134 Goniometrlc Prohlems, 



CDs BDC , ^„ CD.r,DCB , . 



CB = 7;,vTc— and DB = -rpr^ — : also since BA^ = 



s,Lau J, CBD 



BD^ + AD=- 2BD.AD.r, ADB= BC= + AC^ - 2BC. 

 AC, c, ACB ; therefore by substituting the above values ot" 



CD^ ..% DCB CD^ J^ DCA 

 these quantities we get _, ^^^ — + :^AD ~ 



CD* /, DCB s, DCA c, ADB CD* .-. BDC CD« •», ADC 



J, CBD/,CAD ACBD A CaD 

 2CD*/, BDC 5, ADC^, ACB , , , r^nr^ovT^ 



;,-CBD7, CAD ' ^^'* ^^^ '-^"Sl- CBD, CAD 



being the supplements of (CBD + CDB) and (ACD 

 + ADC) respectively, let them be substituted for them, 

 and then dividing the whole equation by CD", we have 



s"-, DCB !;\ DCA 2 5, DCB s, DC a r , ADB 



«',(BDC + BCD) s-,(ADC + ACD) i,(BDC + BCD) 5, (ADC + ACD) 

 •s^BDC 4', ADC 2 s, BDC 5, ADC f, ACB 





s"; (BDC + BCD) ,--, (ADC + ACD) k, (BDC + BCD) .s, (ADC + ACD) 



which cleared of fractions becomes ^% DCB ^% (ACD + 

 ADC) + 5% DCAi% (BDC + BCD) - 2 5, DCB 5, DCA 

 c, ADB s, (BDC + BCD) s, (ADC + ACD) = s\ BDC 

 s\ (ACD + ADC) + s, ADC s\ (BDC + BCD) -2 s, 

 BDC s, ADC c, ACB s, (BDC + BCD) s, (ADC 4- ACD) ; 

 or (5% DCB - s% BDC) s'-, (ACD + ADC) + (.?% DCA 

 _ 5% ADC) 3\ (BDC + BCD) = [<2 s, DCB s, DCA 

 c, ADB - 2 5, BDC s, ADC c, ACB] 5, (BDC + BCD) 

 s, (ADC + ACD) ; but by art. 84, Cagnoli's Trigono- 

 metry; s\ DCB - s\ BDC = s, (BCD + BDC) ,?, (BCD 



— BDC) ; whence by substitution we get s, (BCD+ BDC) 

 s, (BCD - BDC) s\ (ACD + ADC) + s, (ACD + ADC) 

 s, (ACD - ADC) s\ (BDC + BCD) = [25, DCB 5, DCA 

 r, ADB - 2 5, BDC5,ADCc, ACBJ x 5, (BDC + BCD) 

 5, (ADC + ACD), and dividing the whole equation by 

 s, (BCD -f BDC) X s, (ADC +ACD) it becomes s, (BCD 



- BDC) 5, (ACD + ADC) + s, (ACD - ADC) ,9, (BDC 

 + BCD) =2 5, DCB 5, DCA c, ADB - 2 s, BDC 5, ADC 

 c, ACB ; but by Cagnoli, art. 52, s, (BCD - BDC) = 

 5, BCD c, BDC - c,^BCD s, BDC, and 5, (BDC + BCD) 

 = s, BDC c, BCD + c, BDC s, BCD : by substituting 

 these for their equals, then actually multiplying them toge- 

 iher, and dividing the whole equation by 2, we have s, BCD 



r, BDC 



1 



