aireita faaitorferfar (SPHERICAL TRIGONOMETRY) Bto Ho Bo FM (Dr. P. D. Kathal) aaITH SETA et gatadt areata agtfaered, AEST, Toteanette Faatorfegfer SPHERICAL TRIGONOMETRY —by Dr, P. D, Kathal : STM : weanen ferdt aq oeratit, areata are, alae © aeasee ferct ara garett saa deat: 1973 wea: 10 wet qea: aaa oraaara, arefefins fred 65, Wa Uae ara Ue, satgrare-2 Published by Madhya Pradesh Hindi Granth Academy under the Centrally Sponsored Scheme of Production of Books and literatur® in regional languages at the University level, of the Government of India in the Ministry of Education and Social ‘Welfare (Department of Culture); New Delhi.SRATTAT ania faatufate faeafreradt at ones serait far fara crear 1x araTfer TET gers Ss ge ca aera HT faa 2, fant dhe sortie, atthe sortie, site frost og saat qari sie areit # Prerorfits aeara, arate Tete Pres, atthe Pras ar Frater, MTT Tar a ofega, Teta PT at awa ake Tee aifeam, ay frac ot aie Patofate & ace cept ar sfraret fear wat 81 Tfonfes gat tarhas wel et cere ace meet % ofearar 8F gy sete fara at fafee-sergeil & ara ere far wa Zsolt gar wet fare aereiedte sist atte Pasi ar oat free war 21 germ S sea A foie wears wt ate a wt & fad frat at farraeg & afar H afar at sto mom at favafearaa wae ae afte F year aT aaarit ater 21 ara f sast ag gee fant F atafer ait t Ye gm hated (ste year afrertt) wares neqrem fedt aq saraettreat faoa-aat 1. ae sarfafa 1.1 1.2 1.3 aeat tat A TAT fam ata faaa sty a aa (ae) FT 2. atera sorta 7 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 arat are er ame Tae ae oz feast acti F et HIT wa avg att gah fa & arafad sta ayaa a alee ate stata Prat fades aa aria FT a aaah st afeeeet 2.9 syF 2.10 ag ate Saeco eat aT 3. Weta Frat 3.4 3.2 3.3 3.4 3.5 3.6 3.7 3.8 mete Frys aie frye % ata & avahaa she aia fags att faa aro sfrerraifrs frat aeegn frat aa fags, sfromtaifae Pros ate agen Prgat & farina shearer ate Pgs after Pager at serfirfet(ii) 4. iret Boys ot UaTaH atte ete H florida weeE 54 4.1 fret 4.2 greet at - ‘ 4.30 aye stares qa 44° ag 4.5 ater ga 4.6 ada gt 4.7 a gna 4.8 fafae vereea 4.90 artatant a7 4.10 anger anata aw 4.11 cava meant 4.12 Ffraz area? 4.13 saree 3.0 waste wisita fas \ 5.1 afer 5.2 aay gt 5.3 agate ror 5.4. afrax ata saaa frat - 5.5 fafirr gereer ~ + 5.6 atta rg & vite fags 8 GT nk are are gant 3 ave 5.7 attr Pager at att aaa 5.8 agatait frysr 6. ate faaa or fate 6.1 afar 6.2 anata atta frgst ar frafeat 6.3 faa tee fags at Frater 7. ania qe ate afc 7.1 ona ga - 54 55 78 89 102 lll 114 124 133 136 137 138 142 154 154 154 159 162 163 174 178 179 190 190 190 203 219 2197.2 afesa 73° faa 74 aeige Prasat % oferta aa 7.5 gar faysit & wenta ste afta 8. asta fogs TT aes alt Teta onfaeT 8.1 atefer fargo at Garter 8.2 tafe aifaaa (Z) 8.3 £, 40 oR LE frofadte cei ® ae 9. wea-fraer 10. Weta feetofefa F ave aya 10.1 gedt & erat az 10.2 amfaat % agqeatr oifearies great at srgarafive 224 228 232 233 245 245 248 267 277 277 290 305areata 1 ara satiate (SOLID GEOMETRY) sf Ll. gage H a ate onfaie > ge oferta pert att farer art sarfafe atx mete fratofafe & meq FH aeara aaa & | ara Tar ate aaa afcarar 1 wa CHa Tar uHR aaa a feet fg ae wedl Bat va area Par at ‘afaeset Vay ate va fag st sfaeest Var sr “aT? wet El afar 2 : aa sfireteh tet & gax oA ant, gram F fea, sets Par, sfreeet Vat FH oaET gay Ret wfreBel Vat aaTaT ge aT wean & 1 aforn 3 wa afreeel Tar wAaT IT aT at aT SaaT wfaewe farg ‘orea- Tie? FeeTAT zg |2 atta Prrkafinfs bia saa) : afe wm acer Var, Pred at sewed ae tanit F sea fry oz, oat & cate Var oz oe Vat ag Ba aHaa Te AT Ha A TH TT sfreaet Vana TF ae wT st [feerait—ste at sachet % Preafafiaa sonfrdta ofa aT site frat var & frre aerat at ga aet steifce art we ae T |] fags ABC % afe AC Herne Rat & a, - AB? +. AC? = 2AE*4. CE? arate 1 fart far tat OA, afrsdat tart OB ak OC 1 FAT ae are & arate, ZA0C=90° ZAOB=90° aK, fea wert % 1 ava tat AO, aaa XY, at a gk sfreset vail OB az OCR dae oat 3, 1 TAZ I wat O feg a, aaa XY H, Wed ue aa wi OD ect | fag D& OB ate OC F aaa ava Va alaaz, gHKAT apis OBDC aarn | ar at aanae ATT F fawat (Diagonals) avert E fag oe wed FAB, AE we AC A fara |are sentir 3 wafa—fays ABC aie OBC % wag: AE ok OF wee ang & | (caat &) “ AB*+-AC? = 2AEF?2.2CE* soe @ a OB? + OC? = 20E?42CE* ». . i) req fagt AOB H ZLAOB=90° os feta o AB’ = 0A?+ O0B* wat sare, fragt AOC #. AC? = OA?+ OC? AB? +. AC?=204?+ OB*4- OC? = 2042+ 20E?42CE* + (ily a, 2AE?+2CE2= 20424 20E'4-2CE® (i) .. AE?= 04?-40E2 ie ii) LAOE=90° sek, OEmODLOA, afm ava tat OD, O fag & diet we ww ee tare ae: ava tat AO, farg OR oA art se aaae Fea, wets aa Rar rt we 8, gafay aaaa XY at at aad | (sfowret—2) 1-2 feta FT (Dihedral Angle) er oma gfe ore ataca tant ws gat at sredt & aaiq aa waft 2 sail cw site Fae on a fing owattss dare at ee ware ana ret (Plane Angles) a Tat ht 2 | (arafr 2)4 aiyeita Presrferfer aferrat 4 wa a ance fad ce aor tart ow gat at aed fata ae wat um sie Sau us gt aa tar saairs ail gt ea Ta wT arefe at fare stot (Dihedral Angle) #@t @1 atafa 3% fear arg at araaa: A-BC-D ar auta g | arefa. 3 . anat XY site MN, ae Ret BC H oH Zar AY TET ZI area & ene 2 fe ea sare are face ate at car Stet 2 TT at tae fae ety A-BC-D at cam % vert gatea deat fae areher Hardt Vans aver eat ware | guafies act tat BC, fara try A-BC-D at az (Edge) agar ¢ at ance AB atk DB sak warm (Faces) sear F | 1-21 fara sto BT ATT fare ater at atx Te Gam Pett fees Fae aaa HF aha aa art ari fafza anadtr ata arate fect ary stam gta 8 Fe aft 3 % ax BC % fez Qe QP mH QR warm: wae XY ae MNH, BC We! TI THT AAAs sty POR at eat ge freer am fire ety ABC-D stare ¢ | wa: fart at vt asada stats ear aa, arat afas, faacta ary aeelae safes 5 fecuvtt 1 gadtes Pafer & ferarert va facet Frey aT aie wa fea gat t 1 safe ate oe aie zat FTE A fang GA FF At aT POR Wie ater EFG veers aerat it | [xe waa st aeraT wT 87 ae Be editer eget sorfafer (at afiete-XI1, 10) & eeanngia enfrs a a aadt & 1] facqoit 2 arefa 3%, BC| QP ait BCL QR gafat sta 1% BC, QP sik QR Hat a at TAIT arg ett | ara: af oon rarer Fret fart ater at gaat ae HB TTET HE at ea sare ora sfreeet Vani 8 at anata ate a1 an, feat ary at are sats | Fanaa PQR fers ay ABCD wt ware peat 8 fe ate BC, sae ara S| Ba THT HT feeeeT vant QP ae QR ant fatha anacita sta POR, fers sre at am él afer 5 wa 2 ance are fataa fare eho oH aaaY F Gea eat 2 aT ¥ amae TWIT TST FATT z | 4-3. faa Fo (Trihedral Angle) at a (ta) BT (Solid Angle) : afearat 6 wa ate at ofa aaa, fret flere wr H, UH gat At Ee THT and fir gaat sferebat rea Vere at feet ce feg ge UH gat aT6 aireirr Prtenfirfer : are Tea ware a orefr aadt 2st aa sta, ate sta. (Solid: Angle) a1 agast atu wet ZI 13.1 faqa FT e ages ater at afer ¥ aa Fae de sai (Concurrent) aa aa a ot o& Grow at at | mata, : afew 7 ° # aa dt ance, fret fafa mrt, oH Rare gat at ea are are fix carat afaeeet Tare wt frat ww fag oe cH gat at ae at Ba THC gat aes at fas wot (Trihedral Angle) sg #1 Racy Faso ena Cc staf 4 (A) mats 4 (B) agra ttm (S-ABCDEF) — faa atu (S-ABC) areft 4-B'% da dani area, SAB, SBC, vik SCA, faa Fram ate at Frafey eet 2 se ga (S-ABC) arer awit & | aad F aie cant aractt are are Pree stay at caer diet & reg set TT WTA Sat TAH F Fae Frae ste (S-ABC) oz st Higa TST | frat atu (S-ABC) # auat SAB, SBCs SCA sa® woe (Faces) eget & | axaatl at drat fry (Point of Concurrence),ae sarfarfr 7 S faa ater at itd (Vertex) agerat @ | weet at sfreddt Tard SA, SB ate SC faa star at aT (Edges) wget & | fae star (S-ABC) % carat & a3, att fact ata (ASC-B, B-SA-C ate ASB-C) aavt & ae SRE MT ata