Recall that the area of a triangle can be found using the sine of one of the angles.

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1 Nme lss Dte 14.1 Lw of Sines Essentil Question: How n you use trigonometri rtios to find side lengts nd ngle mesures of non-rigt tringles? Resoure Loker Explore Use n re Formul to Derive te Lw of Sines Rell tt te re of tringle n e found using te sine of one of te ngles. re = 1_ 2 sin() You n write vritions of tis formul using different ngles nd sides from te sme tringle. Rewrite te re formul using side lengt s te se of te tringle nd. Rewrite te re formul using side lengt s te se of te tringle nd. Hougton Mifflin Hrourt Pulising ompny D Wt do ll tree formuls ve in ommon? Wy is tis sttement true? 1_ 2 sin()= 1_ 2 sin() = 1_ 2 sin() E Multiply e re y te expression 2. Write n equivlent sttement. Module Lesson 1

2 Reflet 1. In te se of rigt tringle, were is te rigt ngle, wt ppens to te re formul? 2. Disussion In ll tree ses of te re formul you explored, wt is te reltionsip etween te ngle nd te two side lengts in te re formul? Explin 1 pplying te Lw of Sines Te results of te Explore tivity re summrized in te Lw of Sines. Lw of Sines Given: _ sin () sin () = _ = _ sin () Te Lw of Sines llows you to find te unknown mesures for given tringle, s long s you know eiter of te following: 1. Two ngle mesures nd ny side lengt ngle-ngle-side (S) or ngle-side-ngle (S) informtion 2. Two side lengts nd te mesure of n ngle tt is not etween tem side-side-ngle (SS) informtion Exmple 1 Find ll te unknown mesures using te given tringle. Round to te nerest tent. Step 1 Find te tird ngle mesure. m R + m S + m T = 180 Tringle Sum Teorem m T = 180 Sustitute te known ngle mesures. m T = 107 Solve for te mesure of T. R t Step 2 Find te unknown side lengts. Set up proportions using te Lw of Sines nd solve for te unknown. _ sin (T) = _ sin (R) t r _ sin (107 ) = _ sin (35 ) t sin (107 ) t = sin (35 ) Lw of Sines _ sin (S) sin (R) s = _ r Sustitute. _ sin (38 ) sin (35 ) s = _ Solve for te unknown. (sin38 ) s = sin (35 ) t 25 Evlute. s 16.1 s T S Hougton Mifflin Hrourt Pulising ompny Module Lesson 1

3 Step 1 Find te tird ngle mesure. m R + m S + m T = 180 Tringle Sum Teorem + + m T = 180 Sustitute te known ngle mesures. m T = Solve for te mesure of T. R t s 18 T S Step 2 Find te unknown side lengts. Set up proportions using te Lw of Sines nd solve for te unknown. _ sin(t) = _ sin(r) t r sin ( ) sin( ) = _ t Lw of Sines Sustitute. _ sin(s) s = _ sin(r) r sin ( ) _ sin( ) = _ s t = sin( ) sin ( ) t Solve for te unknown. Evlute. sin( ) s = sin( ) s Reflet 3. Suppose tt you re given m. To find, wt oter mesures do you need to know? Hougton Mifflin Hrourt Pulising ompny Module Lesson 1

4 Your Turn Find ll te unknown mesures using te given tringle. Round to te nerest tent Explin 2 Evluting Tringles Wen SS is Known Informtion Wen you use te Lw of Sines to solve tringle for wi you know side-side-ngle (SS) informtion, zero, one, or two tringles my e possile. For tis reson, SS is lled te miguous se. miguous se Given,, nd m. is ute. is rigt or otuse. < No tringle = One tringle = No tringle Hougton Mifflin Hrourt Pulising ompny < < Two tringles One tringle > One tringle Module Lesson 1

5 Exmple 2 Find m. Design E tringulr wing of model irplne s one side tt joins te wing to te irplne. Te oter sides ve lengts = 18 in. nd = in. Te side wit lengt meets te irplne t n ngle wit mesure of 30, nd meets te side wit lengt t point. Find e mesure. Step 1 Determine te numer of possile tringles. Find. sin (30 ) = _, so = 18 sin (30 ) = 9 18 euse < <, two tringles re possile. 18 in in. in in in. Step 2 Determine m 1 nd m 2. _ sin () sin () = sin (30 ) = sin () = _ sin () sin (30 ) Lw of Sines Sustitute. Solve for sin (). Let 1 e te ute ngle wit te given sine, nd let 2 e te otuse ngle. Use te inverse sine funtion on your lultor to determine te mesures of te ngles. -1 m 1 = sin 18 sin (30 ) ( ) 36.9 nd m 2 = = Determine m nd lengt. Solve for m 1. Solve for m 2. Hougton Mifflin Hrourt Pulising ompny sin () _ sin ( ) m 1 = m 2 = 180 sin ( 1 ) = _ 1 _ = 1 = m 1 = m 2 = sin ( ) 1 sin ( ) sin ( ) Lw of Sines Sustitute. Solve for te unknown. sin () _ sin ( ) _ = 1 in. Evlute. 2 in. 2 = sin ( 2 ) = _ 2 sin ( ) _ 2 sin ( ) sin( ) Module Lesson 1

