PPLY SSIGNMENT GUIE SI 1: pp. Es. 8 even : pp. Es. 11 9 o, 47 1,, 4, 0, Quiz Es. 1 7 VERGE 1: pp. Es. 8 even, 40, 4, 4 : pp. Es. 11 9 o, 4, 44, 47 1,, 4, 0, Quiz Es. 1 7 VNE 1: pp. Es. 8 even, 40 4, 4 : pp. Es. 11 9 o, 4, 44, 47 0, Quiz Es. 1 7 LOK SHEULE pp. Es. 8 even (with 9.4) pp. Es. 11 9 o, 4, 44, 47 1,, 4, 0, Quiz Es. 1 7 (with 9.) EXERISE LEVELS Level : Esier 7 Level : More ifficult 8 4 Level : Most ifficult HOMEWORK HEK To quickl check stuent unerstning of ke concepts, go over the following eercises: Es. 1, 4, 0, 4, 8, 48. See lso the il Homework Quiz: lckline Mster (hpter 9 Resource ook, p. 84) Trnsprenc (p. 8) GUIE PRTIE Voculr heck oncept heck Skill heck 1. The e of is the rtio of the length of the leg opposite to the length of the hpotenuse. The coe of is the rtio of the length of the leg jcent to to the length of the hpotenuse. The tngent of is the rtio of the length of the leg opposite to the length of the leg jcent to. PRTIE N PPLITIONS STUENT HELP Etr Prctice to help ou mster skills is on p. 80.. R 0.8491; cos R 0.8; tn R 1.071; S 0.8; cos S 0.8491; tn S 0. 11. = 0.8; cos = 0.; tn 1.; = 0.; cos = 0.8; tn = 0.7 STUENT HELP HOMEWORK HELP Emple 1: Es. 1, 8 Emple : Es. 1, 8 Emple : Es. 4 Emple 4: Es. 4 Emple : Es. 1 7 Emple : Es. 7 4 Emple 7: Es. 7 4 In Eercises 1 n, use the igrm t the right. 1. Use the igrm to eplin wht is ment the e, the coe, n the tngent of.. ERROR NLYSIS stuent ss tht > ecuse the sie lengths of EF re greter thn the sie lengths of. Eplin wh the stuent is incorrect. In Eercises 8, use the igrm shown t the right to fin the trigonometric rtio.. } 4 4. cos } = 0.8 } } = 0.. tn } 4. } 1. } } = 0. 7. cos } 4 8. tn } = 0.8 } 4 } = 0.7 9. ESLTORS One erl escltor uilt in 189 rose t n ngle of. s shown in the igrm t the right, the verticl lift ws 7 feet. Estimte the istnce person trvele on this escltor. out 17 ft FINING TRIGONOMETRI RTIOS Fin the e, the coe, n the tngent of the cute ngles of the tringle. Epress ech vlue s eciml roune to four plces.. R 11. 1. X 8 T 4 S Z Y 1. E 14. G 1 J 1. L 4 7 F J 4 K H LULTOR Use clcultor to pproimte the given vlue to four eciml plces. 1. 48 0.741 17. cos 1 0.9744 18. tn 81.18 19. 7 0.440 0. cos 70 0.40 1. tn 0.049. 78 0.9781. cos 0.8090 4. tn 0.44. cos 0.440. 0.890 7. tn.40 8 7 7 4 1 F E 7 ft hpter 9 Right Tringles n Trigonometr
USING TRIGONOMETRI RTIOS Fin the vlue of ech vrile. Roun ecimls to the nerest tenth. 8. 9. 0. 4 s 4 t 7 s r s 1.; t 1. r 4.9; s.9.0; 8.0 1.. 9. 8 u w 70 v t v 9.; w. 1.0; 14.9 t 7.; u.4 PPLITION NOTE EXERISE 8 Trigonometr is pplie etensivel in surveing pplictions. For emple, the Gret Trigonometric Surve of Ini ws one in the 19th centur to mp out the countr n to mesure the height of Mount Everest, the tllest mountin in the worl. In 18, the surve mesure Mount Everest to e 9,00 feet, which is ver close to the height of 9,0 feet mesure in 1999. WTER SLIE Even though riers on wter slie m trvel t onl 0 miles per hour, the curves on the slie cn mke riers feel s though the re trveling much fster. REL FOUS ON PPLITIONS LIFE FINING RE Fin the re of the tringle. Roun ecimls to the nerest tenth. 4. 4 cm. 41. m. 4.9 m 4 4 cm 8 m 7. WTER SLIE The ngle of elevtion from the se to the top of wterslie is out 1. The slie etens horizontll out 8. meters. Estimte the height h of the slie. out 1.4 m 0 8. SURVEYING To fin the istnce from house on shore to house on n isln, surveor mesures from the house on shore to point, s shown in the igrm. n instrument clle trnsit is use to fin the mesure of. Estimte the istnce. out m 0 11 m 9. SKI SLOPE Suppose ou stn t the top of ski slope n look own t the ottom. The ngle tht our line of sight mkes with line rwn horizontll is clle the ngle of epression, s shown elow. The verticl rop is the ifference in the elevtions of the top n the ottom of the slope. Fin the verticl rop of the slope in the igrm. Then estimte the istnce person skiing woul trvel on this slope. 48 ft; out 1409 ft 1 m h 40 m 4 8. m 1. The vlue of trigonometric rtio epens onl on the mesure of the cute ngle, not on the prticulr right tringle use to compute the vlue. 1. X 0.81; cos X 0.47; tn X = 1.; Y 0.47; cos Y 0.81; tn Y 0.7 1. = 0.8; cos = 0.9; tn 0.917; F = 0.9; cos F = 0.8; tn F.48 14. G 0.8944; cos G 0.447; tn G = ; H 0.447; cos H 0.8944; tn H = 0. 1. J 0.87; cos J 0.14; tn J 1.7; K 0.14; cos K 0.87; tn K = 0. ngle of epression 0 elevtion 00 ft elevtion 018 ft verticl rop 9. Trigonometric Rtios
