,, nd re given. 2.,, nd g re given. 3.,, nd re given. 4.,, nd re given. 5.,, nd g re given. 6.,, nd re given. 7.,, nd re given. 8.,, nd re given.

25 394 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors In Exerises 9 44, solve eh tringle. 9. 4, 3, , 10, g , 2, , 6, g , 5, , 7.3, g , 12, , 13, , 12, g , 4.4, , 82, , 20, , 8, , 118, , 5, , 9, , 4, , 20, , 6, , 19, , 2.9, , 8, , 7.1, , 2001, , 5, , 2.7, , 5, , 5, , 35, , 19.0, g , 5, , 12, g , 18, , g 43, , 2, , 6, 9 APPLICATIONS 45. Avition. A plne flew due north t 500 mph for 3 hours. A seond plne, strting t the sme point nd t the sme time, flew southest t n ngle 150 lokwise from due north t 435 mph for 3 hours. At the end of the 3 hours, how fr prt were the two plnes? Round to the nerest mile. 46. Avition. A plne flew due north t 400 mph for 4 hours. A seond plne, strting t the sme point nd t the sme time, flew southest t n ngle 120 lokwise from due north t 300 mph for 4 hours. At the end of the 4 hours, how fr prt were the two plnes? Round to the nerest mile. 150º 120º

26 7.2 The Lw of Cosines Bsell. A sell dimond is tully squre tht is 90 feet on eh side. The pither s mound is loted 60.5 feet from home plte. How fr is it from third se to the pither s mound? Third se 45º Home plte 48. Airrft Wing. Given the ute ngle (14.4 ) nd two sides (18 feet nd 25 feet) of the stelth omer, wht is the unknown length? 49. Sliding Bord. A 40-foot slide lening ginst the ottom of uilding s window mkes 55 ngle with the uilding. The ngle formed with the uilding with the line of sight from the top of the window to the point on the ground where the slide ends is 40. How tll is the window? 50. Airplne Slide. An irplne door is 6 feet high. If slide tthed to the ottom of the open door is t n ngle of 40 with the ground, nd the ngle formed y the line of sight from where the slide touhes the ground to the top of the door is 45, then how long is the slide? 6 ft 90 ft 90 ft 45º Seond se Pither's mound?? 18 ft ft 40º 60.5 ft 40º 55º 40 ft 90 ft 90 ft? ft First se Ross Hrrison Koty/ Getty Imges, In. For Exerises 51 nd 52, refer to the following: To quntify the torque (rottionl fore) of the elow joint of humn rm (see the figure to the right), it is neessry to B identify ngles A, B, nd C s well s lengths,, nd. Mesurements performed on n rm determine tht the A mesure of ngle C is 105, the length of C Joint the musle is 25.5 entimeters, nd the length of the forerm from the elow joint to the musle tthment is 1.76 entimeters. 51. Helth/Mediine. Find the length of the upper rm from the musle tthment to the elow joint. 52. Helth/Mediine. Find the mesure of ngle B. 53. Blloon. A hot-ir lloon floting in the ir is eing tethered y two 50-foot ropes. If the ropes re stked to the ground 75 feet prt, wht ngle do the ropes mke with eh other t the lloon? 54. Blloon. A hot-ir lloon floting in the ir is eing tethered y two 75-foot ropes. If the ropes re stked to the ground 100 feet prt, wht ngle do the ropes mke with eh other t the lloon? 55. Roof Constrution. The roof of house is longer on one side thn on the other. If the length of one side of the roof is 27 feet nd the length of the other side is 35 feet, find the distne etween the ends of the roof if the ngle t the top is Roof Constrution. The roof of house is longer on one side thn on the other. If the length of one side of the roof is 29 feet nd the length of the other side is 36 feet, find the distne etween the ends of the roof if the ngle t the top is 135. For Exerises 57 nd 58, refer to the following: The term tringultion is often used to pinpoint lotion. A person s ell phone lwys uses the losest tower. As the ell phone swithes towers, uthorities n lote the person s lotion sed on known distnes from eh of the towers. Similrly, firefighters n determine the lotion of fire from known distnes. 57. Tringultion. Two led firefighters (Beth nd Tim) re 300 yrds prt nd they estimte eh of their distnes to the fire s pproximtely 150 yrds (Beth) nd 200 yrds (Tim). Wht ngle with respet to their line of sight with eh other should eh led their tem in order to reh the fire? 58. Tringultion. Two ell phone towers re 100 meters prt. When the ell phone swithes from tower A to tower B, it is estimted tht the phone is 60 meters from tower A nd 50 meters from tower B. Authorities looking to lote the owner of the ell phone will hed 50 meters from tower B t wht ngle (with respet to the line of sight etween towers A nd B)? Musle

27 396 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors CATCH THE MISTAKE In Exerises 59 nd 60, explin the mistke tht is mde. 59. Solve the tringle 3, 4, nd 30. Solution: Step 1: Find. Use the Lw of Cosines os Let 3, 4, nd 30. Solve for. Step 2: Find g. Use the Lw of Sines. Solve for sin g. Solve for g. Let 2.1, 4, nd 30. Step 3: Find (3)(4) os sin sin g sin g sin g sin 1 sin g 72 g Solve for , 3, 4, 30, 78, nd g 72. This is inorret. The longest side is not opposite the lrgest ngle. Wht mistke ws mde? 60. Solve the tringle 6, 2, nd 5. Solution: Step 1: Find. Use the Lw of Cosines os Solve for os Let 6, 2, nd Step 2: Find. Use the Lw of Sines. Solve for. sin sin sin 1 sin Let 6, 2, nd Step 3: Find g. g g 180 Solve for g. g 94 6, 2, 5, 68, 18, nd g 94. This is inorret. The longest side is not opposite the lrgest ngle. Wht mistke ws mde? CONCEPTUAL In Exerises 61 64, determine whether eh sttement is true or flse. 61. Given three sides of tringle, there is insuffiient informtion to solve the tringle. 62. Given three ngles of tringle, there is insuffiient informtion to solve the tringle. 63. The Pythgoren theorem is speil se of the Lw of Cosines. 64. The Lw of Cosines is speil se of the Pythgoren theorem. 65. In tringle, the length of the shortest side of the tringle 1 is 2 the length of the longest side. The other side of the 3 tringle is 4 the length of the longest side. Wht is the size of the lrgest ngle? 66. In tringle, the length of one side is 4 the length of n djent side. If the ngle etween the sides is 60, how does the length of the third side ompre with tht of the longer of the other two? 1

28 7.3 The Are of Tringle 397 CHALLENGE os 67. Show tht os os g Hint: Use the Lw of Cosines. 68. Show tht os os g. Hint: Use the Lw of Cosines. 69. In n isoseles tringle, the longer side is 50% longer thn the other two sides. Wht is the size of the vertex ngle? 70. In n isoseles tringle, the longer side is 2 inhes longer thn the other two sides. If the vertex ngle mesures 80, wht re the lengths of the sides? TECHNOLOGY For Exerises 71 74, let A, B, nd C e the lengths of the three sides with X, Y, nd Z s the orresponding ngle mesures. Write progrm using the TI lultor to solve the given tringle. X 71. B 45, C 57, nd X B 24.5, C 31.6, nd X A 29.8, B 37.6, nd C A 100, B 170, nd C 250 B C Z A Y SECTION 7.3 THE AREA OF A TRIANGLE SKILLS OBJECTIVES Find the re of tringle in the SAS se. Determine the re of tringle in the SSS se. CONCEPTUAL OBJECTIVES Understnd how to derive formul for the re of tringle (SAS se) using the Lw of Sines. Understnd how to derive formul for the re of tringle (SSS se) using the Lw of Cosines. In Setions 7.1 nd 7.2, we used the Lw of Sines nd the Lw of Cosines to solve olique tringles, whih mens to find ll of the side lengths nd ngle mesures. Now, we use these lws to derive formuls for the re of tringle (SAS nd SSS ses). Our strting point for oth ses is the stndrd formul for the re of tringle: A 1 2 h Study Tip If the tringle given is not n SAS or SSS se, then the Lw of Sines or Lw of Cosines n e used to determine the side length nd/or ngle needed to use either of the two re formuls derived in this setion.

29 398 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors The Are of Tringle (SAS Cse) We now n use the generl formul for the re of tringle nd the Lw of Sines to develop formul for the re of tringle when the length of two sides nd the mesure of the ngle etween them re given. WORDS Strt with n ute tringle (given,, nd ). MATH h Write the sine rtio for ngle. Solve for h. Write the formul for re of tringle. sin h h sin A tringle 1 2 h For the SAS se, sustitute h sin. A SAS 1 sin 2 Now, the re of this tringle n e lulted with the given informtion (two sides nd the ngle etween them:,, nd ). Similrly, it n e shown tht the other formuls for SAS tringles re A SAS 1 sin g 2 nd A SAS 1 sin 2 THE AREA OF A TRIANGLE (SAS) For ny tringle where two sides nd the ngle etween them re known, the re for tht tringle is given y one of the following formuls (depending on whih ngle nd sides re given): A SAS 1 2 sin A SAS 1 2 sin g A SAS 1 2 sin when,, nd re known. when,, nd g re known. when,, nd re known. In other words, the re of tringle equls one-hlf the produt of two of its side lengths nd the sine of the ngle etween them.

