# Essential Question What are the Law of Sines and the Law of Cosines?

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1 9.7 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Disovering the Lw of Sines Work with prtner.. opy nd omplete the tle for the tringle shown. Wht n you onlude? Smple Segments = 3.16 = 6.3 = 5.10 ngles m = 9.74 m = m = SELETING TOOLS To e profiient in mth, you need to use tehnology to ompre preditions with dt. m sin m sin m sin. Use dynmi geometry softwre to drw two other tringles. opy nd omplete the tle in prt () for eh tringle. Use your results to write onjeture out the reltionship etween the sines of the ngles nd the lengths of the sides of tringle. Disovering the Lw of osines Work with prtner.. opy nd omplete the tle for the tringle in Explortion 1(). Wht n you onlude? m + os. Use dynmi geometry softwre to drw two other tringles. opy nd omplete the tle in prt () for eh tringle. Use your results to write onjeture out wht you oserve in the ompleted tles. ommunite Your nswer 3. Wht re the Lw of Sines nd the Lw of osines? 4. When would you use the Lw of Sines to solve tringle? When would you use the Lw of osines to solve tringle? Setion 9.7 Lw of Sines nd Lw of osines 511

2 9.7 Lesson Wht You Will Lern ore Voulry Lw of Sines, p. 513 Lw of osines, p. 515 Find res of tringles. Use the Lw of Sines to solve tringles. Use the Lw of osines to solve tringles. Finding res of Tringles So fr, you hve used trigonometri rtios to solve right tringles. In this lesson, you will lern how to solve ny tringle. When the tringle is otuse, you my need to find trigonometri rtio for n otuse ngle. Finding Trigonometri Rtios for Otuse ngles Use lultor to find eh trigonometri rtio. Round your nswer to four deiml ples.. tn 150. sin 10. os 95. tn sin os Help in English nd Spnish t igidesmth.om Use lultor to find the trigonometri rtio. Round your nswer to four deiml ples. 1. tn 110. sin os 165 ore onept re of Tringle The re of ny tringle is given y one-hlf the produt of the lengths of two sides times the sine of their inluded ngle. For shown, there re three wys to lulte the re. re = 1 sin re = 1 sin re = 1 sin Finding the re of Tringle Find the re of the tringle. Round your nswer to the nerest tenth. re = 1 sin = 1 (17)(19) sin The re of the tringle is out 114. squre units. Help in English nd Spnish t igidesmth.om Find the re of with the given side lengths nd inluded ngle. Round your nswer to the nerest tenth. 4. m = 60, = 19, = m = 9, = 38, = hpter 9 Right Tringles nd Trigonometry

3 Using the Lw of Sines The trigonometri rtios in the previous setions n only e used to solve right tringles. You will lern two lws tht n e used to solve ny tringle. You n use the Lw of Sines to solve tringles when two ngles nd the length of ny side re known (S or S ses), or when the lengths of two sides nd n ngle opposite one of the two sides re known (SS se). Theorem Theorem 9.9 Lw of Sines The Lw of Sines n e written in either of the following forms for with sides of length,, nd. sin = sin = sin Proof Ex. 51, p. 50 sin = sin = sin Using the Lw of Sines (SS se) Solve the tringle. Round deiml nswers to the nerest tenth. Use the Lw of Sines to find m. sin = sin sin 11 = sin sin 115 sin = 0 m 9.9 Lw of Sines Sustitute. Multiply eh side y 11. Use lultor. y the Tringle Sum Theorem (Theorem 5.1), m = Use the Lw of Sines gin to find the remining side length of the tringle. sin = sin sin 35.1 = 0 sin sin 35.1 = sin Lw of Sines Sustitute. Multiply eh side y sin Use lultor. In, m 9.9, m 35.1, nd 1.7. Help in English nd Spnish t igidesmth.om Solve the tringle. Round deiml nswers to the nerest tenth Setion 9.7 Lw of Sines nd Lw of osines 513