GH BT (2 ASB, 7 BSC siz Z. CSA) aaa & | feet froa aro & cla fee ato ake de cee she fr aT sah Bs: wrt aeaTT gs | 13.2 a adaraa faqa Fit (Identically equal trihedral angles) aa & faa stat a oa gat ge TeaTefrT (Superposed) Firat * oT at, wate wat gw ATH gaStT wr (fitted) dt ae ae 3 aa'ar an seed & 1 a adarnaa facw agi (aa stat) Fo wes cae Fra AT faa atu, we fafeaa sala aH, gat were ae feat att & AAT AAT FTA BAF | aad fardta, aff ce foe sty & nae ae fara at, OF aaa am % gat fae ate gaa wk fore Arete aavaT aay a at & ae fae ste atar an gi sage weet Haale ar al sfvara gafet araeas 2 fe an Ge at fram thy ma Fe aad 2 foal ep pete ate feo ster gat H gare ate eae That emer (THY TE wate wR H Ae) were reg fax ae fram ata aa atan adi 1 aT8 atte Fratfirfer Fran ary (S, ABC) ei (ee & 8) eat B eR Hep arg (S, A’B’C’) at waar gg frat ere & Fe (i) (S, ABC) ® waa sty (S, A'B'C) & wan stat west aTaAT S| AT, LASC= £ ASC’, wane | (ii) (S, ABC) & fea are (S, A'BIC’) & fea stat % TAT AAT FAT (BSA-C)=(B'-SA'-C') gate | wa am at ay fear stg (S, ABC) at (S, A'B'C’)) ax aearifia wer & feat farg S eaafts tae, ate SA state SA! oe a cas ASB at ore ASB 1 warps at ae cae TTT ASB, aq A'SB’ % qeaz 8 afay Oat war CH Zar wt TET ae at | veg ea Rafe H ahi fro sty ea eras Ta H eT aie (avez) ait | aa afe en faaa atu. (S, ABC) at (S, A’B'C’) at Fa ware wearer at fie Gere SAB, Gee SA’B’ at gaistay ae ak ahi fra ata we oaaits oes B at ate a Hat oH ak di oa farafafed feafa srr at 1 aie SA, Be SB! oz te ae SB, BR SA! oe Fa | a Shs feat ate (B-SA-C), (A’-SB'-C’) & goa agi 8 wate Gare SAC, gas SBC! oz wet T3at ate get THT Gate SBC, wars SA'C’ Te wat 8a | wa: frat sty (S, ABC), (S, ABC) % qaista srargan wet stat g ate gate F gaat aa wl se TT THA | orefa 5 at cart & tat oe aa etm fe facet sta (S, ABC) site (S, A'B'G') % oars ate atte feos ate, wae A, B, CH aH acrae & Teg at aTet BT A (S, ABC) F avatag (Anticlock-wise) site (S, A'B'C’) ¥ efirearat (Clock-wise) | safe ea Prat rat Saae ak fers ate sale ceraaT sat ETSY CH TANT aT H aca wa cafe a faa sty ofr } agar ada vetare snfafr 9 @ lea ware % A faa Hat Mt eafea (Symmetrical) Praoa tre rete | - . 13.3 FAT2 afe om fraw ater (S, ABC) & gaa ata, ZASB, / BSC, LCSA, am: Ft frat sq (S’, A’B'C’) F oes Fa, LAS'B, LBSC' aR (OSA Bawa & A say for are & feo sty vt gat F daa fea argh F aera AAT | Ss s > fam & few ata (S, ABC) site (S', A'B‘C') & ae arr arm: Tat F twats, Z ASB= 7 A'S'B’, 2 BSC= ZB'S'C’ a LCSA=/C'S'A faa wen g—feat sta (B-SA-C)=(B'S'A'-C’) (GSB-A) =(C'-S'B'-A’) atk (A-SC-B) = (A'-S'C'-B’) war —SA ie SUA! Br TA TT: SX ste SX! aera aT at art |“10 ater Pate { ara ASB site ASC #, XY WIR XZ wa: SAR ara ey ait Lett sare ante A'S'B! site A'S'C'H, X'Y" aie XZ! arm: S'A! & araeq alst | sata, LSXV= /S'X'Y' Sree aaa ak LSXZ= L8'KZ’ “2 gate arate srdar Gaal ar, waar at ZYXZ ak XZ! aaa: faaa ary (B-SA-C) siz (B'-S'A-C’) & art F 1 wa, Y, Zatz Y', 2 facgsit Bi frerstt | “saafa—frgal & freafafad ote aeaat & ager fee fee ot aaa gt ASXY=AS'X'Y’ — [afras, Book 1-26] ASKS RZ (7) “ StS seae atte is see @ ¥z=x'2" ASYZ=AS'Y'Z — fafers, Book 1-4] a. rZ="'2' 2. Q (1) aR (2) 8, A-XYZ=A X'Y' Z' [afrre, Book I-18] a ZL YXZ= 7 V'X'2Z' - fae ery (B-SA-C)= face ater (B'.S’A’-C’) Bawa Fe faa stat at wt aca< fag feat at asad | [ate go] samy sta 2% at fraw arate aie ger saw wal at ager (Sequence), feat us & aaa wrt a, Asa asa faa ag adores a | aafa,ore sanfrfat 11 afe ca fram Sho Ss ceH sty, UH TAT mH, ae at frost ate F aaa sats, vet wate ar HF aerate at ata fae arg adatom at | : waa 3 faa ator & ME Wha Gow Sli eT ait atat Sow eT a agt ater at [fecret—afe frow str & dh cee sho owere TUT eat srr ga aan F Rreete 2 | oro: a os det fae sty at freer ow sae BT fe Aa as ray & wer 2 1] Reo arefe 7 feat &—fraat star (S, XYZ) ii, LXSY> / XSZ a, LXSY> ZS fag vor g— LASZ+L 28> LXSY12 atte Fratarfirfer AT OX sk OY Bet oe wag: As BRE fry at ate ere ASB & aaae ¥, LASD= £ XSZ at waar we | ma BR SL 7 fay C xa ware att fe, SC=SD fg Cat Ack Ba fram | wanfer A ASD== A ASC [afave, 1-4] “ AC=AD ee. "A ABCH, AC+BC> AB Tey AB=AD+DB =AC+DB (1) oe AC+BC> AB=AC+DB a, BC> DB » es Q) wa, A CSB aA DSB, SC=SD Strat SB arate . itz, BC> BD “"(2) ZCSB> / DSB 2. 8) . [afeere, 1-24 facia] si, ZLASD= ZASC . . . (4) (3) ar (4) &, , ZLASC+ZCSB> / ASD-+-/ DSBate safiate 13 m, ZASC+ ZCSB> / ASB m, LXSZ4+/28Y> XST (ate Bo) saa 4 feat free stot} atet Gee stot eT att ada 360° 8 wa Cr mafia 8 feat {—faoe ate (5,472). fea wert {sas Tat aT ar 360° F aH FI wah, L&SY+ LZ USZ+LXSZ < 360°. want—SX, SY ste SZ art oe AT: A, Bike Come farg fat atx oe otra # fire feat | waafa—eaat aren ara safe a er an fraet arot (S, ABQLAA, SBC), (B, SCA) ai (C, SAB) ey Hq aga g | wa wie 3 B, : (A, SBC) #, /SAB+ /SAC> / BAC (B, SCA) %, ZSBA-+ / SBC> “atc qa (C, SAB) #, ZSCA+-ZSCB> Z ACB14 aitetter Rrptafinfar Fras ata (S, ABC) % att ware, we SF are de frga aot & ite er at Fgh & sie: att att 3 x 180°= 540° | wate, LASB+ £ BSC+ / CSA-+(ZSAB+ / SAC) ACLSBA+ 2 SBC)-+(/ SCA+-/ SCB)= 540° wy (1) &, LASB+ £ BSC+ /CSA-+(/ BAC+ ZABC+ ZACB) <540° rE A ABCH, © Z BAC+- ZABGC+ ZACB= 180° Z ASB+ / BSC+ / CSA <540°— 180° < 360° a, ZXSY+ £ WSZ+ L ZX < 360° (ate ge) wet aug 1 . 1, fare ardt fr avaraa: dt amet on fargo fra aa at ait feafeat at aaa a | . 2. fire wa fis ante (Space) % feu Praft farg & ada on tet ara tar Stet ot aadt dat at at ge fauna (Skew) avait 3 sete at ated 21 3. fret fg S& de aa Rand SX, SV ate SZ sit we Rew et amar t awe) ak ow stk ae ta SO ware a arr fiz ag SX, SY wR SZ are fafaa free ary (S, AYR) % seat & ot Frew Pe (@) LXS8O+ 218042 280>4 (7 XS + LYSZ+ 2 2SX) @) LXSO+ 72 280< 2 XSY+/ Z8F @) LXSO+ 2 Y80+/ 280< 2 XSY4+7 YSZ+ , LRSKara satiate 15 4, frdt fag AS oefe Hania agar, a sheet aaah a AP a AQ wea ae | fre ed fe aaa ® ate ar fare Sty at at weal & ate R ste F acae M1 aT sear QTE (Supplementary) ere grt |. atata 9 5. ary afet arart fe aa ae (Inclined plane) 9 fea frat farg & HEat-ata-tat (Line of greatest slope) (sate ag ava tar ot aft anae F ave aad ast ate aT 8) Ha date Tsareata 2 Metta sarfata _ (Spherical Geometry) QL. fare sare are oe eet TE Tare site orafeat (Ga Pras, qag's ate) s guaat at aeaat ace senfafe F eT SH eet wT H FT THE sat wart Ga ate afer ox At a§ Rana aie areal et aerat eH ater ounfits Banta eth | afonat 8 wa UH Taqa, TAR sae at quia (afepaat) at (Axis of revolu- tion) HTTHE, GIS WT ae ofewAT Hea g aT EAT TTT fre Ste at faater dat 3 3% Tor (Sphere) Fer FI arafa 10 aw at amiga APB (aeft 10), ae AB at ofemat azar a aa aa at af ofefe APB a % afeera at fang wet f | sateaieter omnfttar 17 wa ad afer om got ofemn seat & at fire cfomra-ge (Surface of revolution) @t fraty gate ag Wa wt afgers seer 8, fore ay feoa sete fag ae H Gaz Saat TA ge Mag | sa: MI eT afgens aafee (Space) Hag farganqe 2 foreat wette fig Ue eax farg 8 ada oH aT et Te ERT 2 feaz farg a A HT eq Ate eat ge aT ae a Pro wee SL ae ar ore va a Pegs ate at gt a faa eB aac TT anit ag aca Var aes afeera at Bacto) at: TA FB aver ong dast & Teg saat ware sar (from at gt) are! araraa: “ater Her F gare areas sat ageaer F WAT | 22 wet FT THAT TSE (Plane Section of a Sphere) . sriq 5 ” TS wt Tete eHAT atest (mm) sean ar a arefa—11 ue a, ance ABC, va ate Ft ateat @ feraat Ae Os ae fraat fast 2 | ata a, aaa oiese am ax A aE ees fag 2 | ax fg OR amaa ABC oe ra erat fraat gt ON=p 1 OA ste AN at frat! 2 Toe fae18 atrettar frratrayfirfer ffs ON, ana ABC a2 wea & o , ON LNA a ZLONA=90° wa, fran OAN %, : LONA=90° NA? = 042 ON? “2 tani eer 7, NA? = 1? p? a, NA=V/(?