30 7.3 The Are of Tringle 399 EXAMPLE 1 Finding the Are of Tringle (SAS Cse) Find the re of the tringle 7.0 feet, 9.3 feet, nd g 86. Solution: Use the re formul where,, nd g re given. A 1 sin g 2 Tehnology Tip Use lultor to find A. Sustitute 7.0 feet, 9.3 feet, nd g 86. Use lultor to pproximte. Round to two signifint digits. A 1 (7.0) (9.3) sin 86 2 A A 32 ft 2 YOUR TURN Find the re of the tringle 3.2 meters, 5.1 meters, nd 49. Answer: 6.2 m 2 The Are of Tringle (SSS Cse) We used the Lw of Sines to develop formul for the re of n SAS tringle. Tht formul, severl trigonometri identities, nd the Lw of Cosines n e used to develop formul for the re of n SSS tringle, lled Heron s formul. Clssroom Exmple Find the re of tringle with mesurements 16 entimeters, 213 entimeters, nd Answer: 1.95 m 2 WORDS Strt with ny of the formuls for SAS tringles. Squre oth sides. Isolte sin 2 g. Use the Pythgoren identity. Ftor the differene of the two squres on the right. Solve the Lw of Cosines, os g, for os g. MATH A 1 sin g 2 A sin 2 g 4A sin2 g 4A os2 g 4A 2 (1 os G)(1 os G) 2 2 os G Clssroom Exmple 7.3.1* Find the re of n equilterl tringle with side length. Answer:

31 400 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Sustitute os G into 2 4A 2 (1 os G)(1 os G). 2 2 Comine the expressions in rkets. Group the terms in the numertors. Write the numertors s the differene of two squres. Ftor the numertors: x 2 y 2 (x y)(x y). Simplify. 2 2 Multiply y 4. The semiperimeter s is hlf the perimeter of the tringle. Mnipulte eh of the four ftors. Sustitute these vlues in for the four ftors. Simplify. 4A A A 2 ( ) A ( ) d d 2 d d 2 d ( )2 2 d 2 d (2 2 2 ) 2 d 2 4A 2 [ ( )][ ( )] [( ) ] [( ) ] d d A 2 ( )( ) ( )( ) d d A 2 ( )( )( )( ) A 2 1 ( )( )( )( ) 16 s 1 ( ) 2 2 2s 2 2(s ) 2 2s 2 2(s ) 2 2s 2 2(s ) 2s A 2 1 2(s ) 2(s ) 2(s ) 2s 16 A 2 s(s )(s )(s ) Solve for A (re is lwys positive). A 1s(s )(s )(s ) THE AREA OF A TRIANGLE (SSS CASE, HERON S FORMULA) For ny tringle where the lengths of the three sides re known, the re for tht tringle is given y the following formul: A SSS 1s(s )(s )(s ) where,, nd re the lengths of the sides of the tringle nd s is hlf the perimeter of the tringle, lled the semiperimeter: s 2

32 7.3 The Are of Tringle 401 EXAMPLE 2 Finding the Are of Tringle (SSS Cse) Find the re of the tringle 5, 6, nd 9. Solution: Find the semiperimeter s. Sustitute: 5, 6, nd 9. Simplify. Write the formul for the re of tringle in the SSS se. Sustitute 5, 6, 9, nd s 10. Simplify the rdind. s s 2 s 10 A 1s(s )(s )(s ) A 110(10 5)(10 6)(10 9) A Tehnology Tip Use lultor to find A. Evlute the rdil. A sq units YOUR TURN Find the re of the tringle 3, 5, nd 6. Answer: sq units Clssroom Exmple Find the re of tringle with mesurements 217, 417, nd 317. Answer: SECTION 7.3 SUMMARY SMH In this setion, we derived formuls for lulting the res of tringles (SAS nd SSS ses). The Lw of Sines leds to three re formuls for the SAS se depending on whih ngles nd sides re given. A SAS 1 2 sin A SAS 1 2 sin g A SAS 1 2 sin The Lw of Cosines ws instrumentl in developing formul for the re of tringle (SSS se) when ll three sides re given. where s. 2 A SSS 1s(s )(s )(s ) SECTION 7.3 EXERCISES SKILLS In Exerises 1 36, find the re (in squre units) of eh tringle desried. 1. 8, 16, , 413, , 12, , 4, 45

33 402 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors 5. 6, 8, g , 10, , 10, , 6, , 6, , 1.9, , 7, , 4.8, g , 150, , 0.7, , 515, g , 111, , 15, , 1, , 13, , 33, , 0.6, , 50, , 151, , 40, , 10, , 50, , 15.7, , 146.5, ,000, 16,500, 18, , 13, , 75, , 23, , 11 4, , , , 8 3, , 17 6, 11 6 APPLICATIONS 37. Bermud Tringle. Clulte the re (to the nerest squre mile) of the so-lled Bermud Tringle, desried in Exmple 4 of Setion 7.2, if, s some people define it, its verties re loted in Norfolk, Bermud, nd Sntigo. LOCATION LOCATION DISTANCE (NAUTICAL MILES) Norfolk Bermud 850 Bermud Sntigo 810 Norfolk Sntigo 894 Agin, ignore the urvture of the Erth in your lultions. 38. Bermud Tringle. Clulte the re (to the nerest squre mile) of the Bermud Tringle if, s some people define it, its verties re loted in Mimi, Bermud, nd Sn Jun. 39. Tringulr Trp. A lrge tringulr trp is needed to over plyground when it rins. If the sides of the trp mesure 160 feet, 140 feet, nd 175 feet, then wht is the re of the trp (to the nerest squre foot)? 40. Flower Seed. A tringulr grden mesures 41 feet y 16 feet y 28 feet. You re going to plnt wildflower seed tht osts $4 per g. Eh g of flower seed overs n re of 20 squre feet. How muh will the seed ost? (Assume you hve to uy whole g you n t split one.) 41. Murl. Some students re pinting murl on the side of uilding. They hve enough pint for 1000-squre-foot re tringle. If two sides of the tringle mesure 60 feet nd 120 feet, then wht ngle (to the nerest degree) should the two sides form to rete tringle tht uses up ll the pint? LOCATION LOCATION DISTANCE (NAUTICAL MILES) Mimi Bermud 898 Bermud Sn Jun 831 Mimi Sn Jun Murl. Some students re pinting murl on the side of uilding. They hve enough pint for 500-squre-foot re tringle. If two sides of the tringle mesure 40 feet nd 60 feet, then wht ngle (to the nerest degree) should the two sides form to rete tringle tht uses up ll the pint?

34 7.3 The Are of Tringle Inset Infesttion. Some very destrutive eetles hve mde their wy into forest preserve. The rngers re trying to keep trk of their spred nd how well preventive mesures re working. In tringulr re tht is 22.5 miles on one side, 28.1 miles on the seond, nd 38.6 miles on the third, wht is the totl re the rngers re overing? 44. Rel Estte. A rel estte gent needs to determine the re of tringulr lot. Two sides of the lot re 150 feet nd 60 feet. The ngle etween the two mesured sides is 43. Wht is the re of the lot? 48. Prking Lot. A prking lot is to hve the shpe of prllelogrm tht hs djent sides mesuring 250 feet nd 300 feet. The ngle etween the two sides is 55. Wht is the re of the prking lot? Round to the nerest squre foot. 49. Regulr Hexgon. A regulr hexgon hs sides mesuring 3 feet. Wht is its re? Rell tht the mesure of n ngle of regulr n-gon is given y the formul 180 (n 2) ngle. n For Exerises 45 nd 46, refer to the following: A ompny is onsidering purhsing tringulr piee of property (see the figure elow) for the onstrution of new fility. The purhse is going to e sight unseen nd the ompny is using old surveys to pproximte re nd osts. A 3 ft 5 in. C 45. Business. If the survey indites tht one side is 275 feet, seond side is 310 feet, nd the ngle etween the two sides is 79,. Find the re of the property.. If the ompny wnts to offer the seller $2.13 per squre foot, wht is the totl ost of the property? 46. Business. If the survey indites tht one side is 475 feet, seond side is 310 feet, nd the ngle etween the two sides is 118,. Find the re of the property.. If the ompny wnts to offer the seller $1.97 per squre foot, wht is the totl ost of the property? 47. Prking Lot. A prking lot is to hve the shpe of prllelogrm tht hs djent sides mesuring 200 feet nd 260 feet. The ngle etween the two sides is 65. Wht is the re of the prking lot? Round to the nerest squre foot. B 50. Regulr Degon. A regulr degon hs sides mesuring 5 inhes. Wht is its re? 51. Pond Plot. A survey of pond finds tht it is roughly in the shpe of tringle tht mesures 250 feet y 275 feet y 295 feet. Find the re of the pond. Round to the nerest squre foot. 52. Field Plot. A field is prtly overed in mrsh tht mkes it unusle for growing rops. The usle portion is roughly 1 3 in the shpe of tringle tht mesures 2 mile y 8 mile y 2 3 mile. Wht is the re of the usle portion of the field? Round to the nerest squre foot. 53. Mrine Biology. A mrine iologist mesures the dorsl fin on shrk (roughly in the shpe of tringle) nd finds tht two of its sides mesure 12 inhes nd 15 inhes. If the ngle etween the sides mesures 42, find the re of the shrk s fin. 54. Mrine Biology. A mrine iologist mesures the til fin on shrk (roughly in the shpe of tringle) nd finds tht two of its sides mesure 34 inhes nd 37 inhes. If the ngle etween the sides mesures 15, find the re of the shrk s fin. 200 ft 65º 260 ft 260 ft 65º 200 ft

