# OVERVIEW Prove & Use the Laws of Sines & Cosines G.SRT.10-HONORS

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1 OVERVIEW Prove & Use te Lws of Sines & osines G.SRT.10-HONORS G.SRT.10 (HONORS ONLY) Prove te Lws of Sines nd osines nd use tem to solve prolems. No interprettion needed - prove te Lw of Sines nd te Lw of osines nd ten solve prolems using tem. Of ourse idden inside tis very sort ojetive is te miguous se for te Lw of Sines. Tt n e diffiult onept for students to see nd understnd. Tis is out extending rigt tringle trigonometry to olique tringles. (1) Te student will e le to solve tringle using te Lws of Sines nd/or te Lw of osines. (2) Te student will e le to explin te ses of S 1 S 2. (3) Te student will e le to determine wi reltionsip (Lw of Sines/Lw of osines) is required for solving te tringle. Tere is lot inside of tis ojetive. Te Lw of Sines requires n understnding of ongruene riteri, otuse ngle trigonometry, nd different ses. Tese items trditionlly re ll diffiult for students. 1 Te miguous se Tis is so mu esier to understnd if te S 1 S 2 se ws disussed during te investigtion of ongruene in tringles. If students lredy know tt in prtiulr se more tn one nswer is possile, te onept is ndled MUH esier. 2 Provide Plenty of Time Te different ses e ve some ig onepts nd issues nd students need to e provided enoug time to lern tem. gsrt10h ProvendUseLwofSinesosines Pge 1 of 7 4/29/2014

2 NOTES Prove & Use te Lws of Sines & osines G.SRT.10-HONORS ONEPT 1 Prove te Lws of Sines. s mentioned in te previous ojetive, te previous fous s een rigt tringles, ut now re strting to expnd te use of trigonometry to ll tringles, even olique tringles. In tis ojetive sow tt if ny tree of te six mesures of tringle re given (provided t lest one mesure is side), ten te oter tree mesures n e found. One ting to onsider wen doing tis is if te tree piees of informtion fore ongruene or not. So for exmple, if given SS, S, S/SS, SSS, nd some ses of S 1 S 2 te informtion is enoug to gurntee ongruene in te tringle. Te first new reltionsip in tis ojetive tt sould e derived is lled te Lw of Sines. Follow similr logi s tt used to derive te new re formul. sin (sin )( ) sin (sin )( ) Te two eigts re equl so set tese vlues equl to e oter. (sin )( ) (sin )( ) sin sin In similr wy, y onstruting te perpendiulrs from oter verties, it n e sown tt: sin sin sin sin Tus te Lw of Sines is: sin sin sin gsrt10h ProvendUseLwofSinesosines Pge 2 of 7 4/29/2014

3 NOTES Prove & Use te Lws of Sines & osines G.SRT.10-HONORS ONEPT 2 Prove te Lws of osines. In te previous onept two ses did not work for te Lw of Sines. Review te following rt S S/S S 1 S 2 SSS SS If given 2 s te 3 rd n e found. Lw of Sines will work (rtio present). If given 2 s te 3 rd n e found. Lw of Sines will work (rtio present). S will lwys e opposite te. Lw of Sines will work (rtio present). No info. Lw of Sines will NOT work (no rtio). No or side opposite e oter. Lw of Sines will NOT work (no rtio). It is tose lst two reltionsips, SSS nd SS tt must e ddressed using different lw, te Lw of osines. x - x x - x x x x os x (os ) ( x) 2x x 2 Sustitute x 2 from te first reltionsip into te seond. 2x x 2 2x 2 2 2x Finlly, eliminte te x y mking one more sustitution. 2x 2 (os ) 2 ( s o Notie now fter te seond sustitution only sides nd ngles of te originl tringle re in te newly found reltionsip. Susequent ses would result in te sme pttern. Tus te Lw of osines is: 2 (os ) ) 2 (os ) 2 (os ) Notie te Pytgoren Teorem in te Lw of osines? gsrt10h ProvendUseLwofSinesosines Pge 3 of 7 4/29/2014