—#*) =Ue Wat Ug, Sora p wR Et afr farg A amare fete am ae oH ase firg 8 ma feg Aat sae feof S faq seis ware at 2) ae: aaa festa ow aa 2 frat Fg WV 2 ate Past (7267) 8 a WS & anaes afeest my _ (ate at farsaTt)2@— { at fasar (ai & tex & aes ae wea)? 221 ae ae ay aT oe TT BY PEA aT THA TA H Ae B gw ora 2 at ae were FY anata ofeese aa aatat 8 ag atdga sama 2 | wha 5 afe po &, aaa TAF Hy a ange st oer gd ya a at afesda ow get dtr faraat fasat y/(r?—0%)= 7, ae at fava. F aerat Bait site Beg aA aT A Fey at | aa: Shae st feat aT at fasar & acrac ait {1 gat wit ance ofeese agra wera E waif gaat Prom, /(r?—p?) <7, Ae at Broan & aH et waft 10 % MM’ Waa e dhiae site ABC gaa tt ere & fr, (1) ww & Te oe AF wa Age wrt aerae BI FE aie saat froar aie at farsa B ger at 2 |atreter sertate 19 (2) ww gt Ae oe AT aa a he ae aH ea Rega ax qeert anfaatiore gta & 1 [aver (2°8), saa-8 HF gaat gerat carts at Teg 1] 22.2 Wea i feta a fag ame ot feae at fagat at em aden ag acl ah are fret aaa Satie ga at faegal a dee seer ance ae oT ase 8 oT aie at fafire ay get H aret site fed eu fry oa ay adit ox fae aft | reg fea ge at fag ake AH Seg T Gee or TT OH ak daa wm dt ance atat ot aaa 8 at ofearat & agar We at aan diga ¥ wet) ae: ate ae fern tt farget wt wea ot fargat & dere art TR atiqa B are are gt frend Fr LO Ap Zs , atafa 12 22.3 sfeeqratfas fareg (Antipodal points) (a) aie & fret care % ote sfrearaifrs faeg seat 2 I sfrge fag & Ae a ae shige GT oT wae EI : TE - (@) AY ox fears A faegui & oft sfeerraifrw farg at es we ae Fae oH dteige afar oT aaa 2 |20 after Paty fife sfroovattins farguit & ghee ant aie weir dhiga @ aad facie vet faegatt at rere art weer Rar TA at ow sata ait g fare 7x EN Sy fa-gA ste B se otc Wg B cer srk Far oe Ota dae wee OP. actas ga SaAse Buea fay rete apf 13 (A) srefr 13 (B) afr at Sex Mt fea eat 8 | we: aferoeratfire fag site ae aT HT wa gt aeat tat eaTa) Te feos FTI we ae tara weer TAIT Pa at ase e frat wate at Ma ce cheese CH Shige eT | ara: sfroarattre fargat & die aT Te eT deat at frat fem ar asat 2, ta—arefe 13 (A) # caterer ge) Teg ate a oe feaa aE aT fae sfrorraifee fag vate a aah S ewe oT aren aaa om a ae ae af are 2 1 Fa, wet 13 (B)H A sie Ba fag ét A shrorraifor farg. (Pret cara ® wiz) wat ZA we Biagit Bawc at Ta age FT anata Fa, O Ft ariert | Teg A, Bate O, (sit ow area Ren Haat @) civ fargst & dec ot Tra ow Whe Fae OH et TATA gt anatg awa: Aw BR ot ae Fae ww ot Sh ea ATE 224 wa ox fea A faegetl at frat area TET ama ane re fee fenret &t Farge at feria ate waa ATT ag EAT faaat waar eqan grit | gat weal Hae se ge HT aTT eh Fraatsiete oat - a fasat afer at | er wree & fie ae oe feast eet F aftr fasat ad ge at Dats TT Te st faeat F rcraz set Bl ga: MA oe feast at fargat at frat aren ama, oa feegat & dex ort are ah ae at art sar I (i) af & farg sfrooreifre farg agf fav a fg de ae at at aaa weit F fries wee g feat ster wer aya ate wet amt dtd-are agerat 2 | taf 13 (B) ¥ Asie Bat fant arr aQeant ag Vat are euiat ware sie AM NB, att ary aufat 1 (ii) af 3 fry sftomaifes feg Fa Sed ae a aafoniee awa el arava: Be gare Hwa oe fea at fared at free aver TT & gant ceed, aa, dat B ag-ara & VT | 225 tet gx fea at fargail & tw at gat sata xa (Spherical Distance) ae oe fea St faegall Bate a aeta ger aaa fen ae agara at ararg at 21 Fa, weft 13 (B) HA ae B fast & abe at mei ga, aa AB at aearg B acaz 2 | Brita gv (Angular Distance) ated oe feet faegait at Prone are are gre ae HE TT ay arat & gaat are fargat F dry at atalia go seaet F 1 TI spf 13 (B) HA site B fargit & ata at staita ga, Tq AOB eat ée 23 Wat ot feam Tat F set (axis) AX 7 (Poles) afcrrat 9 areata 1 faa aett aot (og oT Ste) wr er, wR aT ag SATE dar 2 ot ae Fanaa a aa TI22 ater Fretteafrfr Sree oe Fere neh ger aot oer, we at fort St forge HF wrect at qas ya ert gl : arate 14 arafa ¥, care PP’ dtd ga AB aie ayaa A’B’, AY BY" at swatres met | get ware faeg Pte Po ea ait acl 8 saa waél are & fe (@) fret ger site ate & arg at frat are azar ta, ag aT wet eit 21 : Gi) feat ge} qa 8 sah aaae oe sre aT ae TT & dat ga S| (ii) ga oat get at oat um Gt ata 2 fas ara TART AT Zl (iv) ayaa qa gat area a aaa gt ATs TT are a Mt dae ga ae ge ave a gee 7a FATT | (v) ae oz fera sete sfrorraiftre faegut at sitet Fret ge aya ate &, farat frat ater sate sae get eT wer stat SIattra sorter 23 231 ga k gm at 1, at ve feew ft gu (og re) at a, ae at afer az feaa sete fies & aaa ga ae sats | arate 15 ara a fg P, wg aa AB a wm 4a el aa at ee OO’ Bl ad at ofeft 7¢ D ua eee farg (Arbitrary point) & | O'D at fravt | aa ff wea PP! gate PO’ act % anaa 1 ara 8 He: AE BA TATA 5 Feet sees eet ete ae | PO'LO'D APO! D 4, weraria ¥ ater &, PD? = PO"+0'D? TL PO =a % ara For at oe =Us wae fg (fer eT AY Ta F fre) ak, O'D=aea a Fasat =UR wae TT PD=% wat it {fe D, ae at afer oe ow eee fang 8 wafe D att wet feof & fat PD xt um waz ute att |24 afters Pasiterfifer wa Pitt D feet & Gar wa aa daa, PDP’ et watt wa] arr PD, sferen (§ 2°2.5) F agar wa Psite frg Di ata at ater get & | fe stat (Chord) PD ue wax ofa & safe aq (Arc) PD at avant at wax 3 | wate, gaat offs oz feat sete farg at, sah ya 8 atte gt aa ae | ate dfs aa get } at ware Fare, at Be Te AAT ary atafea aed & garter faeg D at (gat at ofefa ox) mea fafa ¥, ZPOD=¢=79 uta, wats, ae at ofefa oz fear sete fag at sam ofa a atte ett gar eet 1 : 2. Te Te fea Feet Seige F eps ate saat ofehir oe Hs fe farg & ata ate gat oe again eet z | am, Feat ae pe a wet afer Ft arg at ret are at aang os agate att gt avatar 16 art at P, dhiga ABC at a ate arg O ez 2 | fig A oe ABC at ofeft 1x om eee farg 2) Paite A fergsit & dex ot arn aiga APB dist |attra sentir 25 fag exar 2 fs ae PA ow agate # avis 2 | farg O ahige 31 ag faa: ae TA ST ft aeq 2 ate PO Rat aa ABC wt a & | fe PO, a ABC % ana Te os EI LPOA=7/2 +2 eat OA, aa F aaa F feast Vt amt PA=wa agatz | a4 AT ae TAH a A aeafeaT TAT fava sive at oder we cata 2 fs fier Faas afefeaferat & ater ax feaat oar fag Fret fat ge any aT act at Ae at THT g | saa 6 afe me ox eae faci fag At atta ga, ae a Farge &, sit afromraifie fang vat 8, ow aqui Facet a dt ae fg TA fargatt at Frere ave are a gee grwe THY ATA Sta wT wa gate | am at fag C at, Ast B fagat a, ete a again * aware 2 | sata, . wet gt CA=cs agate at, LCOA=n/2 ai, atta gat CB=us agate a, LCOB=7/2. fag were fr, fig C, fg A ate Ba Sta a TA aia at aaa! .26 after fararea fire fe 7COA ak LCOB sete aeraiey % azraz & re: oc, OA wire OB 1, sab wen fag O oe, wr Ewa 18 OC, OA ate OB & Sere oF art arae ae Vt ATE | dfs ag ance Te % aq 8 dae om 2 cafe east afesde, aaa ABD 21 aa: OC aga ABD a1 ae 8 site feg C sam oH ga 8 | sia 7 a 7 A fire fry B af Stare, aft Fel cw a Shige Baa a ai, wr rare & AS oT ae fe sak anae feet RE gt aa (7g aT ate) & arrest ve area ai ot ag fag fat ge ga aT Oe a aT atafa 18 at ot aie oz feaa Bes fg PS ATA a PQ site PR seam: atiga PQP’ ote PRP’ % aa | fem & fe ga aril at we aad Fanaa, agsa ABC Fanaa ae cea 2 gata eat amit & dhiget at afasee-ter, PP’ vt aa ABC & anaa ox aa @rt (atk afe PP’ itt & ez a dar out Baik ae ABC oz ara? gam aa ABC & tag Bit dae oP! wa: PP’ aa ABC ah set Ren & site farg P sae om ra 2 |aPrettar sanfirfer 27 5 TITS Ft ares ate HAT farsat afcarat 10 feat agact exe wile saat ofefer oe & Pareht fag at atetrr ot sraira go Ht AM: Saat Tete ost sreta Pasa Ter S| gat wed , fret ayaa at aieta fron, cet ya ak saat afefr ox % fart farg & dae oa are Stiga at eat farguy at fray are att at aang att 21 ere & fr, sete abige at ater fase om agate} acaz art 21 aefa 15%, am PD agqaa ABD at ata Pasa sie ate POD=$¢ saat Frater