35 404 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors CATCH THE MISTAKE In Exerises 55 nd 56, explin the mistke tht is mde. 55. Clulte the re of the tringle 2, 6, nd 7. Solution: Find the semiperimeter. s Write the formul for the re of the tringle. A 1s(s )(s )(s ) Let 2, 6, 7, nd s 15. Simplify. A 115(15 2)(15 6)(15 7) A 114, This is inorret. Wht mistke ws mde? 56. Clulte the re of the tringle 2, 6, nd 5. Solution: Find the semiperimeter. Write the formul for the re of the tringle. Let 2, 6, 5, nd s 6.5. Simplify. s A s(s )(s )(s ) A 6.5(4.5)(0.5)(1.5) A 22 2 This is inorret. Wht mistke ws mde? CONCEPTUAL In Exerises 57 nd 58, determine whether eh sttement is true or flse. 57. Heron s formul n e used to find the re of right tringles. 58. Heron s formul n e used to find the re of isoseles tringles. 59. Find the re of the tringle in terms of x. x 60º 2 x Find the re of the tringle in terms of x. x 45º 3 x 5 CHALLENGE 61. Show tht the re for n SAA tringle is given y A 2 sin sin g 2 sin Assume tht,, nd re given. 62. Show tht the re of n isoseles tringle with equl sides of length s is given y 63. The segment of irle is the region ounded y hord nd the interseted r. Find the re of the segment shown in the figure. 10 m 80º O 10 m A isoseles 1 2 s2 sin u where u is the ngle etween the two equl sides. 64. The segment of irle is the region ounded y hord nd the interseted r. Find the re of the segment shown in the figure. 2 m 50º 2 m O

36 7.4 Vetors 405 TECHNOLOGY For Exerises 65 68, let A, B, nd C e the lengths of the three sides with X, Y, nd Z s the orresponding ngle mesures. Write progrm using the TI lultor to find the re of the given tringle. X 65. A 35, B 47, nd Z A 1241, B 1472, nd Z A 85, B 92, nd C A 1167, B 1113, nd C 1203 B C Z A Y SECTION 7.4 VECTORS SKILLS OBJECTIVES Find the mgnitude nd diretion of vetor. Add nd sutrt vetors. Perform slr multiplition of vetor. Express vetor in terms of its horizontl nd vertil omponents. Find unit vetors. Find resultnt vetors in pplition prolems. CONCEPTUAL OBJECTIVES Understnd the differene etween slrs nd vetors. Relte the geometri nd lgeri representtions of vetors. Vetors: Mgnitude nd Diretion Wht is the differene etween veloity nd speed? Speed hs only mgnitude, wheres veloity hs mgnitude nd diretion. We use slrs, whih re rel numers, to denote mgnitudes suh s speed nd weight. We use vetors, whih hve mgnitude nd diretion, to denote quntities suh s veloity (speed in ertin diretion) nd fore (weight in ertin diretion). A vetor quntity is geometrilly denoted y direted line segment, whih is line segment with n rrow representing diretion. There re mny wys to denote vetor. For exmple, the vetor shown in the mrgin n e denoted s u, AB, u or AB 1, where A is the initil point nd B is the terminl point. It is ustomry in ooks to use the old letter to denote vetor nd when hndwritten (s in your lss notes nd homework) to use the rrow on top to represent vetor. A u B

37 406 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors In this setion, we will limit our disussion to vetors in plne (two-dimensionl). It is importnt to note tht geometri representtion n e extended to three dimensions, nd lgeri representtion n e extended to ny higher dimension, s you will see in the exerises nd in lter setions. Geometri Interprettion of Vetors The mgnitude of vetor n e denoted one of two wys: u or ƒƒuƒƒ. We will use the former nottion. Study Tip The mgnitude of vetor is the distne etween the initil nd terminl points of the vetor. MAGNITUDE: ƒ U ƒ The mgnitude of vetor u, denoted u, is the length of the direted line segment, whih is the distne etween the initil nd terminl points of the vetor. Two vetors hve the sme diretion if they re prllel nd point in the sme diretion. Two vetors hve the opposite diretion if they re prllel nd point in opposite diretions. EQUAL VECTORS: U V Two vetors u nd v re equl ( u v) if nd only if they hve the sme mgnitude ( ƒ u ƒ ƒ v ƒ ) nd the sme diretion. Equl Vetors u v Sme Mgnitude ut Opposite Diretion u v Sme Mgnitude ƒ u ƒ ƒ v ƒ Different Mgnitude It is importnt to note tht vetors do not hve to oinide to e equl. In ft, vetors re movle with mgnitude nd diretion unhnged.

38 7.4 Vetors 407 VECTOR ADDITION: U V Two vetors, u nd v, n e dded together using either of the following pprohes: The til-to-tip (or hed-to-til) method: Sketh the initil point of one vetor t the terminl point of the other vetor. The sum, u v, is the resultnt vetor from the til end of u to the tip end of v. u u + v v [or] The prllelogrm method: Sketh the initil points of the vetors t the sme point. The sum u v is the digonl of the prllelogrm formed y u nd v. v u + v u The differene, u v, is the Resultnt vetor from the tip of v to the tip of u, when the tils of v nd u oinide. [or] The other digonl formed y the prllelogrm method. v u v u v u v u Algeri Interprettion of Vetors Sine vetors tht hve the sme diretion nd mgnitude re equl, ny vetor n e trnslted to n equl vetor with its initil point loted t the origin in the Crtesin plne. Therefore, we will now onsider vetors in retngulr oordinte system. A vetor with its initil point t the origin is lled position vetor, or vetor in stndrd position. A position vetor u with its terminl point t the point (, ) is denoted: y (, ) u H, I u x where the rel numers nd re lled the omponents of vetor u.

39 408 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Study Tip H, I denotes vetor. (, ) denotes point. Notie the sutle differene etween oordinte nottion nd vetor nottion. The point is denoted with prentheses, (, ), wheres the vetor is denoted with ngled rkets, H, I. The nottion H, I denotes vetor whose initil point is (0, 0) nd terminl point is (, ). Rell tht the geometri definition of the mgnitude of vetor is the length of the vetor. MAGNITUDE: ƒ U ƒ The mgnitude (or norm) of vetor, u H, I, is y u u (, ) x y EXAMPLE 1 Finding the Mgnitude of Vetor Find the mgnitude of the vetor u H3, 4I. Solution: x Write the formul for mgnitude of vetor. u u = 5 Let 3 nd 4. Simplify. u 23 2 (4) 2 u (3, 4) Note: If we grph the vetor u H3, 4I, we see tht the distne from the origin to the point (3, 4) is five units. Answer: 126 YOUR TURN Find the mgnitude of the vetor v H1, 5I. Clssroom Exmple Find the mgnitude of. : u H1,1I.* : u H 1 2, 3 4 I Answer: DIRECTION ANGLE OF A VECTOR The positive ngle etween the positive x-xis nd position vetor is lled the diretion ngle, denoted u. or tn u u tn 1 for u tn where 0 u in qudrnts I or IV for u in qudrnts II or III y u (, ) x

40 7.4 Vetors 409 EXAMPLE 2 Finding the Diretion Angle of Vetor Find the diretion ngle of the vetor v H1, 5I. Solution: Tehnology Tip Use lultor to find. Strt with tn u nd let 1nd 5. tn u 5 1 Use lultor to find tn 1 (5). tn 1 (5) 78.7 The lultor gve qudrnt IV ngle. The point (1, 5) lies in qudrnt II. ( 1, 5) v y 78.7º x Clssroom Exmple Find the diretion ngle of. : u H1,1I.* : u H 1 2, 3 4 I, ssuming 0 Answer: Add 180. u u YOUR TURN Find the diretion ngle of the vetor u H3, 4I. Answer: Rell tht two vetors re equl if they hve the sme mgnitude nd diretion. Algerilly, this orresponds to their omponents ( nd ) eing equl. EQUAL VECTORS: U V The vetors u H, I nd v H, d I re equl ( u v) if nd only if nd d. Vetor Opertions Vetor ddition geometrilly is done with the til-to-tip rule. Algerilly, vetor ddition is performed omponent y omponent. VECTOR ADDITION: U V If u H, I nd v H, d I, then u v H, d I.

