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1 6. SOLVING RIGHT TRINGLES In the right tringle B shwn in Figure 6.1, the ngles re dented y α t vertex, β t vertex B, nd t vertex. The lengths f the sides ppsite the ngles α, β, nd re dented y,, nd. Nte tht ngles α nd β re mplementry ute ngles, tht ngle is right ngle, nd tht is the hyptenuse f right tringle B. Therefre, sinα = s β = sα = sin β = sα = se β = seα = s β = β B tnα = t β = tα = tn β = α Figure 6.1 If the lengths f tw sides f right tringle re given, r if ne side nd n ute ngle re given, then these frmuls sn e used t slve fr the remining ngles nd sides f the tringle. The predure is lled slving the right tringle. Exmple Slve the right tringle B leled s in Figure 6. if β = 30 nd = 4. Figure 6. α 30 B 4 We must find, nd α. Beuse α nd β re mplementry, α = 90 β = = 60. T find, we use the ft tht tn β =, s = tn β. Therefre, = tn β = 4 tn 30 = = 8 3. T find, we use the ft tht se β =, s = se β. Therefre, = se β = 4 se 30 = 4 Of urse, we uld hve insted used the Pythgren therem t find = 3 3 = Unless the speil ngles 30, 45, r 60 re invlved, it is neessry t use lultr r tle f trignmetri funtins t slve right tringle. Yu shuld lwys keep in 41
2 mind tht the slutins tined in these wys re pprximtins. In this setin, whenever suh pprximtins re invlved, we will rund ff ll ngles t the nerest hundredth f degree, nd ll side lengths t fur signifint digits. Exmple In eh se, ssume tht right tringle B is leled s in Figure 6.1. Then slve the tringle nd sketh it. ( i ) = 31, α = We must find,, nd β. Beuse α nd β re mplementry, β = 90 α = = T find, we ntie tht tn α =, s = tnα. Using lultr, we find tht tnα = tn = Therefre, runded ff t fur signifint digits, = tnα = 31 tn = 9.4. T find, we ntie tht s α =, s =. sα Therefre, runded ff t fur signifint digits, s = = = = 4.6. sα s =4.6 α = Figure 6.3 =31 β = B =9.4 One gin, we uld hve insted used the Pythgren therem t slve fr : = + = (.4) = The disrepny in the lst deiml ple ws used y using the runded ff vlue fr. The result = 4.6 is tully mre urte. The tringle is shwn in figure 6.3. ( ii ) = 4, = 3. By the Pythgren therem: = + = = 5. Nw we hve sin α = = 5 4 = 0.8. It fllws tht α = rsin 0.8 = Figure 6.4 =5 β = B =4 β = 90 α = = The tringle is shwn in Figure 5.4. α = =3 4
3 Exmple nsider hurh with steeple, s shwn in the Figure 6.5. The prlem is t lulte the height f the steeple while stnding n the grund. T find the height, mrk pint B n the grund diretly elw the steeple nd nther pint tht is 00 feet wy n the grund. t, mesure the ngles f elevtin α nd β t the tp nd ttm f the steeple. This is ll the infrmtin yu need. Figure 6.5 Let x e the height f the steeple nd y e the distne frm the grund t the ttm f the steeple. Suppse tht α = 35 nd β = 6. x + y y Then tn 35 = nd tn 6 = Slving fr y in the send equtin nd sustituting int the first yields: x + 00 tn 6 tn35 =. 00 α = 35 x y S x = 00 tn tn 6 =00 ( ) = 4.5 feet. β = 6 00 ft B Setin 6 Prlems In prlems 13, ssume tht the right tringle is leled s in the figure shwn. In eh se, slve the tringle nd sketh it. When pprximtins re invlved, rund ff ngles t the nerest hundredth f degree nd side lengths t fur signifint digits. 1. = 10, =10. =100, β = = 91, β = 30 α β B 43
4 Hin 4. Find the re f the shded regin f the right tringle B shwn in the figure. 5. The huse in the figure is 4 ft. wide. The ridge is 9 ft higher thn the side wlls, nd the rfters prjet 1.5 ft eynd the sides f the huse. Hw lng re the rfters? 6. Find the distne B etween the enters f the tw rllers in the figure. 7. highwy uts rner frm prel f lnd. Find the numer f res in the tringulr lt B. (Nte: 1 re = 43,560 ft ). (Hint: First use the Lw f sines, pge 54, t find the the mesure f ne ngle in B.) 44
5 8. em B is supprted y tw rssed ems. Find the distne x. 9. Frm the tp f lighthuse 10 feet ve se level, the ngle f depressin (the dwnwrd ngle frm the hrizntl) t t drift in the se is 9.4. Hw fr frm the ft f the lighthuse is the t? 10. Frm windw in n ffie uilding, I m lking t televisin twer tht is 600 meters wy (hrizntlly). The ngle f elevtin f the tp f the twer is 19.6 nd the ngle f depressin f the se f the twer is 1.3. Hw tll is the twer? 11. Find x in the figure. 1. The vertil distne frm the first t the send flr f ertin deprtment stre is 8 feet. The esltr, whih hs hrizntl reh f 86 feet, tkes 5 sends t rry persn etween flrs. Hw fst des the esltr trvel? 13. nsider tw irles th f rdius r nd with the enter f ne lying n the rim f the ther. Find the re f the mmn prt f the tw irles. 45
6 14. The Gret Pyrmid is ut 480 feet high nd its squre se mesures 760 feet n side. Find the ngle f elevtin, β, f ne f its edges. 15. Find the ngle etween prinipl dignl nd fe f ue. 16. regulr hexgn (6 equl sides) is insried i irle f rdius 4. Find the perimeter, P, nd the re,, f this hexgn. 17. Find the re f the regulr 6pinted Str f Dvid tht is insried in irle f rdius Find the re f the regulr 5pinted str (the pentgrm) tht is insried in irle f rdius 1. 46
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