fasat eatat ze | 2-6 fed ad (Secondary Circles) afearar 11 ale oz fea fret gat (ag at ate) & eqat & glee a ara aT fea gu ga an ates agerat 8, Ga mafe-15 Fad PDP’, ag aa ADB ar facta & | gat weal %, feat ga aa F wet & Mat TTT, are amar arr ate at festa, fat ge ae at factor ae datz | aa: eqe 2 fe fadtas acl Hance fer ga a, fk Ae aa (Primary circle) WY #@o &; ¥ TTA TX aT wT Ae S| 26.1 fadtrn get & ZT 1. mete factors ger oe dtdge det @, wife ag fer ge ae ge & gat & a aT eI 2. un Ae aa F aeter fadtae ae et wae atife qa at a peat a rere arett Terr aA aT TH ore aah & Frae See oF aR ana HA ot aad B faras oieese aeET arize at |28 ania Frsterfifer 3. & dhiget & gare afe ara wr af ata TTT UH GAT 3% factors ght 8, at ow ett ae ga ae fetta aeii a waaay 1 Heat & (Gaia say TAIT ara |T ATF )I ifs Gat fea gdke dhe at met gat hig & aaa # fera dt, gated 3 ow gat & fattaw got ~ 4. > & dhiadt a1 tae ce gaafas faces ate 2 aaife at atdaet 8 at well (AA Hae oe ew GET A aeT amt dea &) & amt ae sat oH a anat AL avant & frrat a & ofeede oie fatter ert | 5. afe qa qa ga (ag aaat ae) & ah fadtaw Peet fag PR dar we A ag fg Ga aa aT Te As ata, atte qe aa at net dal fatal Fanaa F faa gaat art factrs aa ge gas ot F ered Ee: fag P Wi aa ot fers 2 aie cafe ag qa ae aT Oe TL 2-7 Weita FT (Spherical Angle) ferret 12 meat fert at yfaeBel-arai (Intersecting Arcs) are gay Up. CY P atefe 19 . afaste-farg ox faite Ft at atta weet 2 | sfreBet ar, weiter wey at gon seedt F ga waesefarg we seta 2 | Saera sonfirfer 29 orpfa ¥ ata QPR, PQ site PR anata fafa aera st aafat 21 fag P saa aie got ary PQ aa PR god | 27a Tia BT aT ATT arate stay A saat gos B acai ever fafa feat F aaa & | sefel9 Ham Mt, a ABC, wa PQ we PR & ated at suafaes fadtas & 1 wa: at ABC a anam, aT PQ wie PRE aaa 1 wea ST S| aT aT am PQ te PR, wah Sadi a gel 23 oz, at ABC at A site B fargut wred & | ate at QPR, saat garit (are PQ ste PR) & erat art fafaa fear ata QOP-R a A-PP'-B & quar % | Teg ataty A-PP’-B at ance Fre AOB art art stat 21 (12.1) ste anata AOB at am AB tt wang & vat H amas F | ar atera str at, cast opreil e dhigd? ara, sae caafne falas ox atiafer arg ate Aaa F a, Fa ATG, St AMAT SIT TS ee ee a, siete ster QPR st saat gait (art PQ sk PRE aed PQP’. site PRP! sre wah swafass fattas, at ABC a arafer amt AB ara at, 7 AOB art aaa 2 | 27.2 qaae enfats Fat aati ar ofreBersry (Angle of Intersection) ga STR & ofearfte Feataat B—at wat wt Hfresea- Sto, or awl wea farg ae F erat B ate or sto gta Be aet att am os gt ance F a F ake eafae sah cael ht at anaa H fea gta Fl art sta (sfearat-12, are 2-7) ¥ saat yori & arr ca gt aaa F era ag a cog ae wea fag oT BA erat a Meee ot ae aaae F oat eral F abe a ater ATT OT THAT Fb wet Ere seer ag fee wears fe at are ay ee TEN om) nkesbenter re weere ree '30 ater Prorat avefa-19 ¥, PQ wk PRA sfeeeet am Fla wt, Fa | (afe araermat & cheat dig at eter ox) wae scarfs fads aa, ABC a A site B farsi ¥ area sie ara at PM site PN, 3a arat , AH seat farg oz, erat Ted Z| fe, PM ate PN at mam: aa PA ie PBR aaa #%, OP x wad wat OP sci at suas fsa aat anaei at sfaeear-tear 2 | aa: ere tani Fale at ata aa PA site PB Banat F dee sty F ea St veg at PA she PBR anadt & ata at ata, frawarg, A-PP'-B, at amas | Aa amt & eqyit & ate at atu, feat ete A-PP'-B & gga @ sit anit ane fafa atte ate att ated | wate amit at sfreterate aie sat aver fafa ate ate, art ww a EI fina H, er frrafafaa sfegrat at aaa 81 a sfeeeet arat aver fafae ate ater (&) sat dhaett & enact art fafa Gara F acae ware (8) sat sfaede frg ox sae erat F ate F ah aeraT Sat 21 (a) sfaeeeh atet & ata arr, saw Tras fetes gz aiafer are arer aTaT ot aaT BI 28 @ ahigai at sfaeder wiz 8 Me ot fera at dhigqa cs gat et anfawfrs wea Ft aa ana 2 fie Ta ax feos cee Shae ST ama Te HA & dae sat 2 | wa. fea ga Sa Stat F ane Wa HA HTapteite setter: 31 Saez Te E aa: Tat afroBet Vat At At BRA gee oe atsfa 20 sata ag aa at ee sare abit ate ave at wee Shae ar at eer ett | aa: 8 ow gat at aafefrs SUT | sig 9 . Rt ahiget & seeder fig, on Tet Bata gee at Te dhige & ga dag32 attettat Praregfarft art #, O wa at tee Ram PD oe PE & tiga Ot qeere 0 farg 7¢ PO at PP! tan ow gat at wed eet A HK Bwam: aga PD si PE ® qr 8 fag A ate BU AK ara ateiger effet | fre wert & fe fang Pad AB oem aa eI ifs fag A, daa PD a qa et OAOP 2. GY ~ opeiga PDR ewnt tl eat sare, fn farg B, dtiga PE wt e “ OB OP Le. QD we OP dvige PE % ena #4 (ae area 8, vile aiife tet OP, PD aie PE ae at wfeesertat ®1) (2), (2) ate sR] 3, ger tar OP, OA ste OB Ravi #8 Gre mt aa ane (Grom a aes, aga AB a) 1 TEI OP, eriqa AB at 7H | seg P, hig AB sto oe 81 mere Fre st ae & fs quer afee’er fig P’ dhiga AB ot geet 47 eT! saa 10 att oe fea a vfereBet ara & ate ar ater ster sat aT ay fren ant ate are Bex TT AAT BIT B ger der’ | gat eat H, at oat B ate at state A Ta aaadt & sfeeser st Raat ASI - argfe-21 #, aH at eae AB, sfireeet ga PD te PE at M ate N fargait & erent 2 | sta 9-H ent fers Frat 2 fe farg P ga AB wt om 4a g | Aa aa PD wk PE, aa AB & fadtaw aaaieita eniPaf : 33 gt | sae ate or ete sty sah are get AB ot wet wa a MN aver eg 9% ffir ara ster MON % gra att (§ 2°7.1) 1 aa, , am AM=qmagiin *: Aga PD am waz) aa see, amt BN=ca agate AB=AM—BM =BN—BM AM=BN =MN sata, att AB ste MN om gat & aca | te fe 8 oe a aT AB % arr & gah ez at atrax sty aiafer Fw | Tala, LAOB= 2 MON ae: sfrosat Aqat & ata we ste sas seat fort Te are are Bex re artearfcr HTH ger grat Ft onde sia at soafea Ft at freafafiad free at ava dare: sete : a fadtas adi & dra a ae ata, (sfeeBer Sty) aay art Be sat Te He ATT are Fez IT fafae sg ges Mat 2 | 2-9 %$% (LUNE) fedt ance # faa aa tar ga aaae a at weit F fens weet 21 a afaeeat ace Vand anaes ae ATT H fare weet F 1 ue sfreeet ana ater e afgere FT at aM A fits sxe 2 | afe ana, we & Gey A eee Tats at ae MA st ow Sige F areat 21 wa: uw atta we & afgeae at at aeat anti F fern 3 te fro34 afreter Prrdteyhafi. RATT | EAT HTT ATA Te a ahereved Sigel a mera eel | aah a eqs g fie ga feafa F Me at afgeaa are ATi H fafa ata 2 I @ (iy arafa 22 afearat 13 aie % afgera at ag air ot dt aed dhiadi & ata feast dar 2, gm (Lune) agaran é | airefa 22 (i) sie (ii) i ddga PBP’ sie PAP’ % da feat afeeat PAP'BP om equ aia 2 | ere 2 fe 2 sfeeBet oe ae & afeera ot are ggat at waar aed E | mals 22 (ii) ¥ dae PAP'C site PBP’D, ait ot are eg (i) PAP’BP, (2) PBP’ CP, (3) PGP'DP atx (4) PDP’AP, a frata axa 81 fag Pate P’ arai igat & aaftes farg 8, wd Egat F ete sea 21 29.1 ESE BT FIT Sea gar at ad dhigal & dba an aeie ate, ige at ate HeeTaT & | ease & fe, (i) egw H aeeit eat oz & ary aca HI ez) (i) ow & ate (at atrat Wet) 9< fener at ign adivaa ae & afe was Sie TUT a, aaiife ga fea HF ow Tat F aan a wy F1areter sorter 35. 29.2 35H BT AawT om et ate ee (at aerae Tel ot) faa Ft egal F Geraa ga arat % oferat aarginit ae = (gfers, IV, 1) 1 aR ge aferaa 1 vit an oH ige aM ase & freer tO ae TART aT Qe Hawa Z | Hr a ga F sty Hr ada ant 6 8, oa age a aaa _ [ea ae law ives axes | Qn | m, epee 1, WR fro ign oer = 267? x 47? 