41 410 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Clssroom Exmple Let : u H12, 12 I nd : v H12, 12 I. Find : u : v. Answer: H0,212 I EXAMPLE 3 Adding Vetors Let u H2, 7I nd v H3, 4I. Find u v. Solution: Let u H2, 7I nd v H3, 4I in the ddition formul. Simplify. u v H2 (3), 7 4I u v H1, 3I Answer: u v H4, 2I YOUR TURN Let u H1, 2I nd v H5, 4I. Find u v. We now summrize some vetor opertions. Addition nd sutrtion of vetors re performed lgerilly omponent y omponent. Multiplition, however, is not s strightforwrd. To perform slr multiplition of vetor (to multiply vetor y rel numer), we multiply omponent y omponent. In Setion 7.5, we will study form of multiplition for two vetors tht is defined s long s the vetors hve the sme numer of omponents; it gives result known s the dot produt nd is useful in solving ommon prolems in physis. SCALAR MULTIPLICATION: KU If k is slr (rel numer) nd u H, I, then k u k H, I Hk, k I Study Tip Slr multiplition orresponds to: 2u A slr multiple of u, ku, is lwys prllel to u. Inresing the length of the vetor: k 1 Deresing the length of the vetor: k 1 Chnging the diretion of the vetor: k 0 u 1 u 2 u The following ox is summry of some si vetor opertions: VECTOR OPERATIONS If u H, I nd v H, d I nd k is slr, then u v H, d I u v H, d I k u k H, I Hk, k I

42 7.4 Vetors 411 The zero vetor 0 H0, 0I is vetor in ny diretion with mgnitude equl to zero. We now n stte the lgeri properties of vetors: ALGEBRAIC PROPERTIES OF VECTORS u v v u (u v) w u (v w) (k 1 k 2 )u k 1 (k 2 u) k(u v) ku kv (k 1 k 2 )u k 1 u k 2 u 0u 0 1u u 1u u u (u) 0 Horizontl nd Vertil Components of Vetor The horizontl omponent nd vertil omponent of position vetor u re relted to the mgnitude of the vetor ƒ u ƒ through the sine nd osine of the diretion ngle. os u u sin u u y (, ) u x HORIZONTAL AND VERTICAL COMPONENTS OF A VECTOR The horizontl nd vertil omponents of position vetor u, with mgnitude u nd diretion ngle, re given y horizontl omponent: ƒ u ƒ os u vertil omponent: ƒ u ƒ sin u The vetor u n then e written s u H, I H ƒ u ƒ os u, ƒ u ƒ sin ui. EXAMPLE 4 Finding the Horizontl nd Vertil Components of Vetor Find the position vetor tht hs mgnitude of 6 nd diretion ngle of Solution: Write the horizontl nd vertil omponents of vetor u. u os u nd Let u 6 nd u os 15 nd Evlute the sine nd osine funtions of nd 15. u sin u 6 sin Clssroom Exmple 7.4.4* Find vetor of mgnitude 2 with diretion ngle. Answer: H2 os, 2 sin I Let u H, I. u H5.8, 1.6I YOUR TURN Find the position vetor tht hs mgnitude of 3 nd diretion ngle of 75. Answer: u H0.78, 2.9I

43 412 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Unit Vetors A unit vetor is ny vetor with mgnitude equl to 1, or u 1. It is often useful to e le to find unit vetor in the sme diretion of some vetor v. A unit vetor n e formed from ny nonzero vetor s follows: Study Tip Multiplying nonzero vetor y the reiprol of its mgnitude results in unit vetor. FINDING A UNIT VECTOR If v is nonzero vetor, then u v 1 v ƒ v ƒ ƒ v ƒ is the unit vetor in the sme diretion s v. In other words, multiplying ny nonzero vetor y the reiprol of its mgnitude results in unit vetor. It is importnt to notie tht sine the mgnitude is lwys slr, then the reiprol of the mgnitude is lwys slr. A slr times vetor is vetor. Clssroom Exmple Find unit vetor with the sme diretion s. H1,1I.* X 2, 1 Y 2 EXAMPLE 5 Finding Unit Vetor Find the unit vetor in the sme diretion s v H3, 4I. Solution: Find the mgnitude of the vetor v H3, 4I. v 2(3) 2 (4) 2 Simplify. v 5 Answer:.. X 12 12, 2 2 Y X , 1 Y 2 X , Y Multiply v y the reiprol of its mgnitude. Let v 5 nd v H3, 4I. Simplify. Chek: The unit vetor H 3 5, 4 5I should hve mgnitude of 1. 1 v ƒ v ƒ 1 H3, 4I 5 X 3 5, 4 5 Y 2 B B Answer: H 5 13, 12 13I YOUR TURN Find the unit vetor in the sme diretion s v H5, 12I.

44 7.4 Vetors 413 Two importnt unit vetors re the horizontl nd vertil unit vetors i nd j. The unit vetor i hs n initil point t the origin nd terminl point t (1, 0). The unit vetor j hs n initil point t the origin nd terminl point t (0, 1). These unit vetors n e used to represent vetors lgerilly: H3, 4I 3i 4j. (0, 1) y Resultnt Vetors j i (1, 0) x There re mny pplitions in whih vetors rise. Veloity vetors nd fore vetors re two tht we will disuss. For exmple, you might e t the eh nd think tht you re swimming stright out t ertin speed (mgnitude nd diretion). This is your pprent veloity with respet to the wter. But fter few minutes you turn round to look t the shore, nd you re frther out thn you thought nd you lso pper to hve drifted down the eh. This is euse of the urrent of the wter. When the urrent veloity nd the pprent veloity re dded together, the result is the tul or resultnt veloity. EXAMPLE 6 Resultnt Veloities A ot s speedometer reds 25 mph (whih is reltive to the wter) nd sets ourse due est (90 from due north). If the river is moving 10 mph due north, wht is the resultnt (tul) veloity of the ot? Solution: Drw piture. Resultnt veloity 25 mph N W E 10 mph Clssroom Exmple A ot s speedometer reds 35 mph (whih is reltive to the wter) nd sets ourse due west ( 90 from due north). If the river is moving 12 mph due north, wht is the resultnt veloity of the ot? Answer: The tul veloity of the ot is 37 mph heding 71 west of north. Lel the horizontl nd vertil omponents of the resultnt vetor. Determine the mgnitude of the resultnt vetor. H25, 10I mph S Determine the diretion ngle. Solve for u. The tul veloity of the ot hs mgnitude 22 north of est or 68 est of north. 27 mph tn u u 22 nd the ot is heded In Exmple 6, the three vetors formed right tringle. In Exmple 7, the three vetors form n olique tringle.

45 414 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors EXAMPLE 7 Resultnt Veloities A speedot trveling 30 mph hs ompss heding of 100 est of north. The urrent veloity hs mgnitude of 15 mph, nd its heding is 22 est of north. Find the resultnt (tul) veloity of the ot. Solution: Drw piture. 100º Resultnt veloity 22º 15 mph 30 mph N W E S Lel the supplementry ngle to 100 nd its equl lternte interior ngle with prllel north/south lines. 100º 80º W Resultnt veloity 30 mph N S E 80º 22º 15 mph Drw nd lel the olique tringle. The mgnitude of the tul (resultnt) veloity is. The heding of the tul (resultnt) veloity is º = 30 mph = 15 mph N W E S Use the Lw of Sines nd the Lw of Cosines to solve for nd. Find : Use the Lw of Cosines os Let 15, 30, nd (15)(30) os 102 Solve for. 36 mph