OMMON ERROR EXERISE 40 Stuents m hve ifficult ppling the correct rtio when solving this prolem. Remin them tht the e or coe rtios cnnot e use unless the hpotenuse of the tringle is known. EXERISE 4 Stuents m wnt to ppl the right tringle rtios to n tringle or to the right ngle of the tringle. Remin stuents to check tht tringle is right efore ug the rtios n tht the rtios ppl onl to the cute ngles. 4. må = 9, so is not right tringle, s require the efinition of the tngent rtio. To fin, rw n ltitue,, from to. is 0-0 -90 tringle with hpotenuse 18, so the shorter leg,, hs length 9. Then use right n the eqution = } } to fin.. Given: = } c }, cos = } c }, tn = } } Prove: tn = }} Sttements (Resons) 1. = } c }, cos = } c }, tn = } } (Given). }} = } c } } c } (Sustitution prop. of equlit). } c } } c } = } } (iviing n simplifing frctions) 4. tn = }} (Sustitution prop. of equlit) LIFE LUNR RTERS ecuse the moon hs no tmosphere to protect it from eing hit meteorites, its surfce is pitte with crters. There is no win, so crter cn remin unisture for millions of ers unless nother meteorite crshes into it. 4. = } c } ; cos = } c }; REL FOUS ON PPLITIONS tn = } }; = } c }; cos = } c } ; tn = } } 44. The tngent of one cute ngle of right tringle is the reciprocl of the tngent of the other cute ngle. The e of one cute ngle of right tringle is equl to the coe of the other cute ngle, n the coe of one cute ngle is equl to the e of the other cute ngle. 4. Proceures m vr. One metho is to reson tht ce the tngent rtio is equl to the rtio of the lengths of the legs, the tngent is equl to 1 when the legs re equl in length, tht is, when the tringle is 4 4 90 tringle. Tn > 1 when m > 4, n tn < 1 when m < 4, ce increg the mesure of increses the length of the opposite leg n ecreg the mesure of ecreses the length of the opposite leg. 40. SIENE ONNETION Scientists cn mesure the epths of crters on the moon looking t photos of shows. The length of the show cst the ege of crter is out 00 meters. The sun s ngle of elevtion is. Estimte the epth of the crter. out 714 m 41. LUGGGE ESIGN Some luggge pieces hve wheels n hnle so tht the luggge cn e pulle long the groun. Suppose person s hn is out 0 inches from the floor. out how long shoul the hnle e on the suitcse shown so tht it cn roll t comfortle ngle of 4 with the floor? out 1.4 in. 4. UYING N WNING Your fmil room hs sliing-glss oor with southern eposure. You wnt to u n wning for the oor tht will e just long enough to keep the sun out when it is t its highest point in the sk. The ngle of elevtion of the sun t this point is 70, n the height of the oor is 8 feet. out how fr shoul the overhng eten? out.9 ft RITIL THINKING In Eercises 4 n 44, use the igrm. 4. Write epressions for the e, the coe, n the tngent of ech cute ngle in the tringle. 44. Writing Use our results from Eercise 4 to eplin how the tngent of one cute ngle of right tringle is relte to the tngent of the other cute ngle. How re the e n the coe of one cute ngle of right tringle relte to the e n the coe of the other cute ngle? 4. TEHNOLOGY Use geometr softwre to construct right tringle. Use our tringle to eplore n nswer the questions elow. Eplin our proceure. For wht ngle mesure is the tngent of n cute ngle equl to 1? For wht ngle mesures is the tngent of n cute ngle greter thn 1? For wht ngle mesures is the tngent of n cute ngle less thn 1? 4. ERROR NLYSIS To fin the length of Æ in the igrm t the right, stuent writes 18 tn =. Wht mistke is the stuent mking? Show how the stuent cn fin. (Hint: egin rwing n ltitue from to Æ.) in. 4 8 ft sun s r 00 m 70 c 18 0 0 in. sun s r 4 hpter 9 Right Tringles n Trigonometr 4
