MATHEMATICAL PRACTICES In the Solve It, you used what you know about triangles to find missing lengths. Key Concept Law of Sines

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1 Lw of Sines ontent Stndrds G.SRT.11 Understnd nd ppl the Lw of Sines... to find unknown mesurements in right nd nonright tringles... lso G.SRT.10 Ojetives To ppl the Lw of Sines 66 ft ft 68 MTHEMTIL PRTIES In the Solve It, ou used wht ou know out tringles to find missing lengths. Lesson L VoulrV Lw of Sines Essentil Understnding If ou know the mesures of two ngles nd the length of side (S or S), or two side lengths nd the mesure of noninluded otuse ngle (SS), then ou n find ll the other mesures of the tringle. Ke onept Lw of Sines For n, let the lengths of the sides opposite ngles,, nd e,, nd, respetivel. Then the Lw of Sines reltes the sine of eh ngle to the length of the opposite side. sin sin sin Here s Wh It Works Drw the ltitude from to nd lel it h. D nd D re right tringles. sin h nd sin h Definition of sine sin h nd sin h Multiplition Propert of Equlit h D sin sin Trnsitive Propert of Equlit sin sin Division Propert of Equlit 1 ommon ore
2 Prolem 1 Using the Lw of Sines (S) In, m 48, m 93, nd 15. To the nerest tenth, wht is the length of? How will drwing digrm help ou solve the prolem? Drwing digrm will help ou to visulize the prolem. refull drw digrm nd lel it with ll of the given informtion. Drw nd lel. You re given two ngle mesures nd the length of noninluded side (S). Use the Lw of Sines to write n eqution Solve for. Use lultor to find. The length of is out In ove, wht is to the nerest tenth? You n lso use the Lw of Sines to find missing ngle mesures. Prolem 2 Using the Lw of Sines (SS) In RST, RT 11, ST, nd mr 120. To the nerest tenth, wht is ms? Wht unknown ngle should ou use? Use S euse it is opposite known side length. Step 1 Drw nd lel digrm. Step 2 Use the Lw of Sines to set up n eqution. sin 120 Step 3 Find ms. sin S 11 S R T sin S 11 sin 120 Solve for sin S. ms sin 1 11 sin Use the inverse. ms is out In KLM, LM 9, KM 14, nd ml 105. To the nerest tenth, wht is mk? 5 Lw of Sines 2
3 You n ppl the Lw of Sines to relworld prolems involving tringles. Wht other informtion do ou need to know to use the Lw of Sines? You n use the Lw of Sines to find the distnes from the ship to eh lighthouse if ou n find the ngle with the ship s its verte. Prolem 3 Using the Lw of Sines to Solve Prolem ship hs een t se longer thn epeted nd hs onl enough fuel to sfel sil nother 42 miles. Port it Lighthouse nd ove Town Lighthouse re loted 40 miles prt long the ost. t se, the ptin nnot determine distnes oservtion. The tringle formed the lighthouses nd the ship is shown. n the ship sil sfel to either lighthouse? Step 1 Step 2 Find the mesure of the ngle formed Port it, the ruise ship, nd ove Town. The sum of the mesures of the ngles of tringle is 0. Sutrt the given ngle mesures from Use the Lw of Sines to find the distnes from the ship to eh lighthouse. Port it ove Town sin 58 sin 52 sin 58 sin 70 Lw of Sines 40 d 40 d d sin sin 52 Find the ross produts. d sin sin sin 52 d sin 58 Divide eh side sin sin 70 d sin 58 d Use lultor. d The distne from the ship to Port it Lighthouse is out 37.2 miles, nd the distne to ove Town Lighthouse is out 44.3 miles. The ship n sil sfel to Port it Lighthouse ut not to ove Town Lighthouse. 3. The rightfielder fields softll etween first se nd seond se s shown in the figure. If the rightfielder throws the ll to seond se, how fr does she throw the ll? Port it Lighthouse 2nd se 40 mi ove Town Lighthouse ft Rightfielder 40 1st se ship Lesson hek Do ou know HOW? 1. In, 7, 10, nd m 80. To the nerest tenth, wht is m? 2. Wht is? 3. Wht is? G H K Do ou UNDERSTND? MTHEMTIL PRTIES 4. Resoning If ou know the three side lengths of tringle, n ou use the Lw of Sines to find the missing ngle mesures? Eplin. 5. Error nlsis In PQR, PQ 4 m, QR 3 m, nd mr 75. Your friend uses the Lw of Sines to sin 75 write 3 sin P 4 to find mp. Eplin the error. 3 ommon ore
4 Prtie nd ProlemSolving Eerises MTHEMTIL PRTIES Prtie Use the informtion given to solve. See Prolems 1 nd In, m 70, m 62, nd 7.3. To the nerest tenth, wht is? 7. In XYZ, my 80, XY 14, nd XZ 17. To the nerest tenth, wht is mz? Use the Lw of Sines to find the vlues of nd. Round to the nerest tenth. ppl The min sil of silot hs the dimensions shown in the figure t the right. To the nerest tenth of foot, wht is the height of the min sil? 13. portion of it mp is shown in the figure t the right. If ou wlk long Mple Street etween 2nd Street nd Elm Grove Lne, how fr do ou wlk? Round our nswer to the nerest tenth of rd. 14. Nvigtion The ermud Tringle is historill fmous region of the tlnti Oen. The verties of the tringle re formed Mimi, FL; ermud; nd Sn Jun, Puerto Rio. The pproimte dimensions of the ermud Tringle re shown in the figure t the right. Eplin how ou would find the distne from ermud to Mimi. Wht is this distne to the nerest mile? 15. Think out Pln n irplne took off from n irport nd strted fling towrd its destintion 210 miles due est. fter fling 80 miles est, it enountered storm nd ltered its ourse turning left 22. When it ws pst the storm, it turned right 30 nd flew in stright line until it rehed its destintion. How fr ws the plne from its destintion when it mde the 30 turn? Would drwing digrm help ou visulize the prolem sitution? Wht re ou eing sked to find? Wht mesures do ou know? h ft d Elm Grove Lne Mimi nd St See Prolem Sn Jun Mple St 62 ermud 960 mi 5 Lw of Sines 4
5 16. Zipline zipline is onstruted over rvine s shown in the digrm t the right. Wht is the horizontl distne from the ottom of the ldder to the pltform where the zipline ends? Round our nswer to the nerest tenth of foot. 17. If mdeg md mg 43, wht is mefg? E 45 ft pltform 250 ft 60 ft 50 ldder ground 8 in. 8 in. 7.5 in. D F G hllenge. You n use the formul re 1 2 sin to find the re of the tringle shown t the right. Show how this formul eomes the more fmilir formul for the re of tringle if is right tringle nd m 90. Stndrdized Test Prep Refer to the tringle t the right to solve Eerises 19 nd 20. ST/T 19. Find the vlue of to the nerest hundredth of n inh. 20. Wht is the vlue of to the nerest hundredth of n inh? 5 in. 21. In right TRS, mr 90, TR 3, RS 4, nd TS 5. Wht is sin S? Mied Review 22. Indiret Mesurement hotir lloon pilot sights the lnding field from height of 2000 ft. The ngle of depression is 24. To the nerest foot, wht is the ground distne from the hotir lloon to the lnding field? 23. Wht is the vlue of? See Lesson 84. See Lesson Get Red! To Prepre for Lesson 86, do Eerises Evlute eh epression. Round our nswers to the nerest hundredth. 24. os os os os See Lesson ommon ore
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