4 Using the Lw of Sines (S se) Solve the tringle. Round deiml nswers to the nerest tenth y the Tringle Sum Theorem (Theorem 5.1), m = = 48. y the Lw of Sines, you n write sin 48 = 15 sin 5 = sin 107. sin 48 = 15 Write two equtions, eh sin 5 with one vrile. sin 107 = 15 sin 5 15 sin sin 107 = Solve for eh vrile. = sin 5 sin Use lultor In, m = 48, 6.4, nd surveyor mkes the mesurements shown to determine the length of ridge to e uilt ross smll lke from the North Pini re to the South Pini re. Find the length of the ridge. Using the Lw of Sines (S se) South Pini re In the digrm, represents the distne from the North Pini re to the South Pini re, so represents the length of the ridge. y the Tringle Sum Theorem (Theorem 5.1), m = = 49. y the Lw of Sines, you n write sin 71 = 150 sin 49 = sin 60. sin 60 = 150 Write n eqution involving. sin sin 60 = sin Multiply eh side y sin 60. Use lultor. The length of the ridge will e out 17.1 meters. Help in English nd Spnish t igidesmth.om Solve the tringle. Round deiml nswers to the nerest tenth North Pini re WHT IF? In Exmple 5, wht would e the length of ridge from the South Pini re to the Est Pini re? m 60 Est Pini re 514 hpter 9 Right Tringles nd Trigonometry

5 Using the Lw of osines You n use the Lw of osines to solve tringles when two sides nd the inluded ngle re known (SS se), or when ll three sides re known (SSS se). Theorem Theorem 9.10 Lw of osines If hs sides of length,, nd, s shown, then the following re true. = + os = + os = + os Proof Ex. 5, p. 50 Using the Lw of osines (SS se) NOTHER WY When you know ll three sides nd one ngle, you n use the Lw of osines or the Lw of Sines to find the mesure of seond ngle. OMMON ERROR In Exmple 6, the smller remining ngle is found first euse the inverse sine feture of lultor only gives ngle mesures from 0 to 90. So, when n ngle is otuse, like euse 14 > (7.85) + 11, you will not get the otuse mesure. Solve the tringle. Round deiml 14 nswers to the nerest tenth Use the Lw of osines to find side length. = + os Lw of osines = (11)(14) os 34 Sustitute. = os 34 Simplify. = os 34 Find the positive squre root. 7.9 Use lultor. Use the Lw of Sines to find m. sin = sin Lw of Sines sin 11 = sin os 34 Sustitute. 11 sin 34 sin = os 34 Multiply eh side y 11. m 51.6 Use lultor. y the Tringle Sum Theorem (Theorem 5.1), m = In, 7.9, m 51.6, nd m Help in English nd Spnish t igidesmth.om Solve the tringle. Round deiml nswers to the nerest tenth Setion 9.7 Lw of Sines nd Lw of osines 515

6 Using the Lw of osines (SSS se) OMMON ERROR In Exmple 7, the lrgest ngle is found first to mke sure tht the other two ngles re ute. This wy, when you use the Lw of Sines to find nother ngle mesure, you will know tht it is etween 0 nd 90. Solve the tringle. Round deiml nswers to the nerest tenth. First, find the ngle opposite the longest side,. Use the Lw of osines to find m. = + os Lw of osines 7 = (1)(0) os Sustitute = os (1)(0) Solve for os. m 11.7 Use lultor. Now, use the Lw of Sines to find m. sin sin 1 = sin = sin sin 11.7 sin = 7 m 4. Lw of Sines Sustitute for,, nd. Multiply eh side y 1. Use lultor. y the Tringle Sum Theorem (Theorem 5.1), m = In, m 4., m 11.7, nd m Solving Rel-Life Prolem n orgnism s step ngle is mesure of wlking effiieny. The loser the step ngle is to 180, the more effiiently the orgnism wlked. The digrm shows set of footprints for dinosur. Find the step ngle. 155 m 197 m 316 m = + os Lw of osines 316 = (155)(197) os Sustitute = os (155)(197) Solve for os m Use lultor. The step ngle is out Help in English nd Spnish t igidesmth.om Solve the tringle. Round deiml nswers to the nerest tenth hpter 9 Right Tringles nd Trigonometry