47. PROOF Use the igrm of. omplete the proof of the trigonometric ientit elow. ( ) + (cos ) = 1 GIVEN = c, cos = c c. Q 48. ( 0 ) + (cos 0 ) = } 1 + } } = } 1 4 } + } 4 } = 1 49. ( 4 ) + (cos 4 ) = } +} = } 4 } + } 4 } = 1 PROVE ( ) + (cos ) = 1 Sttements 1. = c, cos = c. + = c c. + c = 1 4. c + c = 1 Resons 1.? Given.? Pthgoren Thm..? ivision prop. of equlit 4. propert of eponents Q P 4 P R 8 R 0. ( 0 ) + (cos 0 ) = } +} 1 } = } 4 } + }1 4 } = 1. ( ) + (cos ) = 1.? Sustitution prop. of equlit EMONSTRTING FORMUL Show tht ( ) + (cos ) = 1 for the given ngle mesure. 1. ( 1 ) + (cos 1 ) (0.0) + (0.9744) 1 Test Preprtion 48. m = 0 49. m = 4 0. m = 0 1. m = 1. PROOF Use the igrm in Eercise 47. Write two-column proof of the following trigonometric ientit: tn =.. MULTIPLE HOIE Use the igrm t the right. Fin. 8 cos 8 8 8 tn 8 E 8 E cos 4. MULTIPLE HOIE Use the igrm t the right. Which epression is not equivlent to? 70 cos 0 tn 0 tn 0 E tn 70 hllenge EXTR HLLENGE www.mcougllittell.com. PRE You re t pre looking up t lrge lloon floting irectl ove the street. You re 0 feet from point on the street irectl eneth the lloon. To see the top of the lloon, ou look up t n ngle of. To see the ottom of the lloon, ou look up t n ngle of 9. Estimte the height h of the lloon to the nerest foot. 4 ft h 9 0 ft 9. Trigonometric Rtios ITIONL PRTIE N RETEHING For Lesson 9.: Prctice Levels,, n (hpter 9 Resource ook, p. 7) Reteching with Prctice (hpter 9 Resource ook, p. 7) See Lesson 9. of the Personl Stuent Tutor For more Mie Review: Serch the Test n Prctice Genertor for ke wors or specific lessons.
4 SSESS ILY HOMEWORK QUIZ Trnsprenc ville Fin the e, the coe, n the tngent of the cute ngles of the tringle. Epress ech vlue s eciml roune to four plces. 1. 1 7 0.949, cos 0.4, tn.917, 0.4, cos 0.949, tn 0.49 Use the igrm. Roun ecimls to the nerest tenth. MIXE REVIEW. n enlrgement; ; Q R = n P R = 8. SKETHING ILTION PQR is mppe onto P Q R iltion. In PQR, PQ =, QR =, n PR = 4. In P Q R, P Q =. Sketch the iltion, ientif it s reuction or n enlrgement, n fin the scle fctor. Then fin the length of Q R n P R. (Review 8.7) 7. FINING LENGTHS Write similrit sttements for the three similr tringles in the igrm. Then fin QP n NP. Roun ecimls to the nerest tenth. (Review 9.1) MNP ~ MQN ~ NQP; QP.; NP 7.8 PYTHGOREN THEOREM Fin the unknown sie length. Simplif nswers tht re ricls. Tell whether the sie lengths form Pthgoren triple. (Review 9. for 9.) 8. 9. 0. 9 18; es 19 9; no 8.1; no 0 M 1 4.9 7 N q P 70 8 m. Fin the vlue of ech vrile. m, 9.4 m. Fin the re of the tringle. 0 m EXTR HLLENGE NOTE hllenge prolems for Lesson 9. re ville in lckline formt in the hpter 9 Resource ook, p. 80 n t www.mcougllittell.com. ITIONL TEST PREPRTION 1. WRITING escrie how to fin the height of tree if ou know the ngle of elevtion n our istnce from the tree. Fin the tngent of the ngle of elevtion n multipl it our istnce from the tree. ITIONL RESOURES n lterntive Quiz for Lessons 9.4 9. is ville in the hpter 9 Resource ook, p. 81. QUIZ Self-Test for Lessons 9.4 n 9. Sketch the figure tht is escrie. Then fin the requeste informtion. Roun ecimls to the nerest tenth. (Lesson 9.4) 1. The sie length of n equilterl tringle is 4 meters. Fin the length of n ltitue of the tringle.. m. The perimeter of squre is 1 inches. Fin the length of igonl..7 in.. The sie length of n equilterl tringle is inches. Fin the re of the tringle..9 in. Fin the vlue of ech vrile. Roun ecimls to the nerest tenth. (Lesson 9.) 4... 40 18 0 9.;.1 1.; 11.9 8.; 1.9 7. HOT-IR LLOON The groun crew for hot-ir lloon cn see the lloon in the sk t n ngle of elevtion of 11. The pilot rios to the crew tht the hot-ir lloon is 90 feet ove the groun. Estimte the horizontl istnce of the hot-ir lloon from the groun crew. (Lesson 9.) out 4887 ft hpter 9 Right Tringles n Trigonometr Not rwn to scle groun crew 11 90 ft