210 wy ate Shiga F aaa ars aM at, ab aged eae) aya TT EC aaa Pe) amt at O att at arg 2, P farg & dhisea Pad six POB dist ot aa P & dhiga at A ote B fargat H wet & | Ca, Cb, OA wk OB a ferent | aa dfs OP, anae aC ate AOB az ara 8 Ha:36 arate Paareafafea gt aaaat F feat Ca, Ch,OA aie OB ae Wt ara S Lae: THAT Lach st LAOB, avi gt arat POA tx POBR AAR feast ar are eMia &, cafe J reese qUAT F | Ba: ZaCbh=/ AOB wa, aca fratufafe &, LaCh= an AB fasrt0A amt ab fasat Ca amt ab _ fasat Ca amt AB fast OA _ fasat Ca © fasat Oa =sin COa =sin (7/2—4), g=at Aa =cos¢ . ab = AB cos $ ak, AB=ab sec ¢. = ZAOB= *' OA=0aa 3 Metta Pays (SPHERICAL TRIANGLE) 3-1, feet aarere H fora, dra Vat eat SF fa Aa a TAIT Page (Plane Triangle) a1 Fax fags tet & | ae wo reat wel ot aapat & fe cree H fore ata fargat FY, we at a tat ae agi 8; Rat Set aren Prot oz at stata Tact & we frgx set & | eet care, afearar 14 ad & afeeder oz feaa, da ahaa adhe fat ar at distr fang (Spherical triangle) aga € | gat areal Fe Ba axe eae PRAT ar amet & fie ata oz fea cla feegal Gt ow gt aga a feat at & ) a Shdge avet aver frat oe a oft aah f ST TT frga wea @ 1 : . fargat @ fret ard are. rg at gang site aaa Bate ata shy, frye F sty car ara Fares ferg, Pargs ats were EL fecquitatt ox fea a fags a, sit afeearaife farg vel g, am gt Shige & deat (ag ars ae Sth ara) aver firart He E | a A,B,C fargat a Stiga aie aver frat Te ore Tefal aT Frat dard seg af fsa A Fae Shige B ag a arc fret oa at a feat H Fae oH Mette Pras at fray et | meets Te F AGA, Tt ox feast, deal F ate wy arit aret fat ae a dt atte Frage ofonfer feat oat fe38 asta Pratt afore afer vat dae oF attr fags ofemtra at | weara 2 at aver (2°2.4) # ag eqeea: art mat & fe, tt oz fear 3 fegett at front are are om areal, set gsi B dree oy aT, aga & ag am 32) aa: srg'aa Tfomet A a dice F ea wor & fire aaa 8— ith oe fears, ate ah grat fat az st ately Frys eee Ey : srrefe 24 ; arate 8A, B vite C fargat a4 front ara atthe Sys ABC srt art eaia wag iatt AB, BC atk CA atta Pras at garg, aetha ster ABC, BCA sie CAB aa sty oan fry A, B ae C sat wit fag €) wafer afer & agar, atte Pras atu sito are at aint & A, B,C mect aret site st aa at wae qad Hag: a, b, ¢ met ret cate onedt € 1 ga scare att amd (a, 5, ¢) ste at ate (4, B, C) frorere after rg % w: aaa (Elements) & | ans frye mupfa & ere ¢ fe C atx B fare at, oat Bae oA aa Se qa KK’ ea ani, CB ak CRE'B arn fram ot aaa 81atta Fras 39 ga: aM a, Beste C frgat at A fry & reat at at et a arefaat (i) ABC (wre avert frast) site (ii) ABK’ KC sree att | ce are, Aga Ho AS a Herter ay site A are ae afgeder 1% Het Frerrnre wre srTafeet at TaAT NSE {@@t are1(3°6)} | aq gern %, Pras ABC & gat eed sa faye & eT feat anil & eo H dhiga F ay a7 g, gatg write frat at aad ae qa & Stet get Ft 3-2. atta Pras at yoni & water stg er chert & Prrafafen meeage afer eefirr Frat ot arate | saa 11 aeta faye ar rete sor at arate & alee gat Zt A F B b savayfa-25 art at, aa AC, BC ste amt ADB Pras ABC & yard & frat ato C at anaia & ast 2 (arate #, 7C=/1>7)1 arr BC #1 atdaa gal att at aad AB at D fag oe area 2 | siz 8 &, dt dhiga aah sfeeee fargel ce ow gat at aafeatioe awa tian daa ADB ok BCD wrt a Bak D frgat cx anfamina at eafee am BD wiga * gz! tre ¢ fe at ADB, wt fet gu frags at oe Yor s, oe at F TeT40 rete: frei . & veg ag aena agt & vite rahe afeaet & SIAR The rT agora a wR het EL aire ag sew i set Brae BT ate sta at anaty & agra, wafer sheet at fade ate aa: archer Pgs wr adie sty et easter 8 set eat tL 33 ate frye att fara tr smpfe 24 8 afe att & aq at ait Gege & ete Regal & farerrat aa at aaa OAB, OBC ak OAC, farg O 1% uw frat re (0, ABC) wr frakey ae | Frat see (O, ABC) site Pras ABC wm gat & sink er F aeafa & | ae, ate Page ABC Ht gavage ary AB, BC sie CA are eat & 1 0g, aaa sty AOB= SAB fasat OA , am BC galt Taz, anat LBOC= oR amt CA ar wart Z COA= aa qq OA=O0B=0C, aa amt AB, BC, st CA mam: wart ra AOB, BOC site COU % aang & | arte sfuurr &, atetta firys at gornt (4B, BC six CA) at BAe sr Ae oe falar anat stat (ZAOB, / BOC siz ZCOA) are vt ant at E 1 are (13.1) Fa sty, faa stg (0, ABC) & gars ate oferta fet wa & | ga: 1. feet afefta frye at gad, ae daa fae ay oe atot & aargret Met F wa: ate Pryor gt gon a sail at ANAT TT THATS | 2. fra ABC % ay, art (1:3.1) % agar, foe Fro (0,ABC) % fern tai & quay 2, @—aty A fara sto (B-OA-C) & aeraz 8; eee |atest Prger 41 3. art (13.1) a a ae 3 fe, Cefegaeh a aaaa, arte % are Preven ated ar Fratey ea St wre. Pare at H eit aT fog neat AE ter Stet ar aT A ts Paar MT AAA A aT TK are svete (frye) at Frater att | gta foe aT He et ara sat sa frat ary oe Bga wer & frat caw we fart atait Bete 180° & wer a aver (1°3.1) et feoraht Hae bet ge & Ge he oz fea ata Farge A Sige ae acer frat Bat a ams avefeat (frys) fradt 1 veg afe fag at dae F ag sare aver fiver ora at Baer oa aver Prget sree slat faa: TT qe ofeariet acta frga ot tra Prat sty ag stag free Fae ary vite fara sry 180° Baa eI . 33.1 ateite faga sie Prat ate F saeias aearel B, arette frost & freafafian ge awit at serae Prot stat & fe vera 1, amet (1°3.3) # earfirr gu afeftorat at aerrat & eeerangt Fea oT ara sl sae aiette-Brga F (faraat gored site ater 180° & ar g)— 1 fag St germit a ate dterat got & agr ar 8, wife Fre Bry H a vam steel ar ate drat were ster ae At F (AAT 3) | 2, tdi gait arate ada eget a afele (are euate) & am Star 8; waitin Pere Ste aT pe ThE a atte aaa TT TA arg & #m aa 2 (sa 4) | asta Farge & eae gat first 3-4 0 2. @ ate fc td Progr at wet er @ eater Be GT . A+B+C <3 +s (ii) @ ae Gi) a, ac (1) (a—A)--(n9—B) > (an —C) * 4 13 RP afrert a, A+B /B, at fee wor 2 fe gat BC>3a AC | staf 30 AB gn %A fg 1, ABR awa, (BAD H wT amt | aa, AD+DC>AC -5 amet (3°8), (1) AD=BD “7 aver (3'8.2), (2) “ BD+DC>AC at, BC> AC 4, afe Meta ras at oe yor gad yar ag at ast aT a aeqa wry at sal yo F aera Flr A ast eT art at, faa ABC # A, B, C aie a, b,c maT: sae at ait yond Zz 1 art at a>b, at fag wet & fr, LA> LB amt at fags A’B‘C’, frys ABC at aata fargo & frat A’, BY, C', sae ate ae a’, b,c! saat gag & | aa, a>b . fee 7 a—aBweary 4 Tita Frys Bt yore ake stil Fareturfarcite aeaeer 41.0 aha frase & aafes aie ot satiate ofa fret wear eorfia fet oT aed a, Te_ ga veer Far Witt varfafa & afefaar gist wa: qater faaea at aeqaa oats 2 | ara &, fra sar aaae satfefs & afer ate ANI wet at ae fantafafa & gat at agra & aera eer ee aaa E set THT weiter sorfafs seid at aferrd ge at F vice fata (Spherical Trigonometry) age fag att 2 | wa: Ba ETT Her aTiita Pretafafa at aera ye att | aaa sorta site Paarafafe at soatfirat ar era oe attire eer & | areaa H fra qeedt ot ae TANS ae UH TATA aT: yet oz ar aeft afefafedt aie gas wile & aeqaa Fateh sofafea ake faahufafe or a salt date | aeatert (Navigation), aatw-faart (Astronomy), geaife F aefta faarofis ar soit fate oT dare afar % efaera a, faarr ate aster fecrteae (150 B.C.) (Hipparchus) at ict Prptafafe at seaatat ata 2 | 41.1. feed weita frye Ho: saa ae g, tha yor sik ht atu, fire gat fret aera t wat: a, b, att A, B, C ret are cata 2 | af fas F eto: caval a att at are at at er ae and & fe Pras } ears F (sak ae, farsa eee) vt geaiitr faye % Perrot ar 55 amt 2 | araraa: ae et eto: cama FS ae At saaa ate a @ aa da waadiat ae aise fretofafe at aga & oa at avd | ga aera H gant seer TT fas F gaat Fda fafr ftofinta wart enfis ae 8 A frat F fake (wert 5) % saat fre att wl ware a en A ger dal F faarita ax aeT ze) war az H, fags} are waaal & ara aie facts de H aia ae ow: aaa F ara faratafadta aeare cafe BT | aaa ae 420 ax aqaal & arafead qa area faa % wera HY an waa fea wa A GIST aaa & | - (a) da yond ate ce aT I (a) ata ata ak oe ATI (@) tang ak wat eye aig | (@) @ yond; sat ate a ata a oH a AT, aT @ ary, ae Yar wT Te AR ASTI 42.1 aa yore att om arr Ft faatrfita aaa —alarat aa (COSINE FORMULA) aie frst & HR str e stares St saat yout F eet atte stares % oat Fi oewe wea am at ABC ue attr fram 3 fraat qari BC, CA ok AB & @aita art HAM: a, b, c Ft wate anaet ar AOC=b BOC=a AOB=c56 So aftefter Praitagfifir aa oft A, Bate C fare & Raye tA fag oe aa AB oz AC & erat Rand eis st wAM: aHaT AOB six AOC 4 OB are 31 aR OC axa tani a (mererTATGATe ast Iz) aan: D ate E fargait A Bred 8 | (ag taaT aa aeaa faa YSTTE AB ae AC sae oe saga ter & ar at (Beit erat (4°2.2) ait fecret) | se: : LDAE= /.A +5 art (311) ste (27.2) (a) wa, anat Pras ADE atk ODE ¥, DE*= AD?