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Winifred Beasley
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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 1-3, 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 6-pinted Str f Dvid tht is insried in irle f rdius Find the re f the regulr 5-pinted str (the pentgrm) tht is insried in irle f rdius 1. 46

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right