SECTION 72 Law of Cosines


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1 516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished nd the ngles,, nd n e mesured. Show tht d SETION 72 Lw of osines Lw of osines Derivtion Solving the SAS se Solving the SSS se If in tringle two sides nd the inluded ngle re given (SAS) or three sides re given (SSS), the lw of sines nnot e used to solve the tringle neither se involves n ngle nd its opposite side (Fig. 1). Both ses n e solved strting with the lw of osines, whih is the sujet mtter for this setion. FIGURE 1 () SAS se () SSS se Lw of osines Derivtion Theorem 1 sttes the lw of osines. Theorem 1 Lw of osines os os os All three equtions sy essentilly the sme thing.
2 72 Lw of osines 517 The lw of osines is used to solve tringles, given: 1. Two sides nd the inluded ngle (SAS) 2. Three sides (SSS) We will estlish the first eqution in Theorem 1. The other two equtions then n e otined from this one simply y releling the figure. We strt y loting tringle in retngulr oordinte system. Figure 2 shows three typil tringles. For n ritrry tringle loted s in Figure 2, the distneetweentwopoints formul is used to otin (h) 2 (k 0) 2 2 (h ) 2 k 2 h 2 2h 2 k 2 Squre oth sides. (1) FIGURE 2 Three representtive tringles. (h, k) (h, k) (h, k) (, 0) h k h () () () k (, 0) k (h, 0) h 0 (, 0) From Figure 2, we note tht 2 h 2 k 2 Sustituting 2 for h 2 k 2 in eqution (1), we otin h (2) But os h h os Thus, y repling h in eqution (2) with os, we reh our ojetive: os [Note: If is ute, then os is positive; if is otuse, then os is negtive.] Solving the SAS se For the SAS se, strt y using the lw of osines to find the side opposite the given ngle. Then use either the lw of osines or the lw of sines to find seond ngle. Beuse of the simpler omputtions, the lw of sines will generlly e used to find the seond ngle.
3 518 7 Additionl Topis in Trigonometry EXPLOREDISUSS 1 After using the lw of osines to find the side opposite the ngle for SAS se, the lw of sines is used to find seond ngle. Figure 2 () shows tht there re two hoies for seond ngle. (A) If the given ngle is otuse, n either of the remining ngles e otuse? Explin. (B) If the given ngle is ute, then one of the remining ngles my or my not e otuse. Explin why hoosing the ngle opposite the shorter side gurntees the seletion of n ute ngle. () Strting with (sin )/ (sin )/, show tht sin sin 1 (D) Explin why eqution (1) gives us the orret ngle only if is ute. (1) The ove disussion leds to the following strtegy for solving the SAS se: Strtegy for Solving the SAS se Step Find Method 1. Side opposite given ngle Lw of osines 2. Seond ngle (Find the ngle Lw of sines opposite the shorter of the two given sides this ngle will lwys e ute.) 3. Third ngle Sutrt the sum of the mesures of the given ngle nd the ngle found in step 2 from 180. EXAMPLE 1 Solving the SAS se Solve the tringle in Figure 3. FIGURE m m
4 72 Lw of osines 519 Solution Solve for Use the lw of osines: os Solve for os (10.3) 2 (6.45) 2 2(10.3)(6.45) os m Solve for Sine side is shorter thn side, must e ute, nd the lw of sines is used to solve for. sin sin sin sin sin sin sin 32.4 sin Solve for sin. Solve for. Sine is ute, the inverse sine funtion gives us diretly Solve for 180 ( ) 180 ( ) Mthed Prolem 1 Solve the tringle with 77.5, 10.4 feet, nd 17.7 feet. Solving the SSS se Strting with three sides of tringle, the prolem is to find the three ngles. Susequent lultions re simplified if we solve for the otuse ngle first, if present. The lw of osines is used for this purpose. A seond ngle, whih must e ute, n e found using either lw, lthough omputtions re usully simpler with the lw of sines. EXPLOREDISUSS 2 (A) Strting with os, show tht os (2) (B) Does eqution (2) give us the orret ngle irrespetive of whether is ute or otuse? Explin. The ove disussion leds to the following strtegy for solving the SSS se.