+AE*—2AD . AE cos A @ a, DE*=0D?+0E*—20D . OF cos a (ii) ara @, arate Prat AOD siz AOE ®, OD*= (OA? AD? aes ak Ont \o4EARA (iti) (ii) # & (i) aY er sie (ill) B ote a, 0= 204?42AD.AE cos A—20D.0Ecos a OA.OA , AE. AD 84-07 OE TOD. OE © 4aitsita Pagar 3 Pretafincte att _ 387 =cos b cos C+sin } sin c cos A Aw C8 4098 b cose ™ cos A= "sin 6 sin ¢ a,b, cote A, B,C a ate aa Ho aaat rz, fryr s eae ara & fare frees Ga sre Prat oe amar 2 | ee ga a Freafefiar at ageaal Here cat sent fers aT cos a=cos b cos ¢+sin b sin ¢ cos A cos b=cos ¢ cos asin ¢ sin a cos B (1) cos c=cos acos b-+sin a sin bcos C atz, cos a—cos b cos¢ cos A = —— 1 sin b sin ¢ ‘OS cos B =2° b—cos ¢ cos a (2) sin ¢ sin a OS ¢—COS OS cos C = 9082608 4 cosh sin a sin 6 Barer ga cieita frarafaia ar ger gt (Fundamental Formula) ara ea & | at ae a FAR aT GT aT fava oz aad € 1 42.2 fecqvit—freci ara at carat AD atc OC & fret Ferg az frat & fr og arses 2 fie anata / AOC St gat b antag, om ama & an gt wate gar AC ws aaain tana | etife, at gmt AC =b= / AOC =2/2 tc AE, a AC Herat tar’ Bahra, LOAE =n/2 | a war eH Baad Bf OC MROD Ft TAAHAT Rand & faa wer ferg aera set | ate afe got AC = =b>a/2% at AE ate OC a frada fert H age a wet Faetet, fret aeaan gat vat at 8 | eet sare, D farg vit ca arena fra got AB um agate & am al ma: vag easter Heaths TrTET GA FT aeaTT freafefert aftrara % aaa afratar @ 1” sfrara—“ dren A at area gard wa agate & erg 1”58 after Prprafafer rt at eatepat exits at & fare wa ahrara-gas seat afrare @ er sog'a afters a ar fate eakirdi % freagan fratine a ad Fs (i) ZA at area gona #8 aa ow goragate & ast a1 Gi) 2.4 a dat aaa god agate & at a - (iii) ZA at ora gon How got agai & awa ath (iv) ZA wt et ora goad agate & aut gi! b>72 a» we cr ON a ¢ avait 32 @) at a, Pgs ABC H gat AC=5 > 2/2 CA she CB at agnit it C’ farg ae frit 8) apt at, ABC’ Paget AC’ =0' aie BO’ =a" am / BAC’ =A’ baa bv=n—b 9 . AB ae AC Stage | at a, FA’ fag 1x fret @1 Prae A'BC i, ABaa—0<5 e>ds feat 31 AC=2—-b<5 . b> 5, fer’ BC=a a, . LAA “sige arg | wa: frye ABC & ay A’ at ares god Cw agatT aa Z| zafay tare at & cos a=cos (4—b) cos (w—c)-+8in (w—b) sin (a—e) x cos A’ =cos b cos ¢-+sin b sin ¢ cos A. ma: Sater aa St gore ea feat H vt wens a ore g | (iii) are at fags ABC %, AC=b=a]260 area Praterfifer A fee © AB sx (aie meereat @ ct AB st ree) AD = 5 ret site CD # frst | arafa 34 met & feafrat sera at aaat 8 @)CD=5 m& () CD43 (a) am at CD=n/2 siz 4c=5 “feat ‘afer ah (6) &, farg C ate AB a GH Aaa | “ CB=a=7/2 ae CD=n/2 ifs, AC = 7/2 feet =n/2 Sowa a fag Aaa CD mt aa e ZA= amt CD=n/2 wat x: stores ge, Faas aerat aanfiga att 8, ATT wat Te, - cos a=cos $ cos ¢-+sin 5 sin ¢ cos Aasia rast a Pasteytadter ae 61 a, cos 7/2=cos 7/2 cos c-fsin 7/2 sin ¢ cos 7/2 a, . 0=0 ara: ae Saga feat & fare wer wath state | (@) aH A CD#¢ 2/2, (a) ¥ a cara we TE fe A, TT DC a Fae ate C, a AB a ATS <. frye BDC % CD=A, /CDB=7/2, aa are BDC % fac starea at TATA I, cos BC =cos BD cos DC+sin BD sin DC cos CDB at cos a=cos (awn/2) cos A-+sin BD sin DC cos 7/2 =sin a cos A site A ® starr qa (1) 8 at afe ba a/2 at gh ad wT faaat 2 | aa: aates Ga st acaat gat feat ae qarat & 1 arafa 35 (iv) ata at fags ABC 4%, b=n/2 We c= 7/2 aig A staat Ea F aA TAA Tz, cos a=cos b cos c-++sin b sin ¢ cos A aI, cos a=cos 7/2 cos 7/2-+sin 7/2 sin 7/2 cos A at, cos a=cos A . a=A ait eer 2 elif geo @ A, a BC ow aT62 atte Pratteafirfer @, Gi), (ii) se (iv) & ae ee aeHe FF StaTET GT oTTTE ET Hare : 42.3. Saver qa aye (1) ae (2) & ease 8 fs (i) ah feet fags # qe atg ate saat cree gait Fa ana ai dt ata at weyS FA HAT ae BC THT g | (ii) ale faye at ae gare art ma af Tt atat & arr art ae aaT ZI rere A A SeefreT ste eave freafafaa Fe Tae & arse dt ae | aa —aea Pratofafes at sheers, a? = B22 2By cos A set a, 8, y Tree frye at gone &, starr aa art eanfier fier OT erat g | . sarge L. atieta Pays % Maret aa at ene aT area faye & faq ears wet frat—ateia Pras a1 stares q, cos a=cos b cos G-+sin b sin ¢ cos A. ar a, atte fags ABC om wea fera 8 fraat Pasar 2 1 art a, goreit & ary AB, BC a CA t reateat mare: a, By fla: aieie firgs at gard, ant, b=Bir, ont, oem Protatife & ener ote Saver Gat & va B, cos a=cos (a/r)= a3 a sin b=sin (6/r)= =hlr— 5 ba wraTet Tt ‘ 7 wt R, (1-8) = (op + (In t) HE )EBBra)edsateita Fra # Pretatieite de 63 wa aera wa fe wa at Paoat war a ae wget eit F wate 100 WR a, B, y WAC wea & oa Keafer H aete PrgW GH ANAT fag at six cae err raat. gost at erat AT: a, B, y et | aa: (i) A ATA FH TY, : ata P42 cos A+ aet kom ofefea ute & fae art Fae are wT at ae sqe él det Flan aaa aa aqe MF HT dala aM at, AAT 7->00, AT TAT, a? = f?+-y?—2By cos A ot arart Pryor & fare tlareT EAS | waren 2. (a1) aeita fags ABC # afe fag D, got BC at wen farg 8 &t fare wet Fr, . cos b-}-cos ¢ a 2 cosy, (@) a (a) at ata aa aaa Pays & fae as FA | cos AD= [arr 64] safe 36 (a) art ata, b,c Pras ABC & yond % 1 D, BC at wen fary = | wate, BD=DC=a/2.64 aii faaatate Pras ABD WADE % waT: / D ie /(7-—D)® fat Flaret 77, cos AB=cos AD cos BD-+-sin AD sin BD cosD... @ ax, cos AC = cos AD cos CD-+-sin AD sin CD cos(—DYii) Tq, DC = BD, cos (a—D)=—cos D .. cos AC=cos AD cos DC—sin AD sin BD cos D (iii) (i) te (il) Hate 8, cos AC-+cos AB=cos AD cos BD-+-cos AD cos DG =cos AD (cos BD-+-cos DC) =cos AD .2.cos (3?) cos (7??5**) . =2 cos AD cos a . BD=DC cos AC - +cos AB 2 cos BC A, A tat I, : cos AD= cos bh-e0s « ae cos AD= 2cos 5 (a) aa at aeata tan AD % ary aT mark 5 2 aK a, By Pago a goat & arat at aeargat g | ae wer at fsa B at, cos b= cos (8) = 1b Bit cos semeos(y) = 15, rr, cos a2 =co0s(a/2r) =" hen (ai2y spn cos AD= 0s (8/r)= 1— aoater frye # Pretfidtr date 65 (a) & canter ofa F art waz 9, ania (GPF) =(I- get) gitins Al gan ey H+ Jet pe gate Wa, 700 WR a, B, y, WAC Wee Baa Meta frye ABC, anda Pras at aie saa gat g | aa: roo ART AM aT ART oy 2{(a/2)P°+83} = 36+] aa: aaae Brae at at ast & aut er ate, atest ast F at F a. att saat serra tar R ae & otras B yd F acraz Hat 3 | sagen 3. waftarg atta frst ABC 3 fra at fe, : tan? a/2=1—2 cos A ze afcara & feet anfaang viet Gry & stat atx yar at afanan sit grat dtd fraffer at | art a, ser fg ABC i, a=b=c, cos a—cosb cose ee 1—2cos A= 1-2 0 my, cos A< 1/2 A>al/3 at ware ge eater ac wate fis B> Fate C>F | Sif feet tettr Pegs at gore site sre ofearet & agers AT: wiga sm & ae ad Fae: et frond frees & fe aafrang arte forge B ster 7/3 site x atte B are ges FI afeure (i) ¥, . (tant 5) any afieast at = (1—2 cos A) a efesiae aT a, axel, aafrag fran at gor ar afascr wm 27/3 a saat 21 RIT: YA BTA ATT YT at awe z | vargen 4. ates ryt ABC # afe A=a & &t fae Ft fe,ater frge % fretyfaite dite 67 fen—ary A & fry staret yA TTA Tz, cos a=cos b cos C-+sin 6 sinc cos A qq, A=a, . cos a=cos 6 cos c-++sin sin ¢ cosa a, : cos @(1—sin b sin ¢)=cos 6 cos¢ cos b cose P \osnbane eee wes Freteats & eet ert ff, cosa= 24 _1—cosa tant T-Fcosa (i) & cos @ #t a Tet oR, cos b cos¢ tan? i—sin b sinc 2 cos b cos¢ 4 gin bsine ~ Lesin } sin c—cos b cose” * i—sin b sin ¢-++cos 6 cos ¢ _ 1=c0s (bw) * T-Feos (+e (6~«) eure t68 fetta frarafafe sin? cos eco! sin & _ 2° 2 272 cos 2 cos — sin’ sin £ 20° 2 2° 2 sia wiz a #1cos $a cos $b Fa BT, tan?~tan £ tan2 2 2 I be 1—tans tang farwen fafa em (i) H etfs az TF E FF, cos b cose . co8 a= 7 "sin b sine 7 Q) ae fratafaia a, 2 j : tan sin 20 = tan soe e (2) —tan? “ 1—tan?