4 SSESSMENT Prove & Use te Lws of Sines & osines G.SRT.10-HONORS 1. Wen m = 39, = 25 m, = 24 m re used to form, te following will result in: ) no tringle ) 1 tringle ) 2 tringles D) depends on digrm 2. Wen m = 39, = 25 m, = 24 m re used to form, te following will result in: ) no tringle ) 1 tringle ) 2 tringles D) depends on digrm 3. Wi group of tree piees of informtion out tringle ould NOT e used in te Lw of Sines? ) S ) S ) SS D) SSS 4. Wi of te following is te Lw of osines? ) os 2 ( ) ) os 2 ( ) ) ( os) D) os 2 ( ) 5. Wi of te following tree piees of informtion work wit te Lw of Sines? ) Yes or No ) Yes or No ) Yes or No m = 33 = 17 m = 24 m m = 53 = 11 m = 7 m m = 24 m = 65 m = 91 d) Yes or No e) Yes or No f) Yes or No m = 75 m = 38 = 9 m m = 66 = 15 m = 15 m = 13 m = 14 m = 24 m 6. If given te following informtion out, m = 34, m = 100 nd = 15 m, n te Lw of Sines e used? Explin. 7. Solve for ll te sides nd ngles of using te Lw of Sines. (Round nswers to te undredts) ) ) m = 51 m = m m = = 12 m 21 m m = 54 m = 83 m = = 21 m gsrt10h ProvendUseLwofSinesosines Pge 4 of 7 4/29/2014

5 SSESSMENT Prove & Use te Lws of Sines & osines G.SRT.10-HONORS 8. On rfting trip, te rft it rok (point ) nd got stuk tere. Te two rfters lvin nd lvin were trown from te rft nd ended up getting out of te river furter downstrem t points (lvin) & (lvin). If te river is 147 m wide nd lvin sees te rft t 73 wile lvin sees it t 68. Wo is loser to te rft, nd ow mu loser re tey? (Round to te nerest meter) m Solve te tringle for ll ngles nd sides. m = 51, = 21 m, = 25 m (Round nswers to te undredts ple) Drw Digrm m = 51 m m = 21 m m = 25 m (If needed) m = 51 m m = 21 m m = 25 m gsrt10h ProvendUseLwofSinesosines Pge 5 of 7 4/29/2014

6 SSESSMENT Prove & Use te Lws of Sines & osines G.SRT.10-HONORS nswers: 1) 2) 3) D 4) D 5) ) No ) Yes ) No d) Yes e) Yes f) No 6) Yes it n given two ngles te 3 rd ngle is found using m - m = m. One is found ten te piring of nd ours. Tus te Lw of Sines n e used. 7) sin 51 sin (sin104) 12 sin sin 51 (sin104) 9.61 m sin104 sin (sin104) 12 sin sin 25 (sin104) 5.23 m sin 43 sin83 21 (sin 43) 21 sin83 21 sin83 (sin 43) m sin 43 sin (sin 43) 21 sin sin 54 (sin 43) m 8) = 25 = m sin 39 sin (sin 39) 147 sin sin 73 (sin 39) m = 39 = m sin 39 sin (sin 39) 147 sin sin 68 (sin 39) m = 43 = m lvin is m wy. lvin is m wy. lvin is loser y 6.8 m or 7 m. gsrt10h ProvendUseLwofSinesosines Pge 6 of 7 4/29/2014

7 SSESSMENT Prove & Use te Lws of Sines & osines G.SRT.10-HONORS 9) (2 solutions) sin 51 sin (sin )21 25 sin sin 51 sin 21 sin sin (0.9252) m = m m sin 51 sin (sin 61.31)21 sin 51 (sin 61.31)21 sin m m = 51 m m = 21 m m = 25 m seond tringle is possile so use nd solve for te supplement of = m = m sin 51 sin (sin16.69)21 sin 51 (sin16.69)21 sin m m = 51 m m = 21 m 7.76 m = 25 m gsrt10h ProvendUseLwofSinesosines Pge 7 of 7 4/29/2014

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