5 520 7 Additionl Topis in Trigonometry Strtegy for Solving the SSS se Step Find Method 1. Angle opposite longest Lw of osines side this will tke re of n otuse ngle, if present. 2. Either of the remining ngles, Lw of sines whih will e ute (why?) 3. Third ngle Sutrt the sum of the mesures of the ngles found in steps 1 nd 2 from 180. EXAMPLE 2 Solving the SSS se Solve the tringle with 27.3 meters, 17.8 meters, nd 35.2 meters. Solution Three sides of the tringle re given nd we re to find the three ngles. This is the SSS se. Sketh the tringle (Fig. 4) nd use the lw of osines to find the lrgest ngle, then use the lw of sines to find one of the two remining ute ngles. FIGURE m 27.3 m 35.2 m Sine is the lrgest ngle, we solve for it first using the lw of osines. Solve for os os os os 1 (27.3)2 (17.8) 2 (35.2) 2 2(27.3)(17.8) Solve for os. Solve for. Solve for We now solve for either or, using the lw of sines. We hoose. sin sin
6 72 Lw of osines 521 sin sin sin sin sin Solve for sin. Solve for. is ute. Solve for ( ) 180 ( ) 29.8 Mthed Prolem 2 Solve the tringle with 1.25 yrds, 2.05 yrds, nd 1.52 yrds. EXAMPLE 3 Finding the Side of Regulr Polygon If sevensided regulr polygon is insried in irle of rdius 22.8 entimeters, find the length of one side of the polygon. Solution Sketh figure (Fig. 5) nd use the lw of osines: FIGURE 5 Atully, you only need to sketh the tringle: d d (22.8)(22.8) os d 2(22.8)2 2(22.8) 2 os entimeters Mthed Prolem 3 If n 11sided regulr polygon is insried in irle with rdius 4.63 inhes, find the length of one side of the polygon.
7 522 7 Additionl Topis in Trigonometry Answers to Mthed Prolems ft, 33.3, , 95.0, in. EXERISE 72 The leling in the figure elow is the onvention we will follow in this exerise set. Your nswers to some prolems my differ slightly from those in the ook, depending on the order in whih you solve for the sides nd ngles of given tringle. A 1. Referring to the figure ove, if 47.3, 11.7 entimeters, nd 6.04 entimeters, whih of the two ngles, or, n you sy for ertin is ute nd why? 2. Referring to the figure ove, if 93.5, 5.34 inhes, nd 8.77 inhes, whih of the two ngles, or, n you sy for ertin is ute nd why? Solve eh tringle in Prolems , 5.32 yrds, 5.03 yrds , 6.08 entimeters, 5.25 entimeters , 5.73 millimeters, 10.2 millimeters , 8.44 inhes, 20.3 inhes B 7. Referring to the figure t the eginning of the exerise, if 13.5 feet, 20.8 feet, nd 8.09 feet, then if the tringle hs n otuse ngle, whih ngle must it e nd why? 8. Suppose you re told tht tringle hs sides 12.5 entimeters, 25.3 entimeters, nd 10.7 entimeters. Explin why the tringle hs no solution. Solve eh tringle in Prolems 9 12 if the tringle hs solution. Use deiml degrees for ngle mesure meters, 10.2 meters, 9.05 meters miles, 20.7 miles, 12.2 miles kilometers, 5.30 kilometers, 5.52 kilometers meters, 29.4 meters, 33.7 meters Prolems represent vriety of prolems involving oth the lw of sines nd the lw of osines. Solve eh tringle. If prolem does not hve solution, sy so , 88.9, 15.2 entimeters , 102.3, 6.4 millimeters , 13.8 inhes, 12.5 inhes , 16.4 yrds, 28.2 yrds meters, 6.9 meters, 31.3 meters inhes, 32 inhes, 53 inhes , 11.5 inhes, 14.0 inhes , 25.5 meters, 25.5 meters meters, 42.4 meters, 20.4 meters entimeters, 5.23 entimeters, 8.66 entimeters , 7.23 meters, 6.54 meters , 18.1 meters, 22.6 meters , 12.5 inhes, 7.31 inhes , 35.2 inhes, 25.5 inhes 27. Show, using the lw of osines, tht if 90, then (the Pythgoren theorem). 28. Show, using the lw of osines, tht if 2 2 2, then Show tht for ny tringle, Show tht for ny tringle, os os os os os 31. Give solution to Exmple 3 tht does not use the lw of d 360 osines y showing tht 22.8 sin. 2 14
8 72 Lw of osines Show tht the length d of one side of n nsided regulr polygon, insried in irle of rdius r, is given y 180 d 2r sin. n 41. Anlyti Geometry. If point A in the figure hs oordintes (3, 4) nd point B hs oordintes (4, 3), find the rdin mesure of ngle to three deiml ples. y APPLIATIONS 33. Surveying. To find the length AB of smll lke, surveyor mesured ngle AB to e 96, A to e 91 yrds, nd B to e 71 yrds. Wht is the pproximte length of the lke? 0 A B x 42. Anlyti Geometry. If point A in the figure hs oordintes (4, 3) nd point B hs oordintes (5, 1), find the rdin mesure of ngle to three deiml ples. 43. Engineering. Three irles of rdius 2.03, 5.00, nd 8.20 entimeters re tngent to one nother (see figure). Find the three ngles formed y the lines joining their enters (to the nerest 10). A B 34. Surveying. Suppose the figure for this prolem represents the se of lrge rok outropping on frmer s lnd. If surveyor finds AB 110, A 85 meters, nd B 73 meters, wht is the pproximte length (to one deiml ple) of the outropping? 