46 7.4 Vetors 415 Find A: Use the Lw of Sines. Isolte sin. Let 15, 36, nd 102. sin sin sin sin 15 sin 102 sin 36 Use the inverse sine funtion to solve for. Use lultor to pproximte. Atul heding: The tul veloity vetor of the ot hs mgnitude 76 est of north. sin mph 15 sin nd the ot is heded Two vetors, u nd v, omine to yield resultnt vetor, u v. The vetor u v is lled the equilirnt. EXAMPLE 8 Finding n Equilirnt A skier is eing pulled up hndle lift. Let F 1 represent the vertil fore due to grvity nd F 2 represent the fore of the skier pushing ginst the side of the mountin, whih is t n ngle of 35 to the horizontl. If the weight of the skier is 145 pounds, tht is, ƒ F 1 ƒ 145, find the mgnitude of the equilirnt fore F 3 required to hold the skier in ple (not let the skier slide down the mountin). Assume tht the side of the mountin is fritionless surfe. F 2 F 3 F 1 35º Clssroom Exmple A skier is eing pulled up slope y hndle lift. Let F 1 represent the vertil fore due to grvity, nd F 2 represent the fore of the skier pushing ginst the side of the mountin t n ngle of 40 to the horizontl. If the weight of the skier is 190 pounds, ƒ F 1 ƒ, find the mgnitude of the equilirnt to hold the skier in ple. Answer: 122 l Solution: The ngle etween vetors F 1 nd F 2 is 35. The mgnitude of vetor F 3 is the fore required to hold the skier in ple. Relte the mgnitudes (side lengths) to the given ngle using the sine rtio. Solve for ƒ F 3 ƒ. Let ƒ F 1 ƒ 145. A fore of pproximtely the hill. 83 pounds F 2 F 3 35º 55º 55º sin 35 ƒ F 3 ƒ ƒ F 1 ƒ ƒ F 3 ƒ ƒ F 1 ƒ sin 35 ƒ F 3 ƒ 145 sin 35 ƒ F 3 ƒ º is required to keep the skier from sliding down F 1

47 416 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Tehnology Tip Use lultor to find. EXAMPLE 9 Resultnt Fores A rge runs ground outside its hnnel. A single tugot nnot generte enough fore to pull the rge off the sndr. A seond tugot omes to ssist. The following digrm illustrtes the fore vetors, F 1 nd F 2, from the tugots. Wht is the resultnt fore vetor of the two tugots? Brge 80º 80º F 1 = 15,000 l 20º F 2 = 18,000 l Tug Tug Use lultor to find. Solution: Using the til-to-tip rule, we n dd these two vetors nd form tringle: N N 80º Resulting Resultnt fore 80º 18,000 l 80º 15,000 l Clssroom Exmple A rge runs ground outside its hnnel. A single tugot nnot generte enough fore to pull the rge off the sndr. A seond tugot omes to ssist. Consult the digrm on the right, nd ssume tht the fores pplied y the tugots remin unhnged.. If the ngle eh tugot mkes with the sndr is hnged from 80 to 75, wht is the mgnitude of the resultnt fore?. Wht hppens to the mgnitude of the resultnt fore s this ngle ontinues to derese? Answer:. 31,885 l. The mgnitude ontinues to derese. Solve for : Use the Lw of Cosines. Let 15,000, 18,000, nd 160. Solve for. Solve for A: Use the Lw of Sines. Isolte sin. 32,503 l Let 15,000, 32,503, nd 160. Use the inverse sine funtion to solve for. = 18,000 l = 160º os 2 15, , (15,000)(18,000) os 160 sin = 15,000 l sin sin sin 15,000 sin 160 sin 32,503 15,000 sin 160 sin 1 32,503 Use lultor to pproximte The resulting fore is 32,503 pounds t n ngle of 9 from the tug pulling with fore of 18,000 pounds, whih is 91 est of north.

48 7.4 Vetors 417 SECTION 7.4 SUMMARY In this setion, we disussed slrs (rel numers) nd vetors. Slrs hve only mgnitude, wheres vetors hve oth mgnitude nd diretion. Vetor: Mgnitude: u H, I ƒ u ƒ Diretion (u): tn u 0 We defined vetors oth lgerilly nd geometrilly nd gve interprettions of mgnitude nd vetor ddition in oth wys. Vetor ddition is performed lgerilly omponent y omponent. H, I H, di H, di The trigonometri funtions re used to express the horizontl nd vertil omponents of vetor. Horizontl omponent: ƒ u ƒ os u Vertil omponent: ƒ u ƒ sin u Resultnt veloity nd fore vetors n e found using the Lw of Sines nd the Lw of Cosines. SECTION 7.4 EXERCISES SKILLS In Exerises 1 6, find the mgnitude of the vetor AB, given the points A nd B. 1. A (2, 7) nd B (5, 9) 2. A (2, 3) nd B (3, 4) 3. A (4, 1) nd B (3, 0) 4. A (1, 1) nd B (2, 5) 5. A (0, 7) nd B (24, 0) 6. A (2, 1) nd B (4, 9) In Exerises 7 18, find the mgnitude nd diretion ngle of eh vetor. 7. u H3, 8I 8. u H4, 7I 9. u H5, 1I u H4, 1I 12. u H6, 3I 13. u H8, 0I u H13, 3I 16. u H5, 5I 17. u H 4 5, 1 3I 18. u H6, 2I u H0, 7I u H 3 7, 1 2I ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ In Exerises 19 30, perform the indited vetor opertion, given u H 4, 3I nd v H2, 5I. 19. u v 20. u v 21. 3u 22. 2u 23. 2u 4v 24. 5(u v) 25. 6(u v) 26. 3(u v) v 2u 3v 28. 2u 3v 4u 29. 4(u 2v) 3(v 2u) 30. 2(4u 2v) 5(3v u) In Exerises 31 42, find the position vetor, given its mgnitude nd diretion ngle. 31. u 7, u u 5, u u 16, u u 8, u u 4, u u 8, u u 9, u u 3, u u 2, u u 6, u u 3, u u 12, u 280

49 418 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors In Exerises 43 54, find the unit vetor in the diretion of the given vetor. 43. v H5, 12I 44. v H3, 4I 45. v H60, 11I v H24, 7I 48. v H10, 24I 49. v H9, 12I v H12, 312 I 52. v H413, 213 I 53. v H216, 316 I 54. In Exerises 55 62, express the vetor s sum of unit vetors i nd j. 55. H7, 3I 56. H2, 4I 57. H5, 3I H1, 0I 60. H0, 2I 61. H0.8, 3.6I 62. v H7, 24I v H40, 9I v H615, 4 I H6, 2I H 15 4, 10 3 I In Exerises 63 68, perform the indited vetor opertion. 63. (5i 2j) (3i 2j) 64. (4i 2j) (3i 5j) 65. (3i 3j) (2i 2j) 66. (i 3j) (2i j) 67. (5i 3j) (2i 3j) 68. (2i j) (2i 4j) APPLICATIONS 69. Bullet Speed. A ullet is fired from ground level t speed of 2200 feet per seond t n ngle of 30 from the horizontl. Find the mgnitude of oth the horizontl nd vertil omponents of the veloity vetor. 70. Weightlifting. A 50-pound weight lies on n inlined enh tht mkes n ngle of 40 with the horizontl. Find the omponent of the weight direted perpendiulr to the enh nd lso the omponent of the weight prllel to the inlined enh. 71. Weight of Bot. A fore of 630 pounds is needed to pull speedot nd its triler up rmp tht hs n inline of 13. Wht is the omined weight of the ot nd its triler to the nerest pound? 75. Heding nd Airspeed. A plne hs ompss heding of 60 est of due north nd n irspeed of 300 mph. The wind is lowing t 40 mph with heding of 30 west of due north. Wht re the plne s tul heding nd irspeed? N 30º 40 mph 60º 300 mph 13º 72. Weight of Bot. A fore of 500 pounds is needed to pull speedot nd its triler up rmp tht hs n inline of 16. Wht is the weight of the ot nd its triler to the nerest pound? 73. Speed nd Diretion of Ship. A ship s ptin sets ourse due north t 10 mph. The wter is moving t 6 mph due west. Wht is the tul veloity of the ship, nd in wht diretion is it trveling? 74. Speed nd Diretion of Ship. A ship s ptin sets ourse due west t 12 mph. The wter is moving t 3 mph due north. Wht is the tul veloity of the ship, nd in wht diretion is it trveling? 76. Heding nd Airspeed. A plne hs ompss heding of 30 est of due north nd n irspeed of 400 mph. The wind is lowing t 30 mph with heding of 60 west of due north. Wht re the plne s tul heding nd irspeed? 77. Heding. An irplne tkes off nd flies t 175 mph for 1 hour on ompss heding of N 135 E. The pilot then turns nd flies for 2 hours t 185 mph on heding of N 80 E. How fr is the plne from the irport, nd wht is its ering from the irport? 78. Heding. An irplne tkes off nd flies t 205 mph for 1 hour on ompss heding of N 255 E. The pilot then turns nd flies for 2 hours t 165 mph on heding of N 170 E. How fr is the plne from the irport, nd wht is its ering from the irport?