7 9.7 Exerises Dynmi Solutions ville t igidesmth.om Voulry nd ore onept hek 1. WRITING Wht type of tringle would you use the Lw of Sines or the Lw of osines to solve?. VOULRY Wht informtion do you need to use the Lw of Sines? nd Modeling with Mthemtis In Exerises 3 8, use lultor to find the trigonometri rtio. Round your nswer to four deiml ples. (See Exmple 1.) 3. sin sin os os tn tn 116 In Exerises 9 1, find the re of the tringle. Round your nswer to the nerest tenth. (See Exmple.) In Exerises 13 18, solve the tringle. Round deiml nswers to the nerest tenth. (See Exmples 3, 4, nd 5.) In Exerises 19 4, solve the tringle. Round deiml nswers to the nerest tenth. (See Exmples 6 nd 7.) ERROR NLYSIS Desrie nd orret the error in finding m sin sin 55 = sin 55 sin = 5 m 79.4 Setion 9.7 Lw of Sines nd Lw of osines 517

8 6. ERROR NLYSIS Desrie nd orret the error in finding m in when = 19, = 1, nd = 11. os = (19)(1) m 75.4 OMPRING METHODS In Exerises 7 3, tell whether you would use the Lw of Sines, the Lw of osines, or the Pythgoren Theorem (Theorem 9.1) nd trigonometri rtios to solve the tringle with the given informtion. Explin your resoning. Then solve the tringle. 35. MODELING WITH MTHEMTIS You re on the oservtion dek of the Empire Stte uilding looking t the hrysler uilding. When you turn 145 lokwise, you see the Sttue of Lierty. You know tht the hrysler uilding nd the Empire Stte uilding re out 0.6 mile prt nd tht the hrysler uilding nd the Sttue of Lierty re out 5.6 miles prt. Estimte the distne etween the Empire Stte uilding nd the Sttue of Lierty. 36. MODELING WITH MTHEMTIS The Lening Tower of Pis in Itly hs height of 183 feet nd is 4 off vertil. Find the horizontl distne d tht the top of the tower is off vertil. d 7. m = 7, m = 44, =14 8. m = 98, m = 37, = m = 65, = 1, = ft 30. m = 90, = 15, = m = 40, = 7, = = 34, = 19, = MODELING WITH MTHEMTIS You nd your friend re stnding on the seline of sketll ourt. You oune sketll to your friend, s shown in the digrm. Wht is the distne etween you nd your friend? (See Exmple 8.) 37. MKING N RGUMENT Your friend sys tht the Lw of Sines n e used to find JK. Your ousin sys tht the Lw of osines n e used to find JK. Who is orret? Explin your resoning. K 17 7 ft 6 ft 110 J 0 48 L 34. MODELING WITH MTHEMTIS zip line is onstruted ross vlley, s shown in the digrm. Wht is the width w of the vlley? 38. RESONING Use XYZ. Z 17 5 ft 10 w 84 ft X 64. n you use the Lw of Sines to solve XYZ? Explin your resoning. Y. n you use nother method to solve XYZ? Explin your resoning. 518 hpter 9 Right Tringles nd Trigonometry