@ 2. Q) ait, cos 20= Tptan®6 am at, x,y eZ oNT: tan fa, tan } we tan fo eae EI (1) ¥ (2) aie (3) & A aR Te, vn Gea) Ma) _ 1 —yttegt ty? 2 Tay he aaraargirt (Componendo et Dividendo) &, (1 +42)— (1-4?) (1-Fx*)-+-(1—2*) a lt tettytt= ye (1p 9222) Tey? h yz 49 HI Py? 2?)aiteira fagor % fraterfeeta date 69 Prey “Ty BDz = Ire? “5 rte aI Pe" 1-yz GU X, y, ZH ATT TAT TT, b ¢ tan 5~tan5 tan 6 ¢ I—tang tan nie varget 5. reer Pegs ABC St Bea, at are So ai Page at caer at wg | ate at Pram at Bare & g gent & ats ar any 0 8 it fag aa fe cos 6=cos A+4 tan $0 tan 4c sin? 0 [atrz, °57, ara, °58] = ‘ z=8 cos a 2 cos? 5= 1+cos a cos A =| 41 c0s.A 1 1 . “T=cos A 22 @ ’ ABC % wate Pras 4, sha 13 8, a=n—A cos a’ = —cos A 2 cos? = 1+-cos a’ =1—cos A 2.» ii) @ ae GH) 8, 4 cos? 5008 1 1reos 4 =T—=cos A* 1 aa’ cos 5 cos gatative firger ® Pretatirdta dat 85 [feoret—atart & fire era wea fore wetaivg (2) wT wer 8 afat 1] sareen 4. ow ate Prge mrt ate Pg} aarti: aetat ai faa wt fe sec? A+-sec? B+sec? C4-2 sec A sec Bsec C=1 aie ea there & fread frrret fe tar fargo anfratg set at aa | [fawa, °59, arr, °62] feo at Pras ABC at acta Prost A’BYC’ 3 | ifs fran ABC sit A'B'C’ aaftaa: aeat 2, A=A’, wey A =7—a, FA 13% “. A=r—-a Rat war, a=n—A B=r—b _b=2—BS vee GY C=n—c | c=7—C) FA, TOT BATT YF IT, cos a= £98 A-+-cos B cosC - ~sin Bsin C cos (n— A) = COS Actos B cos C at sos B cos c :@ . 7 aq, —cos A sin B sin C=cos A-+-cos B.cos Cc cos A cos B cos C a att 24 I, —tan B tan C=sec Bsec C-+sec A > tan? B tan® C =(sec B sec C-+-sec A)*86 attette Paaterfifr am, (sec? B— 1)(sec? C—1)= (sec? B sec? “+sec? A-+2 sac A sec B sec C) at, sec? A+-sec? B+sec? C-+2 sec A sec Bsec C=1 wa at at, frys ABC anfrag @ at ag anatfon at 8, zafac, A=B=C orgs shart ¥ A= B=C wit x, 3 sec? A+2 sec? 4=1 m, * 3.cos A4+2—cos* A=0 mT, cos’ A—3 cos A—2=0 at, (cos A—2)(cos? A+-2 cos A-++-1)=0 a, (cos A—2Xcos A+1)?=0 “ad, cos A= 2, ot waRra & aaa, cos A=—] A=7=B=C A+B4-C=37=6 anata aT fe sraewa 8 aie are (38) & faye & dei stat at ate a: anata & oH at arfet | ~ oma: fags ABC, at anfaaa: aay sata faye & acaz @, aa- faatg wat at erat | [feowwt—garreit & fire eine see & faq get img (2) aT ee 12 aft 1] sagen 5. iter faye ABC % at A® safes wk are aig (Internal and External bisectors) gat BC % ara Sa:atta fags # fratofadte date 87 a ate Pa THe & | fers aa Fe cos C—cos B @) c0sa= “9 eos A/D ¢. sB @) 08 p= Sa am aan AD ok AE #AT: LA ® ata ae are ada €1 /BAD= 1CAD= 4 wat, LADS=a az, LAEC=8B . ZL EAD=7/2 aiat & ata at Se | in—A . LCAE=("5*) . 2. @ ayes SaTEA GT YT AD a am faa ADB ite ACD & ara Heh, aerate TAT Ty, Frys ADB 4, cos B-+-cos4 cosa cos AD=——=—g@gp . sin A/2 sin a Gi)88 after Brea Pry ACD ii, cos C+cos4 cos (1n—a) cos AD=———-| sin Oa sin (7—a) cos C—cos 4 cosa _ Zz ass sin 5 sina Gi) ate (iii) &, cos Brows cos an cos C—cos 4 cos a sing sina / sin 4 sina 2 cos A/2 cos a= cos C—cos B cosa £08 C—cos B 80 cos A/2— at ware, Pras AEB ¥, cos B+-cos, BAE cosp coke sin BAE sin np Te, LBAE= / A+ /CAE = 244754 nA 2 cos B+-cos (=) cos B sin (*4 ata \ sing — 608 B—sin A/2 cos B cos A/2sin B cos AE = ii) - @) ji) (iv)citar Frys Pretend sida 89 ait fag AEC 3, AE= £95 (7—C)-+-cos CAE cos 8 = sin CAE sin B —cos C-+-cos }(m—A) cos 8 sin 4(a— A) sin B _ —cos C+sin A/2 cos 8 =. “eos dQ sin B mM (iv) ate (v) &, cos . A - cos B—sin 5 cos B _ —cos G+-sin 4/2 cos B A cos 4 sin ~ cos A/2 sin B 2 sin A/2 cos B=cos C-+-cosB __ cos C-+cos B cop= “yinap (a) & yond ate gam aeqa sto 1 aea qa (Sine Formula) 4:4, meta fage & atic saat ange yor & Peara fede arer F agra Tet aaTgaie aa g, vata ateta frge ABC H, sinA: sina: sinB: sind :: sinC : sine sin A_sin B sin C sna snb sine waafa—ataret 97 a, cos a—cos b cos ¢ sin 6 sin ¢ at, sin? A= 1—cos? A sin? A= 1-( a—cos b cos sy" ~ sin 6 sin ¢ at, cos A= _ sin? b sin? c—(cos a—cos b cos c)? ~ sin? 6 sin? ¢90 att Sratafirfer (1—cos*b) (1 —cos®c)—(cos*a+-cos®b cos*e—2 cosa cosh Cos ¢) sin? 6 sin? ¢ AaVG=cos?a—cos*h— cos*C-+-2 cosa co cos 5 cose) » sin sin b sin ¢ aet waht ar art sans 8 adits frye at yond atx at 180° & wy aa & fred sin A, sin 4, sin ¢ aanae Z| amt a, n= $4/(1—cos*a—cos*b—cos%e-+2cos a cosh cos c) an Rad Z fis n(a,%, c) et oH aafea GAA (Symmetrical function) 2 Lope ne at atti Prget at gore at were (Norm) wat & | “ sin A= sin b sine sin A 2n at, =a sina sinasin 6 sine aati ware an faa wt aaa z fx, sinB____2n sind sinasin bsine ait, ° sinC Qn sine sina sind sine ay Sin A_ sin B _sin C Qn (5) sina sind sine sinasinbsinc 44.1. Met ga at satiate saat av at, Prat ABC ga we oz feat 2 frre a O 21 O farg at frye % aie faegst & fromtt ote OA ae teat ee wre farg P at aa ga fag X anata OBC 1% PD wa atlaaa Pras # Prarafadts date 91 farz D & ance OBC, OB att OC area Vanit TH AAT: DE ait DF wea ert | OD a frat | araia 44 (tant, PD, anat OBC ax ara & eafere ae eae H feat wars Vat TAS | “ LPDE=90° .. @ ar, ZLPDF=90° ~ ee ii) wat, ZOED=90° 2. Gi) ++ caat att DE, tat OB Te AE site, . ZOFD=90° . oe Gv) , waat ant DF, tat OC ATI aa, (i) Sanaa anary frys PED i, PE*= PD*4+.DE* eq, (iii) § eraT THAT FryT POD &, OP? =0D*+-PD? “ PE®= OP*—(OD*— DE?) atx (iv) & anaa anata frye ODE 5, OD? = OF?+ DE®92 : ciety. fratata fr oo PE*=OP?—OE* TT, OP? = PE*+0E? anaa frga OPE ce anaty frye &, frat LOEP=90° ee @™ aa, wraet araty frye PED &, . PD sin PED= PE a, PD=PE sin PED Tey (v) a, aaae arate frye OPER, PE=OP sin POE =OP sin AOB =OP sinc ak aaa ata PED, arat OAB wie OBC & ata fara atu Gi wafe-ZPED=/B : PD=OP sinc sin B ee (i) sal sor, avaa arate Gras PFD % gates saver art et fra Ht oat & fr . : PD=OPsinbsnC . . . (vii) (vi) a (vii) &, OP sin ¢ sin B= OP sin b sin C sin B_ sin € sin 6 sine art aareat & a @, b,c ate A, B, C at amie wa FH gaat gz, sin A_ sin B_sin sina sind sine ara: Fett atcha Pager shart & fewteniadta ater the gaat aegE gant & fratofinia area & agora vere aargaiet ae eIaftether Prger tt fratefadts de 93 fecquit, safes earent, b,c, Batt C at art us aaaty & ar at red ona 8, vith eH ABS 5 aa B= aaar ate PED 2a 2 sie ance fry PDE at ve ary PDE, CAAT EI CF anergy 21 sa yan am tea fe ance faye PDEF a sty Ft ra a a dts warag t| ae: waa H fag PR anat OBC % wea srt aa a aera & wa / PED=/B << | ma arr (4°4.1) # ean aren Ga Ror sige et a fare arpfe 44 at ge sare Srrafes ser wife fer ag aa feat H aera A ak gaat ae aes | Tea FET % am a LPED= LB>F & aa fea ¥ na me frg D’, OB sik OC Rami & ate Fa aoe OB F Fad site Ft We / PED’ wala & LPED a aye aT! aa: sin PED’ =sin(7—B)=sin B, sata qa ca feats & fac wer senfige dar Fae TaTE GT aT are ofardi & oft gaa fae feat a eat 1 44.2 WES Wet Ia fags ABC % sta Pras A’B‘C’ & aren a7, sin A’ sin B’_ sin C’ snag sind’ sine’ a, (ema cest bce cna cosb’ cos c’ . ~ sina’ sind’ sinc” ) gata fage & mandi am frye ABC % randt & sa H cat 8, sin (n— —4) _ Sin (7b) sin (7—c) sin(s—A) sin (w—B) sin (w—C)94 attetrr Pratrerfirfir Jl 1—cos%a—A)— spams spot eam) +2 cos (®—A) cos (mB) cos (r—C) sin (7— A) sin (7—B) sin (7—c) Je —cos?A—cos*B—cos*C ) sina_sind sinc _ —2 cos A cos B cos C, sn A sinB sin C_ sin Asin B sin C 2N - “tadsnBinc ++ + ©) a, met N= 4/(1—cos? A—cos? B—cos? C—2cos A cos B cos ©), A, B,C atanfar caré | N at ite Pras & eteit ar ance red G1 - : waren i. tet Frys ABC H fire wa fs, sing 2 snd W feot—arar qa (5) 8, sin A_sin B_ sin C_ 2n sna snd sine sinasind sine wegen ater aA (6) 3, sina _sinb sine _ 2N sin A’ sin B’ sin C sin Asin B sin C = 4\. 