The Lws f Sines nd sines I The Lw f Sines We hve redy seen tht with the ute nge hs re: re sin In se is tuse, then we hve re h where sin 80 h 0 h sin 80 S re Thus, the frmu: 0 h sin y the Suppementry nge Identity sin re f Tringe is vid in ses Tht is, prdut f ength sine f f ny tw sides inuded nge re sin sin β sin Mutipy eh side f the ve equity y t tin sin sin β sin Nw divide eh side y t rrive t the tringe identity: sin sin β sin This frmu is knwn s the Lw f Sines Ntie tht the w f sines n e written in the terntive frm: sin sin β sin
Sving Tringe S If tw nges f the tringe re given, then the third nge n e fund y using the retinship: β 80 ; hene, the three denmintrs sin, sin β, nd sin n e fund using utr Nw, if ny ne f the sides,, r is s given, then the equtins sin sin β sin n e sved fr the remining tw sides The fwing exmpe indites the predure fr sving tringe when tw nges nd ne side re given r n e determined frm the infrmtin prvided Uness therwise indited, we sh rund ff nges t the nerest hudredth f degree, nd side engths t fur signifint digits Exmpe ---------------------------- ------------------------------------------------------------ In, suppse tht 4, β 77, nd 74 Sve fr,, nd β 77 74 4 Sutin 80 4 y the w f sines, sin 4 hene, sine 74, nd sin 77 sin 4 sin 4 sin 6 74sin 77 sin 4 sin 4 74 sin 6 77 6 ; sin 77 sin 6 099 999 Nte tht in this se there is wys unique sutin y the nge-side-nge riteri fr ngruent tringes Sving Tringe SS euse there re sever pssiiities, the situtin in whih yu re given the engths f tw sides f tringe nd the nge ppsite ne f them is ed the miguus se
Fr instne, suppse yu re given side, side, nd nge in Yu might try t nstrut frm this infrmtin y drwing ine segment f ength nd ry tht strts t nd mkes n nge with Figure T find the remining vertex, yu ud use mpss t drw n r f ire f rdius with enter If the r intersets the ry t pint, then is the desired tringe β Figure I s figure iustrtes, there re tuy fur pssiiities if yu try t nstrut y the ve methd in se is ute: i The ire des nt interset the ry t nd there is n tringe Figure ii The ire intersets the ry in exty ne pint nd there is just ne right tringe Figure iii The ire intersets the ry in tw pints nd nd there re tw tringes nd Figure iv The ire intersets the ry in exty ne pint nd there is just ne ute tringe Figure d Figure < sin n tringe pssie sin ne right tringe
sin < < tw tringes d > ne ute tringe II In se is tuse, then there re ny tw pssiiities s shwn in Figure 3 Figure 3 > ne tringe < n tringe pssie In the miguus se, yu n wys use utr t sve the tringe Just use the w f sines, sin sin β sin t evute sin β : sin β, 0 < β < 80
sin Re tht the sine f n nge is never greter thn ; hene, if >, then this trignmetri equtin hs n sutin, nd n tringe stisfies the given nditins If sin sin, then the equtin hs ny ne sutin, β 90 If <, then the trignmetri equtin hs tw sutins Nmey, sin β rsin nd β 80 β One yu hve determined β r β nd β, yu knw tw nges nd tw sides f the tringe r tringes, nd yu n sve the tringe y using the methds previusy expined Even if there re tw sutins β nd β f the trignmetri equtin fr β, it is pssie tht ny ne f these sutins rrespnds t n tu tringe stisfying the given nditins see Figure d nd Figure 3 Exmpe ---------------------------- ------------------------------------------------------------ Sve the tringe in eh se 30, 8, Sine is ute nd > there is ny ne ute tringe nstrutie sin β sin 30 8 sin β 8 6 β rsin 8 6 Then 80 30 8 3 79 sin 379 sin 30 Finy, sin 379 8 93 8 sin 30 8 30 β 30,, 8 8 Here is ute nd sin < < there re tw tringes nstrutie sin 30 sin β 8 8 sin β 4 30 8 β β β
Thus, β rsin 4 sin30 3 3 Then sin 9687 T find the mesure f β serve tht sin β sin 80 β sin β 30 33 87 80 sin 9687 sin30 Thus, β 80 β 80 33 6 87 nd 80 30 687 3 3 998 96 nd sin30 sin 33 sin 33 Then 3 98 sin30 II The Lw f sines When tw sides nd the inuded nge SS r three sides SSS f tringe re given, we nnt ppy the w f sines t sve the tringe In suh ses, the w f sines my e ppied Therem: The Lw f sines In the gener tringe, the squre f the ength f ny side is equ t the sum f the squres f the engths f the ther tw sides minus twie the prdut f thse side engths times the sine f the nge etween them s s β s β T prve the therem, we pe tringe in rdinte pne with verties eed unterkwise nd s tht ne side ies n the psitive x xis nd ne vertex is t O Suppse tht is t 0, 0 Then, 0 nd s, sin Thus, s sin s sin s
S s y s, sin β, 0 x Nw rtte the tringe s tht is t the rigin nd is n the psitive x xis n ngus rgument nw gives s β When is t the rigin, we find s Exmpe 3 ---------------------------- ------------------------------------------------------------ SS se: Sve the tringe if 60, 4, 0 Sine s 4 0 40 s60 96 00 40 6 49 4 It is gemetriy evident tht β is ute nd y the w f sines sin 60 49 sin β sinβ 4 4 49 3 7 3 49 60 0 β rsin 7 3 760 49 Then 80 60 760 43 90
Exmpe 4 ---------------------------- ------------------------------------------------------------ SSS se: Sve the tringe if, 6, 7 s 36 49 67 s 60 8 84 s s 84 7 rs 44 4 7 6 Nte: there is n ther nge θ fr whih: 0 < θ < 80 nd s θ 7 7 Then y the w f sines sin 444 sin β 6 sinβ sin 444 6 β rsin sin 444 6 73 ery, β must e ute Then 80 444 73 78 4 Therem: Hern s re Frmu The re f tringe with sides, nd nd semiperimeter s s s s s given y hs re h sin The prf fws frm the w f sines expressed in the frm: s Nte tht h sin sin 4
Nw we my tin the desired frmu y geri mniputin s sin 4 4 s s 6 s s 6 6 6 s s s s