9 39. MKING N RGUMENT Your friend lultes the re of the tringle using the formul = 1 qr sin S nd sys tht the re is pproximtely 08.6 squre units. Is your friend orret? Explin your resoning. Q R 43. MTHEMTIL ONNETIONS Find the vlues of x nd y. E 3 5y 18 3y S 5 D 49 x F 40. MODELING WITH MTHEMTIS You re fertilizing tringulr grden. One side of the grden is 6 feet long, nd nother side is 54 feet long. The ngle opposite the 6-foot side is 58.. Drw digrm to represent this sitution.. Use the Lw of Sines to solve the tringle from prt ().. One g of fertilizer overs n re of 00 squre feet. How mny gs of fertilizer will you need to over the entire grden? 41. MODELING WITH MTHEMTIS golfer hits drive 60 yrds on hole tht is 400 yrds long. The shot is 15 off trget. 15 Not drwn to sle 60 yd 400 yd u x x 10 yd. Wht is the distne x from the golfer s ll to the hole?. ssume the golfer is le to hit the ll preisely the distne found in prt (). Wht is the mximum ngle θ (thet) y whih the ll n e off trget in order to lnd no more thn 10 yrds from the hole? 4. OMPRING METHODS uilding is onstruted on top of liff tht is 300 meters high. person stnding on level ground elow the liff oserves tht the ngle of elevtion to the top of the uilding is 7 nd the ngle of elevtion to the top of the liff is 63.. How fr wy is the person from the se of the liff?. Desrie two different methods you n use to find the height of the uilding. Use one of these methods to find the uilding s height. 44. HOW DO YOU SEE IT? Would you use the Lw of Sines or the Lw of osines to solve the tringle? in in REWRITING FORMUL Simplify the Lw of osines for when the given ngle is right ngle. 46. THOUGHT PROVOKING onsider ny tringle with side lengths of,, nd. lulte the vlue of s, whih is hlf the perimeter of the tringle. Wht mesurement of the tringle is represented y s(s )(s )(s )? 47. NLYZING RELTIONSHIPS The miguous se of the Lw of Sines ours when you re given the mesure of one ute ngle, the length of one djent side, nd the length of the side opposite tht ngle, whih is less thn the length of the djent side. This results in two possile tringles. Using the given informtion, find two possile solutions for. Drw digrm for eh tringle. (Hint: The inverse sine funtion gives only ute ngle mesures, so onsider the ute ngle nd its supplement for.). m = 40, = 13, = 16. m = 1, = 17, = STRT RESONING Use the Lw of osines to show tht the mesure of eh ngle of n equilterl tringle is 60. Explin your resoning. 49. RITIL THINKING n irplne flies 55 est of north from ity to ity, distne of 470 miles. nother irplne flies 7 north of est from ity to ity, distne of 890 miles. Wht is the distne etween ities nd? Setion 9.7 Lw of Sines nd Lw of osines 519

10 50. REWRITING FORMUL Follow the steps to derive the formul for the re of tringle, re = 1 sin.. Drw the ltitude from vertex to. Lel the ltitude s h. Write formul for the re of the tringle using h.. Write n eqution for sin.. Use the results of prts () nd () to write formul for the re of tringle tht does not inlude h. 51. PROVING THEOREM Follow the steps to use the formul for the re of tringle to prove the Lw of Sines (Theorem 9.9).. Use the derivtion in Exerise 50 to explin how to derive the three relted formuls for the re of tringle. re = 1 sin, re = 1 sin, re = 1 sin. Why n you use the formuls in prt () to write the following sttement? 1 sin = 1 sin = 1 sin. Show how to rewrite the sttement in prt () to prove the Lw of Sines. Justify eh step. 5. PROVING THEOREM Use the given informtion to omplete the two-olumn proof of the Lw of osines (Theorem 9.10). Given D is n ltitude of. Prove = + os h STTEMENTS 1. D is n ltitude of. RESONS 1. Given x D x. D nd D re right tringles.. 3. = ( x) + h Expnd inomil. 5. x + h = Sustitution Property of Equlity 7. os = x x = os = + os 9. Mintining Mthemtil Profiieny Find the rdius nd dimeter of the irle. (Skills Review Hndook) Reviewing wht you lerned in previous grdes nd lessons ft ft 50 in. 10 in. 50 hpter 9 Right Tringles nd Trigonometry

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