<prompt> Pointi A, B, C, na D zimepangwa kwa usawa kwenye mstari hivi kwamba AB BC CD. Pointi P inapatikana ili cos angle APC frac45 na cos angle BPD frac35. Amua sin 2 angle BPC.  <response>: Acha a AP, b BP, c CP, na d DP. Acha alpha pembe APC, beta pembe BPD, gamma angle BPC, na delta angle APD. Kisha cos alpha frac45 na cos beta frac35. Tangucos2 alpha cos2 beta 1,na alpha na beta ni kali, ni lazima pembe hizi zikidhi alpha beta 90circ. Pia, sin angle APC frac35 na sin angle BPD frac45.asyukubwa wa kitengo 2 cm;jozi A, B, C, D, P, Q, R;A 0,0;B 1,0;C 2,0;D 3,0;Q 1,3;R 2,2;P sehemu za makutanomduaraA,Q,C,duaraB,R,D0;kuchoraA--D; choramduaraA,Q,C; choramduaraB,R,D;choraA--P--D;choraP--B;kuchoraP--C;choraarcP,0.3,digriiA - P,digriiC - P,nyekundu;choraarcP,0.5,digriiB - P,digriiD - P,nyekundu;choraarcP,0.6,digriiB - P,digriiC - P,nyekundu;choraarcP,0.9,digriiA - P,digriiD - P,nyekundu;leboA, A, SW;leboB, B, S;leboC, C, S;leboD, D, SE;leboP, P, N;leboa, interpA,P,0.2, NW, nyekundu;lebob, interpB,P,0.2, NW, nyekundu;leboc, interpC,P,0.2, W, nyekundu;lebod, interpD,P,0.2, E, nyekundu;leboalpha, P -0.25,-0.35, Ondoa Kujaza;lebobeta, P -0.05,-0.65, Ondoa Kujaza;lebogamma, P -0.35,-0.7, Ondoa Kujaza;lebodelta, P -0.45,-0.95, Ondoa Kujaza;asyKumbuka kuwa pembetatu ABP,BCP, na CDP zina msingi na urefu sawa, kwa hivyo maeneo yao ni sawa. Acha K ABP BCP CDP.Tuna hiyoAPC frac12 ac sin angle APC frac310 ac,hivyo K frac12 APC frac320 ac.Pia,BPD frac12 bd sin angle BPD frac25 bd,hivyo K frac12 BPD frac15 bd. Kwa hiyo,K2 frac3100 abcd.Pia,APD frac12 tangazo sin delta,hivyo K frac13 APD frac16 tangazo sin delta. Tangu K BPC frac12 bc sin gamma,K2 frac112 abcd sin gamma sin delta.Inafuata kwambasin gamma sin delta frac925.Kumbuka kwamba gamma delta alpha beta 90circ, hivyo delta 90 circ - gamma. Kisha sin delta sin 90circ - gamma cos gamma, nasin gamma cos gamma frac925.Kwa hiyo, sin 2 gamma 2 sin gamma cos gamma frac1825. 