35. Geometry. Two djent sides of prllelogrm meet t n ngle of nd hve lengths of 3 nd 8 feet. Wht is the length of the shorter digonl of the prllelogrm (to 3 signifint digits)? 36. Geometry. Wht is the length of the longer digonl of the prllelogrm in Prolem 35 (to 3 signifint digits)? 37. Nvigtion. Los Angeles nd Ls Vegs re pproximtely 200 miles prt. A pilot 80 miles from Los Angeles finds tht she is 6 20 off ourse reltive to her strt in Los Angeles. How fr is she from Ls Vegs t this time? (ompute the nswer to 3 signifint digits.) 38. Serh nd Resue. At noon, two serh plnes set out from Sn Frniso to find downed plne in the oen. Plne A trvels due west t 400 miles per hour, nd plne B flies northwest t 500 miles per hour. At 2 P.M. plne A spots the survivors of the downed plne nd rdios plne B to ome nd ssist in the resue. How fr is plne B from plne A t this time (to 3 signifint digits)? 39. Geometry. Find the perimeter of pentgon insried in irle of rdius 12.6 meters. 40. Geometry. Find the perimeter of ninesided regulr polygon insried in irle of rdius 7.09 entimeters. 44. Engineering. Three irles of rdius 2.00, 5.00, nd 8.00 inhes re tngent to eh other (see figure). Find the three ngles formed y the lines joining their enters (to the nerest 10). 45. Geometry. A retngulr solid hs sides s indited in the figure. Find AB to the nerest degree. A 4.3 m 8.1 m 2.8 m 46. Geometry. Referring to the figure, find AB to the nerest degree. 47. Spe Siene. For ommunitions etween spe shuttle nd the White Snds trking sttion in southern New Mexio, two stellites re pled in geosttionry orit, 130 prt reltive to the enter of the Erth nd 22,300 B
9 524 7 Additionl Topis in Trigonometry miles ove the surfe of the Erth (see figure). (A stellite in geosttionry orit remins sttionry ove fixed point on the surfe of the Erth.) Rdio signls re sent from the trking sttion y wy of the stellites to the shuttle, nd vie vers. This system llows the trking sttion to e in ontt with the shuttle over most of the Erth s surfe. How fr to the nerest 100 miles is one of the geosttionry stellites from the White Snds trking sttion W? The rdius of the Erth is 3,964 miles. 48. Spe Siene. A stellite S, in irulr orit round the Erth, is sighted y trking sttion T (see figure). The distne TS is determined y rdr to e 1,034 miles, nd the ngle of elevtion ove the horizon is How high is the stellite ove the Erth t the time of the sighting? The rdius of the Erth is 3,964 miles. S T Horizon S W S R Erth SETION 73 Geometri Vetors Geometri Vetors nd Vetor Addition Veloity Vetors Fore Vetors Resolution of Vetors into Vetor omponents Mny physil quntities, suh s length, re, or volume, n e ompletely speified y single rel numer. Other quntities, suh s direted distnes, veloities, nd fores, require for their omplete speifition oth mgnitude nd diretion. The former re often lled slr quntities, nd the ltter re lled vetor quntities. In this setion we limit our disussion to the intuitive ide of geometri vetors in plne. In Setion 74 we introdue lgeri vetors, first step in the generliztion of onept tht hs frrehing onsequenes. Vetors re widely used in mny res of siene nd engineering. Geometri Vetors nd Vetor Addition v O FIGURE 1 Vetor OP, or v. P A line segment to whih diretion hs een ssigned is lled direted line segment. A geometri vetor is direted line segment nd is represented y n rrow (see Fig. 1). A vetor with n initil point O nd terminl point P (the end with the rrowhed) is denoted y OP. Vetors re lso denoted y oldfe letter, suh s v. Sine it is diffiult to write oldfe on pper, we suggest tht you use n rrow over single letter, suh s v, when you wnt the letter to denote vetor. The mgnitude of the vetor OP, denoted y OP, v or v, is the length of the direted line segment. Two vetors hve the sme diretion if they re prllel nd point in the sme diretion. Two vetors hve opposite diretion if they re prllel nd point in opposite diretions. The zero vetor, denoted y 0 or 0, hs mgni
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