6 Your Turn In Exerises 6 nd 7, suppose tt for te model irplne in te Exmple, = 21 in., = 18 in., nd m = in How mny tringles re possile wit tis onfigurtion? Explin. 7. Find te unknown mesurements. Round to te nerest tent. Elorte 8. If te se ngles of tringle re ongruent, use te Lw of Sines to sow te tringle is isoseles. 9. Sow tt wen =, is rigt ngle. Hougton Mifflin Hrourt Pulising ompny Module Lesson 1

7 10. Essentil Question ek-in Given te mesures of, desrie metod for finding ny of te ltitudes of te tringle. Evlute: Homework nd Prtie Find te re of e tringle. Round to te nerest tent Online Homework Hints nd Help Extr Prtie Hougton Mifflin Hrourt Pulising ompny 7. Wt is te re of n isoseles tringle wit ongruent side lengts x nd inluded ngle θ? 8. Wt is te re of n equilterl tringle of side lengt x? Module Lesson 1

8 Find ll te unknown mesurements using te Lw of Sines Hougton Mifflin Hrourt Pulising ompny Module Lesson 1

9 Design model irplne designer wnts to design wings of te given dimensions. Determine te numer of different tringles tt n e formed. Ten find ll te unknown mesurements. Round vlues to te nerest tent.. = 7 m, = 9 m, m = = 12 m, = 4 m, m = = 9 m, = 10 m, m = = 7 m, = 5 m, m = 45 Hougton Mifflin Hrourt Pulising ompny Module Lesson 1

10 19. Spe Trvel Two rdio towers tt re 50 miles prt trk stellite in orit. Te first tower s signl mkes 76 ngle etween te ground nd stellite. Te seond tower forms n 80.5 ngle.. How fr is te stellite from e tower? Stellite Tower Tower. How ould you determine ow fr ove Ert te stellite is? Wt is te stellite s ltitude? 20. iology Te dorsl fin of srk forms n otuse tringle wit tese mesurements. Find te missing mesurements nd determine if noter tringle n e formed m Nvigtion s sip pproes te dok, it forms 70 ngle etween te dok nd ligtouse. t te ligtouse, n 80 ngle is formed etween te dok nd te sip. How fr is te sip from te dok? dok 70 sip Hougton Mifflin Hrourt Pulising ompny Imge redits: Mtt9122/ Sutterstok 5 mi 80 ligtouse Module Lesson 1

11 22. For te given tringle, mt e ltitude wit its equivlent expression.. 1 sin() 2. 2 sin(). 3 sin() 1 3 H.O.T. Fous on Higer Order Tinking Use te digrm, in wi ~ DEF. 23. To find te missing mesurements for eiter tringle using te Lw of Sines, wt must you do first? 37º E f 6 65º D F 9 e 24. Find te missing mesurements for. 25. Find te missing mesurements for DEF. Hougton Mifflin Hrourt Pulising ompny Imge redits: Kokki Ng/Flikr/Getty Imges 26. Surveying Two surveyors re t te sme ltitude nd re 10 miles prt on opposite sides of mountin. Tey e mesure te ngle reltive to te ground nd te top of te mountin. Use te given digrm to indiretly mesure te eigt of te mountin. 30º 53º Module Lesson 1

12 Lesson Performne Tsk In te middle of town, Stte nd Elm streets meet t n ngle of 40. tringulr poket prk etween te streets stretes 100 yrds long Stte Street nd 53.2 yrds long Elm Street. West ve Elm Street yd Stte Street 100 yd. Find te re of te poket prk using te given dimensions.. If te totl distne round te poket prk is yrds, find S, te ngle tt West venue mkes wit Stte Street, to te nerest degree.. Suppose West venue mkes ngles of 55 wit Stte Street nd 80 wit Elm Street. Te distne from Stte to Elm long West venue is 40 yrds. Find te distne from West venue to Elm Street long Stte Street. Hougton Mifflin Hrourt Pulising ompny Imge redits: Simon Turner/lmy Module Lesson 1

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