50 7.4 Vetors Wind Speed. An irplne hs n irspeed of 250 mph nd ompss heding of 285. With 30 mph wind, its tul heding is 280. When tking into effet the wind, wht is the tul speed of the plne? 80. Wind Speed. An irplne hs n irspeed of 220 mph nd ompss heding of 165. With 40 mph wind, its tul heding is 173. When tking into effet the wind, wht is the tul speed of the plne? 81. Sliding Box. A ox weighing 500 pounds is held in ple on n inlined plne tht hs n ngle of 30 with the ground. Wht fore is required to hold it in ple? 30º 500 l 82. Sliding Box. A ox weighing 500 pounds is held in ple on n inlined plne tht hs n ngle of 10 with the ground. Wht fore is required to hold it in ple? 83. Bsell. A sell plyer throws ll with n initil veloity of 80 feet per seond t n ngle of 40 with the horizontl. Wht re the vertil nd horizontl omponents of the veloity? 84. Bsell. A sell pither throws ll with n initil veloity of 100 feet per seond t n ngle of 5 with the horizontl. Wht re the vertil nd horizontl omponents of the veloity? For Exerises 85 nd 86, refer to the following: A post pttern in footll is when the reeiver in motion runs pst the qurterk prllel to the line of srimmge (A), then runs 12 yrds perpendiulr to the line of srimmge (B), nd then uts towrd the gol post (C) B A 30º C A + B + C Footll. A reeiver runs the post pttern. If the mgnitudes of the vetors re ƒ A ƒ 4 yrds, ƒ B ƒ 12 yrds, nd ƒ C ƒ 20 yrds, find the mgnitude of the resultnt vetor A B C. 86. Footll. A reeiver runs the post pttern. If the mgnitudes of the vetors re ƒ A ƒ 4 yrds, ƒ B ƒ 12 yrds, nd ƒ C ƒ 20 yrds, find the diretion ngle u. 87. Resultnt Fore. A fore with mgnitude of 100 pounds nd nother with mgnitude of 400 pounds re ting on n ojet. The two fores hve n ngle of 60 etween them. Wht is the diretion ngle of the resultnt fore with respet to the fore pulling 400 pounds? 88. Resultnt Fore. A fore with mgnitude of 100 pounds nd nother with mgnitude of 400 pounds re ting on n ojet. The two fores hve n ngle of 60 etween them. Wht is the mgnitude of the resultnt fore? 89. Resultnt Fore. A fore of 1000 pounds is ting on n ojet t n ngle of 45 from the horizontl. Another fore of 500 pounds is ting t n ngle of 40 from the horizontl. Wht is the mgnitude of the resultnt fore? 90. Resultnt Fore. A fore of 1000 pounds is ting on n ojet t n ngle of 45 from the horizontl. Another fore of 500 pounds is ting t n ngle of 40 from the horizontl. Wht is the diretion ngle of the resultnt fore? For Exerises 91 nd 92, refer to the following: Musle A nd musle B re tthed to one s indited in the figure elow. Musle A exerts fore on the one t ngle, while musle B exerts fore on the one t ngle. Musle B Musle A Bone 91. Helth/Mediine. Assume musle A exerts fore of 900 N on the one t ngle 8, while musle B exerts fore of 750 N on the one t ngle 33. Find the resultnt fore nd the ngle of the fore due to musle A nd musle B on the one. 92. Helth/Mediine. Assume musle A exerts fore of 1000 N on the one t ngle 9, while musle B exerts fore of 820 N on the one t ngle 38. Find the resultnt fore nd the ngle of the fore due to musle A nd musle B on the one.

51 420 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors CATCH THE MISTAKE In Exerises 93 nd 94, explin the mistke tht is mde. 93. Find the mgnitude of the vetor H2, 8I. Solution: Ftor the 1. H2, 8I Find the mgnitude ƒ H2, 8I ƒ of H2, 8I Write the mgnitude of H2, 8I. ƒ H2, 8I ƒ 2117 This is inorret. Wht mistke ws mde? CONCEPTUAL 94. Find the diretion ngle of the vetor H2, 8I. Solution: Write the formul for the diretion ngle of H, I. Let 2 nd 8. Use the inverse tngent funtion. Use lultor to evlute. This is inorret. Wht mistke ws mde? tn u tn u 8 2 u tn 1 4 u 76 In Exerises 95 98, determine whether eh sttement is true or flse. 95. The mgnitude of the vetor i is the imginry numer i. 96. Equl vetors must oinide. 97. The mgnitude of vetor is lwys greter thn or equl to the mgnitude of its horizontl omponent. 98. The mgnitude of vetor is lwys greter thn or equl to the mgnitude of its vertil omponent. 99. Would slr or vetor represent the following? A r is driving 72 mph due est (90 with respet to north). CHALLENGE 100. Would slr or vetor represent the following? The grnite hs mss of 131 kilogrms Find the mgnitude of the vetor H, I if 0 nd Find the diretion ngle of the vetor H, I if 0 nd Find the mgnitude nd diretion of the vetor H3, 4I. Assume Find the mgnitude nd diretion of the vetor H 3 4, 2 3I. Assume Let u H2, 3I nd v H4, 2I. Find the mgnitude nd diretion of u v Let u H2, 3I nd v H4, 2I. Find the mgnitude nd diretion of u v Let u H2, 1I nd v H2, 1I. Find the ngle u etween u nd v Let u H4, 1I nd v H1, 5I. Find the ngle u etween u nd v. TECHNOLOGY For Exerises , refer to the following: Vetors n e represented s olumn mtries. For exmple, the vetor u H3, 4I n e represented s 2 1 olumn 3 mtrix Using TI-83, vetors n e entered s mtries 4 d. in two wys, diretly or vi MATRIX. Diretly: Use lultor to perform the vetor opertion given u H8, 5I nd v H7, 11I Use lultor to find the unit vetor in the diretion of the given vetor u 3v 9(u 2v) u H10, 24I 112. u H9, 40I Mtrix:

52 SECTION 7.5 THE DOT PRODUCT SKILLS OBJECTIVES Find the dot produt of two vetors. Use the dot produt to find the ngle etween two vetors. Determine whether two vetors re prllel or perpendiulr. Use the dot produt to lulte the mount of work ssoited with physil prolem. CONCEPTUAL OBJECTIVE Understnd why the dot produt of two perpendiulr vetors is equl to zero. Multiplying Two Vetors: The Dot Produt There re two types of multiplition defined for vetors: slr multiplition nd the dot produt. Slr multiplition (whih we lredy demonstrted in Setion 7.4) is multiplition of vetor y slr; the result is prllel vetor. Now, we disuss the dot produt of two vetors. In this se, there re two importnt things to note: (1) The dot produt of two vetors is defined only if the vetors hve the sme numer of omponents, nd (2) if the dot produt does exist, the result is slr. THE DOT PRODUCT The dot produt of two vetors u H, I nd v H, d I is given y u v is pronouned u dot v. u v d EXAMPLE 1 Finding the Dot Produt of Two Vetors Find the dot produt H7, 3I H2, 5I. Solution: Sum the produts of the first omponents nd the produts of the seond omponents. H7, 3I H2, 5I (7)(2) (3)(5) Clssroom Exmple Find the dot produt H2, 1I H1, 3I.. *Find the vlue of x suh tht X 2, Y Hx, 21I 0, 3 ssuming 0. Answer:. 5. x 1 for 0 3 Study Tip The dot produt of two vetors is slr. Simplify YOUR TURN Find the dot produt H6, 1I H2, 3I. Answer: 9 The following ox summrizes the properties of the dot produt: PROPERTIES OF THE DOT PRODUCT 1. u v v u 4. k(u v) (ku) v u (kv) 2. u u u 2 5. (u v) w u w v w 3. 0 u 0 6. u (v w) u v u w 421

53 422 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Angle Between Two Vetors We n use these properties to develop n eqution tht reltes the ngle etween two vetors nd the dot produt of the vetors. WORDS Let u nd v e two nonzero vetors with the sme initil point nd let u, 0 u 180, e the ngle etween them. The vetor u v is opposite ngle u. MATH v v u u v u A tringle is formed with side lengths equl to the mgnitudes of the three vetors. v u v u Use the Lw of Cosines. Use properties of the dot produt to rewrite the left side of eqution: Property (2): Property (6): Property (6): Property (2): Property (1): Sustitute this lst expression for the left side of the originl Lw of Cosines eqution. Simplify. Isolte os u. ƒ u v ƒ 2 ƒ u ƒ 2 ƒ v ƒ 2 2 ƒ u ƒƒv ƒ os u ƒ u v ƒ 2 (u v) (u v) u (u v) v (u v) u u u v v u v v ƒ u ƒ 2 2 u v v u ƒ v ƒ ƒ u ƒ 2 2 2(u v) ƒ v ƒ ƒ u ƒ 2 2(u v) ƒ v ƒ 2 ƒ u ƒ 2 ƒ v ƒ 2 2 ƒ u ƒƒvƒ os u 2(u v) 2ƒuƒƒvƒ os u os u u v ƒ u ƒƒvƒ Notie tht u nd v hve to e nonzero vetors sine we divided y them in the lst step.

54 7.5 The Dot Produt 423 ANGLE BETWEEN TWO VECTORS If u is the ngle, 0 u 180, etween two nonzero vetors u nd v, then os u u v ƒ u ƒƒvƒ Study Tip The ngle etween two vetors u os 1 u v is n ngle ƒ u ƒƒvƒ etween 0 nd 180 (the rnge of the inverse osine funtion). In the Crtesin plne, there re two ngles etween two vetors: u nd 360 u. We ssume is the smller ngle, whih will lwys e etween 0 nd 180. EXAMPLE 2 Finding the Angle Between Two Vetors Find the smller ngle etween H2, 3I nd H4, 3I. Solution: Let u H2, 3I nd v H4, 3I. STEP 1 STEP 2 STEP 3 Find u v. Find ƒ u ƒ. Find ƒ v ƒ. u v H2, 3I H4, 3I (2)(4) (3)(3) 17 u 1u u 22 2 (3) v 1v v 2(4) Clssroom Exmple Find the ngle etween H2, 1I nd H1, 3I. Answer: 135 STEP 4 Find u. Use lultor to pproximte u. os u u v 17 ƒ u ƒƒvƒ 5113 u os u 161 Tehnology Tip Use lultor to find, os u STEP 5 Drw piture to onfirm the nswer. Drw the vetors H2, 3I nd H4, 3I. 161 ppers to e orret. ( 4, 3) v 161º y u x (2, 3) YOUR TURN Find the smller ngle etween H1, 5I nd H2, 4I. Answer: u 38 When two vetors re prllel, the ngle etween them is 0 or 180. = 0º u v v = 180º u When two vetors re perpendiulr (orthogonl), the ngle etween them is 90. v = 90º u Note: We did not inlude 270 euse the ngle u etween two vetors is tken to e the smller ngle (i.e., 0 u 180 ).

55 424 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors WORDS When two vetors u nd v re perpendiulr, u 90. Sustitute os MATH os 90 u v ƒ u ƒƒvƒ 0 u v ƒ u ƒƒvƒ Therefore, the dot produt of u nd v must e zero. u v 0 ORTHOGONAL VECTORS Two vetors u nd v re orthogonl (perpendiulr) if nd only if their dot produt is zero. u v 0 EXAMPLE 3 Determining Whether Vetors Are Orthogonl Clssroom Exmple 7.5.3* Determine the vlue of for Determine whether eh pir of vetors is n orthogonl pir.. u H2, 3I nd v H3, 2I. u H7, 3I nd v H7, 3I whih X 21, nd 3 Y X 1 re perpendiulr. 4 1, Y Solution (): Find the dot produt u v. Simplify. u v (2)(3) (3)(2) u v 0 Answer: 0 Vetors u nd v re orthogonl sine u v 0. Solution (): Find the dot produt u v. Simplify. u v (7)(7) (3)(3) u v 58 Vetors u nd v re not orthogonl sine u v 0. Work If you hd to rry either rells with weights or pillows for 1 mile, whih would you hoose? You proly would pik the pillows over the rell nd weights euse the pillows re lighter. It requires less work to rry the pillows thn it does to rry the rell with weights. If sked to rry either of them 1 mile or 10 miles, you would proly pik 1 mile, euse it s shorter distne nd requires less work. Work is done when fore uses n ojet to move ertin distne. The simplest se is when the fore is in the sme diretion s the displement for exmple, stgeoh (the horses pull with fore in the sme diretion). In this se, the work is defined s the mgnitude of the fore times the mgnitude of the displement, distne d. W ƒ F ƒ d Notie tht the mgnitude of the fore is slr, the distne d is slr, nd hene the produt is slr. If the horses pull with fore of 1000 pounds nd they move the stgeoh 100 feet, then the work done y the fore is W (1000 l)(100 ft) 100,000 ft-l

56 7.5 The Dot Produt 425 In mny physil pplitions, however, the fore is not in the sme diretion s the displement, nd hene vetors (not just their mgnitudes) re required. F 2 F 1 We often wnt to know how muh of fore is pplied in ertin diretion. For exmple, when your r runs out of gsoline nd you try to push it, some of the fore vetor F 1 you generte from pushing trnsltes into the horizontl omponent F 2 ; hene, the r moves horizontlly. If we let u e the ngle etween the vetors F 1 nd F 2, then the horizontl omponent of F 1 is where ƒ F 2 ƒ ƒ F 1 ƒ os u. F 2 F 2 F 1 If the mn in the piture pushes t n ngle of 25 with fore of 150 pounds, then the horizontl omponent of the fore vetor F 1 is (150 l)(os 25 ) 136 l WORDS To develop generlized formul when the fore exerted nd the displement re not in the sme diretion, we strt with the formul for the ngle etween two vetors. We then isolte the dot produt u v. MATH os u u v ƒ u ƒƒv ƒ u v ƒ u ƒƒv ƒ os u Let u F nd v d. W F d ƒ F ƒƒd ƒ os u ƒ F ƒ os u mgnitude of fore in diretion of displement ƒ d ƒ e distne W ORK If n ojet is moved from point A to point B y onstnt fore, then the work ssoited with this displement is W F d where d is the displement vetor nd F is the fore vetor.

57 426 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors Work is typilly expressed in one of two units: SYSTEM FORCE DISTANCE WORK Amerin pound foot ft-l SI newton meter N-m Clssroom Exmple Find the work done when fore F H, 1 2I moves n ojet from (0, 0) to (, 3). Answer: EXAMPLE 4 Clulting Work How muh work is done when fore (in pounds) F H2, 4I moves n ojet from (0, 0) to (5, 9) (the distne is in feet)? Solution: Find the displement vetor d. d H5, 9I Use the work formul W F d. W H2, 4I H5, 9I Clulte the dot produt. W (2)(5) (4)(9) Simplify. W 46 ft-l Answer: 25 N-m YOUR TURN How muh work is done when fore (in newtons) F H1, 3I moves n ojet from (0, 0) to (4, 7) (the distne is in meters)? SECTION 7.5 SUMMARY SMH In this setion, we defined the dot produt s form of multiplition of two vetors. A slr times vetor results in vetor, wheres the dot produt of two vetors is slr. H, I H, di d We developed formul tht determines the ngle u etween two vetors u nd v. os u u v ƒ u ƒƒvƒ Orthogonl (perpendiulr) vetors hve n ngle of 90 etween them, nd we showed tht the dot produt of two orthogonl vetors is equl to zero. Work is the result of fore displing n ojet. When the fore nd displement re in the sme diretion, the work is equl to the produt of the mgnitude of the fore nd the distne (mgnitude of the displement). When the fore nd displement re not in the sme diretion, work is the dot produt of the fore vetor nd displement vetor, W F d. SECTION 7.5 EXERCISES SKILLS In Exerises 1 16, find eh of the following dot produts. 1. H4, 2I H3, 5I 2. H7, 8I H2, 1I 3. H5, 6I H3, 2I 4. H6, 3I H2, 1I 5. H1, 3I H4, 10I 6. H2, 3I H3, 5I 7. H1, 8I H2, 6I 8. H10, 8I H12, 6I 9. H7, 4I H2, 7I 10. H5, 2I H1, 1I 11. H13, 2I H313, 1I 12. H412, 17 I H12, 17 I 13. H5, I H3, 2I 14. H4 x, 3yI H2y, 5xI 15. H0.8, 0.5I H2, 6I 16. H18, 3I H10, 300I In Exerises 17 32, find the ngle ( 0 180; round to the nerest degree) etween eh pir of vetors. 17. H4, 3I nd H5, 9I 18. H2, 4I nd H4, 1I 19. H2, 3I nd H3, 4I 20. H6, 5I nd H3, 2I 21. H4, 6I nd H6, 8I 22. H1, 5I nd H3, 2I 23. H6, 2I nd H3, 8I 24. H7, 2I nd H4, 1I

58 7.5 The Dot Produt H2, 213 I nd H13, 1I 26. H313, 3I nd H213, 2I 27. H513, 5I nd H12, 12 I 28. H5, 513 I nd H2, 12 I 29. H4, 6I nd H6, 9I 30. H2, 8I nd H12, 3I 31. H1, 6I nd H2, 4I 32. H8, 2I nd H10, 3I In Exerises 33 44, determine whether eh pir of vetors is orthogonl. 33. H6, 8I nd H8, 6I 34. H5, 2I nd H5, 2I 35. H6, 4I nd H6, 9I 36. H8, 3I nd H6, 16I 37. H0.8, 4I nd H3, 6I 38. H7, 3I nd H 1 7, 1 3I 39. H5, 0.4I nd H1.6, 20I 40. H12, 9I nd H3, 4I 41. H13, 16 I nd H12, 1I 42. H17, 13 I nd H3, 7I 43. H 4 3, 15I 8 nd H12, 1 24I H 5 6, 6 7I nd H 25 36, 49 36I APPLICATIONS 45. Lifting Weights. How muh work does it tke to lift 100 pounds vertilly 4 feet? 46. Lifting Weights. How muh work does it tke to lift 150 pounds vertilly 3.5 feet? 47. Rising Wreks. How muh work is done y rne to lift 2-ton r to level of 20 feet? 48. Rising Wreks. How muh work is done y rne to lift 2.5-ton r to level of 25 feet? 49. Work. To slide rte ross the floor, fore of 50 pounds t 30 ngle is needed. How muh work is done if the rte is drgged 30 feet? Round to the nerest ft-l. 50. Work. To slide rte ross the floor, fore of 800 pounds t 20 ngle is needed. How muh work is done if the rte is drgged 50 feet? Round to the nerest ft-l. 51. Close Door. A sliding door is losed y pulling ord with onstnt fore of 35 pounds t onstnt ngle of 45. The door is moved 6 feet to lose it. How muh work is done? Round to the nerest ft-l. 52. Close Door. A sliding door is losed y pulling ord with onstnt fore of 45 pounds t onstnt ngle of 55. The door is moved 6 feet to lose it. How muh work is done? Round to the nerest ft-l. 30º Diretion drgged 53. Brking Power. A r tht weighs 2500 pounds is prked on hill in Sn Frniso with slnt of 40 from the horizontl. How muh fore will keep it from rolling down the hill? Round to the nerest pound. 54. Brking Power. A r tht weighs 40,000 pounds is prked on hill in Sn Frniso with slnt of 10 from the horizontl. How muh fore will keep it from rolling down the hill? Round to the nerest pound. 55. Towing Power. A semi-triler truk tht weighs 40,000 pounds is prked on hill in Sn Frniso with slnt of 10 from the horizontl. A tow truk hs to remove the truk from its prking spot nd move it 100 feet up the hill. How muh work is required? Round to the nerest ft-l. 56. Towing Power. A r tht weighs 2500 l is prked on hill in Sn Frniso with slnt of 40 from the horizontl. A tow truk hs to remove the r from its prking spot nd move it 120 feet up the hill. How muh work is required? Round to the nerest ft-l. 57. Trvel Vetor. Two irplnes tke off from the sme irport nd trvel in different diretions. One psses over town A known to e 30 miles north nd 50 miles est of the irport. The other plne flies over town B known to e 10 miles south nd 60 miles est of the irport. Wht is the ngle etween the diretion of trvel of the two plnes? 58. Trvel Vetor. Two irplnes tke off from the sme irport nd trvel in different diretions. One psses over town A known to e 60 miles north nd 10 miles west of the irport. The other plne flies over town B known to e 40 miles south nd 20 miles west of the irport. Wht is the ngle etween the diretion of trvel of the two plnes? 59. Push Lwn Mower. How muh work is required to push lwn mower 100 feet if the fore pplied to the hndle is 70 pounds nd the hndle mkes n ngle of 40 with the horizontl? Round to the nerest ft-l. 60. Push Lwn Mower. How muh work is required to push lwn mower 75 feet if the fore pplied to the hndle is 65 pounds nd the hndle mkes n ngle of 45 with the horizontl? Round to the nerest ft-l.

59 428 CHAPTER 7 Applitions of Trigonometry: Tringles nd Vetors CATCH THE MISTAKE In Exerises 61 nd 62, explin the mistke tht is mde. 61. Find the dot produt H3, 2I H2, 5I. Solution: Multiply omponent y omponent. Simplify. H3, 2I H2, 5I H(3)(2), (2)(5)I H3, 2I H2, 5I H6, 10I This is inorret. Wht mistke ws mde? 62. Find the dot produt H11, 12I H2, 3I. Solution: Multiply the outer nd inner omponents. H11, 12I H2, 3I (11)(3) (12)(2) Simplify. H11, 12I H2, 3I 9 This is inorret. Wht mistke ws mde? CONCEPTUAL In Exerises 63 66, determine whether eh sttement is true or flse. 63. A dot produt of two vetors is vetor. 64. A dot produt of two vetors is slr. 65. Orthogonl vetors hve dot produt equl to zero. 66. If the dot produt of two nonzero vetors is equl to zero, then the vetors must e perpendiulr. For Exerises 67 nd 68, refer to the following to find the dot produt: The dot produt of vetors with n omponent is H 1, 2,..., n I H 1, 2,..., n I n n 67. H3, 7, 5I H2, 4, 1I 68. H1, 0, 2, 3I H5, 2, 3, 1I ƒ ƒ In Exerises 69 72, given u H, I nd v H, d I, show tht the following properties re true u v v u 2 u u u 0 u 0 k(u v) (ku) v u (kv), k is slr CHALLENGE 73. Explin why (u v) w for vetors u, v, w does not exist. 74. Let u H2, 3I, v H4, 1I, nd w H3, 5I. Demonstrte tht u (v w) u v u w. 75. Let u H, I nd v H, I. Find ƒ u ƒ 2 (u v). 76. Find the dot produt of u nd v if the ngle etween the vetors is 45 nd ƒ u ƒ 222 nd ƒ v ƒ Find the dot produt of u nd v if the ngle etween the vetors is 30 nd ƒ u ƒ 210 nd ƒ v ƒ Let u H, 0I nd v H, 23I. Find the ngle u etween vetors u nd v. TECHNOLOGY For Exerises 79 nd 80, use lultor to find the indited dot produt. 79. H11, 34I H15, 27I 80. H23, 350I H45, 202I 81. A retngle hs sides with lengths of 18 units nd 11 units. Find the ngle, to one deiml ple, etween the digonl nd the side with length of 18 units. Hint: Set up retngulr oordinte system nd use vetors H18, 0I to represent the side with length of 18 units nd H18, 11I to represent the digonl. 82. The definition of dot produt nd the formul to find the ngle etween two vetors n e extended nd pplied to vetors with more thn two omponents. A retngulr ox hs sides with lengths 12 feet, 7 feet, nd 9 feet. Find the ngle, to the nerest degree, etween the digonl nd the side with length 7 feet.

60 CHAPTER 7 INQUIRY-BASED LEARNING PROJECT When solving tringles using the Lw of Sines, it is importnt to keep in mind the domin nd rnge of the inverse sine funtion, euse it n ply role in determining your nswers. Mke sure to hek your nswers for resonleness, espeilly when solving n otuse tringle, s you will see next. For tringle ABC, two side lengths nd the mesure of the ngle etween them re given elow A How mny tringles re possile with these mesurements? Explin. 2. Mke reful sketh of tringle nd lel the given informtion. Also lel the unknown side, nd the unknown ngles opposite sides nd, s B nd C, respetively. Wht n you sy out ngle B? 3. Explin why the Lw of Cosines is needed for this prolem. 4. Find the length of side. Round to three signifint digits. 5. Suppose fellow student now wnts to find the mesure of ngle B, nd deides to use the Lw of Sines. Shown elow re the steps he wrote out to illustrte his proess for omputing the mesure of B nd C. Look t the hrt he filled in elow. B sin 1 24 sin(15) d C 180 ( ) Sides Angles A B C 15º 22.3º 142.7º Wht is wrong with the nswers given y this student? 6. Beuse your lultor gives positive vlues of sin 1 x only etween 0 nd 90º, the nswer this student got for the mesure of ngle B does not mke sense. Show wht he needs to do to orret his error. 429

61 MODELING OUR WORLD Mny Amerins re onsidering hyrid nd eletri utomoiles s n ttrtive lterntive to trditionl utomoiles. The following tle illustrtes the pproximte gross vehile weight of oth lrge SUV nd smll hyrid, nd the pproximte fuel eonomy rtes in miles per gllon: Automoiles Weight Mpg Ford Expedition 7100 l 18 Toyot Prius 2800 l 45 Rell tht the mount of work to push n ojet tht weighs F pounds distne of d feet long horizontl is W F d. 1. Clulte how muh work it would tke to move Ford Expedition 100 feet. 2. Clulte how muh work it would tke to move Toyot Prius 100 feet. 3. Compre the vlues you lulted in Questions 1 nd 2. Wht is the rtio of work required to move the Expedition to work required to move the Prius? 4. Compre the result in Question 3 with the rtio of fuel eonomy (mpg) for these two vehiles. Wht n you onlude out the reltionship etween the weight of n utomoile nd its fuel eonomy? 5. Clulte the work required to move the Ford Expedition nd Toyot Prius 100 feet long n inline tht mkes 45 ngle with the ground (horizontl). 6. Bsed on your results in Question 5, do you expet the fuel eonomy rtios to e the sme in this inlined senrio ompred with the horizontl? In other words, should onsumers in Florid (flt) e guided y the sme numers s onsumers in the Applhin Mountins (North Crolin)? 430