2n sin A sin B sin C Gre "sina sin 6 sin ¢ Qn _2 sin A\? . ON sna)aieite Prost H farerafadts tat 95 sare 2. aieita fags ABC H fees at fs, tan (=) tan (FE)- tan (7) tan (4) 2 2 2 2 [rige, °57] feat—arer 47 2, sin A_ sin B sina sin b ‘ APIATGIT Fy, sin A—sin a_ sin B—sin b sin A+sin a sin B+sin b A+ta\ . [A— B+b\ . (Bb __ zen) 54) ll ain) | an (en) tan sl saremnt 3, aieita fry ABC aint A, Bait C& saat age gant oe araes, aedge arr ata ae | ale aretha aearg aT: 0, b, o dt at fae wat fr, sin 6 sin a=sin ¢ sin b=sin y sinc TT, = /(1—cos? a—cos*b—cos"¢+2 cosa cosb cose) [fawa, 762] am at ABC% ett A® a AD drama gar BOR wa & | wate, ZADB= / ADC=7/2 frags ABD 3, aren 9H (5) 3 sin B_sin 7/2 sin@~ sine sin 6=sin ¢ sin B96 . attr Prta tif: Frys ABC 4, ang qa 2, . snB_____&n sin 6” sina sin } sine __2n ~ sina sinc 2n sin a sin ¢ sinB. sin 0=sin ¢ \ _ 2a ~ sina A ci carafe 45 *. sind sin a= 2n= /(1—cos?a—cos?b—cos? ¢+2 cos a x cos bcos ¢ afcarr at wreqat &, sin ¢ sin b=2n ak, sin # sin c= 2n ae: sin # sin a=sin ¢ sin b=sin ¢ sin ¢ = V(1—cos*a—cos*b— cos*¢-+-2cos acosb cose) saree 4. cH aafgarg Teta faye ABC at acrax yoni % yea fagat @ ata DE are freen rat | afe gar BC frye at ater &aiete fags H Protofada dae 97 at fag a fe sin DE 4sin Fe sec £ [arrz, 63] anata 46 amt ait frys ABC H, AB= AC, Dax E, ym AB att AC % Hem fare St zram—aid AX aR BC aa srt at BC ate DE at wae: Late M fags oe areaté | woafa—aife frye ABC anfearg 2 atk D vat E, AB ate AC % wer farg 2, safeu fran ADE wt anfeag t vik wa AL, DE qe ara at 2 | aieita srnfafa & ar aeaangdan fre ax and & fr, BC BL=CL="¥" 2. @ ait, pM=EM=28 i aa, frags ALC # area a7 8, nC, cn 7 tito Fao98 aera faastaferfer sin BC a, 2. og AC AC ino AC=2 sin zeosy sin-y sin OTA wm AC . . . iii) sin =~ cos “> faga AME % area 47 &, sin EM _ sin AE sin 8 ~ sin 90° sin DE 2 . AC ae aq, ip AG oe , ne PQ @) in PE _ sin osin 3 sin 37S n in q sinB gin A ay ~ AC 1 . 2 sin J CS = sin 28 se AC = bsinsy"see “5 [fecroht—aa sea at faerat Fae stared Ya Ft aera a ead wT 1] waver 5. fat ateite Pras at om ya a a aay ant FT frnfara feat wat | a aM, arHa gilt Tz AAT: 01, Oy, Os IK Oy after gare 2 | fire at fr, sin (0,99) sin 0, sin 8, sin (0,40,) "sin 0, sin 0, [***5, ’60] aw at fay ABCH, BL=LM=MN=NCmea farga # faerafadta date 99 ak, LBAL=0,, /LAM= 05, {MAN =0y, ZNAC=6, wat LAMB =} . LAMC=(n—$) avait 47 aa, Pago ABM Hare 7 8, sin (0,4+6,)_ sins “sin BM “sine sin (0,40) =" sin BM 2. @ gat sar, fags AMC H, sin (05+-94)_ sin (7—) sin MC sind sin (0,-+ 4) =e Sit MC Le. Gi) Tw, BM= MG, zafar (i) ak (ii) &, sin (0,--0,)_ sin ~ sin BM sin b sin (0,+6,) sine sin % sin MC sin 6 a =ane +. . (iii).100 air farrafefer afeure (ii) at areca &, frys ALB ate ALM &, sin 6, sin AM t sin, sine site fags ANM aie ANC #, sin 0, sin b- sin 6, sin AM sin g, sin 6, sin b sin 6, sin 6, sin¢ “ (iv) (iii) ax (iv) 3, sin (A)-+0,) _sin 0, sin 6, sin (@5+-4,) sin 4, sin 6, saa dag 3 1. alee frys ABC ¥, nae NV mre: gaat ganat ate stat am ara & | fre wet fa, 2n? @ Naa asin sin ¢ 2 @ 2a rae “sin A sin B sin C 2. fas att fe faga ABC at yori ak atait & ae aA: sah acta fags & aha ate yore & arate ae E | 3. fag wa fe ca mete frys Fon aie N sah fret at ag ign Pras F nate N Foe ae ZI 4, afe iteta frat ABCH A=’ Ht faz wd aA B wiz b aerate Bt Taal TH Fat arqes Mt i aadt war Cat ¢ Ht Ol at AAT HAA TLEAT TET ST | 5. farg E site F, ieita fags ABC at gat AC aR ABR ney farg 21 afe EF fagat at frat arat dhige ara agrt ottalata fry # frarafada aaa 101 oy, adt gf gat BC at D fare oe fr a at fag at fe, in DE cos = sin DF cos § sin Cos g= Sin cl a 6. ata frat ABC # ate A ttanfeanina oer aren aaa ara, arge gat BC at D fara oe wreat @ | af fag E, gat BC at weg farg 2 at fre wel fe b4e\ a bo) tan DE tan(?5°) stan 5 tan (5) 7. atte Pagar ABC # wie fags orga gail % meq fargait at fret ae atti at wet fag, ya a aT aafeatar wet aa ATT Ft, o Be Bowie H fauna aeaT 2 | fre Fe fe, ; inp 2 cog 5 : [sera, °59] 8. aista frys ABC % aid C@ arr AB oe ht aa ara eq arg at ararg afe 8 2 ct fas Fat fa, cos? a-+-cos? b—2 cos a cos bcos c= sin? ¢ cos® § at, cos 8=cosec ¢ (cos® a+-cos* b—2 cos a cos 5 cos c)¥2 [fame, 759, atrz, 66] [atart serecer 2 & afeqrr a agate wet 1] 9. afe aiteita frat ABC H b--e= 7 & tt fra ae fe, sin 2B+sin 2C=0 [zeretarz, °52] [ata beam a sin b= sin (7—c) =sin¢ sin B=sin C soar gah aa B=C aaat Baa—C102 meta fratefafer SAW AIT AT —AT aT, B=C on bao *o quar ata tara ga | IRI bc=m J. b=c=n/2 <. A, att BC a aa 21 3 B=b=n/2 a Cac=n/2 & sin 2B+sin 2C=sin 7+sin 7 =0 fedta aeran—anr a, B=n—-C sin 2B-+sin 2G=sin Am7—C)-+-sin 2C =—sin 2C+sin 2C _ =O] [fervoht—ga ser at aret site Pater dat gal & agar & ate Marans a FT awe z |] 10. faa ABC ¥ nate Vo aam saat qa ate stat ama & | fre wat fe . sin Asin B sinC_N® sinasin bsine 73 (@) & yard, sak afta at ato ate ow ate aT wate a BATT aagal FT ararey ®ériee Ya (Cotangent Formula) 5:4, ateita Pras at at aaa, atm ala F ary ak we a aig mais Pays are marie aaaai % dra aeaee eats eA | art at diet Pays ABC % mara (Consecutive) are saaa, B, a, Cate b 2 | Bates fags H fastafacte deta 103 carta BC a fag D ae ea rH agTat fs aT BD=-/2\ AD at faa | avaft 48 satfa—Pays ACD # atargt G7 F, cos AD = cos b cos CD-+sin 8 sin CD cos (7—C) =cos bcos (7/2—a)—sin b sin (7/2—a) cos C =cosb sina—sinbcocacosG . . . (i) gat aarz, frast ABD 4, wares qa a, cos AD=cos ¢ cos BD-++sin¢ sin BD cos B =cos¢ cos 7/2-+sin c sin 7/2 cos B =sinc cosB oes (ii) (i) ae Gi) @, cos b sin a—sin b cos acos C=sine cos B sin 6 & ait 29%, cot 6 sin a=cos a cos CHS cosB . . (iii) are a 3, sin B sin b sin104 attettr Pratertafar (ii) art eat oz, . sin cot b sin a=cos a cosC-+ 9 | cos B + sin B co =cos a cos C+sin C cot B al, cos acos C=sin a cot b—sin Ccot B 45.1. feet 1. saqac qa 8 fretted atdare ager & wal ® safeas aa F are, ere at afaat % fer, et wide aa Teg 2 | 2, seh 48% saudi a, fain feafe H, fda are BF & ieee qa a Hem ws aad 2) G8, aava B, a, C, b# Wn a arg Bate CH ate fears 81 ek ga tere 4aV (Inner side) air ae ast bat gael aa (other side) a¥ gat ge ya & @a Sata Cat oat ta’ died ae ata Bat gat ate ae at aver (4°5) % seers qa A ge Frew wat ence we ana 2: alates (Stee aot) . ater (tae ate) + Sarea (stadt yam). a1éere (grat yar) area (ilaet whe) . eee (gezt ate) at, cos (Inner side).cos (Inner angle) =sin (Inner side). cot (Other angle) —sin (Inner angle). cot (Other angle) Fras ABC & at dt are wane aaa AA oe sae SAT atc gat’ aot sat ster ate Pet ot aad a te cae ASAE ara sag'aa fafa are frat at agat 81 da—aw aw, aR mame FATT ¢, B,a,Ce\ ene 2 fis tat wat a 8 ate ad at Be aa at waraiteia Pras % faratofadta att 105 Nagas wT arafa 49 six ate wae: ged HoT ae geet AT Al wa: HAE ETT sears & frat are cos a cos B=sin a cote—sin B cot C ate afe mata waza, B, a, C, bs aH, wad gst = a; gat wat=b | watda=C, za aig =Bs Ba: HAVE TT, cos a cos B=sin a cot b—sin C cot B aa | 45.2. fear oa waral HA a HAMA FATT BT FT Ts war R anand 2) wa: Aes qa et fer o edt oes T aaa | wears TT cos a cos B=sin a cot c—sin B cot C 7 cos b cos in 6 cot a—sin C cot A > (i) cos ¢ cos A=sin C cot b—sin A cot B cos a cos C=sin a cot b—sin C cot B) cos b cos A=sin b cot c—sin A cot C } (ii) cos ¢ cos B=sin c cot a—sin B cot A sages Teer aa at afe A ana, (i) ste (id), F Fanta F< { , ) |106 airetter Pareptarfirfer & at ote at H ate we at afafthe amex ag qa, aaadt at wale apr HE ace ae sree Prat ot awd & | 45.3. sameml. atete Fras ABC afe AD ata BAC at atafee anfanrae 2 at fra act fr cot AD—£° b+cote _ sin (+e) 2cos $4 ~ 2sin bcos FA sinc [arre, 752] ara 50 am a, arafr % frat ABC # AD, Ata BAC am anates aafgarae & att ara ait, ZADB=6 oe LADC=1—0 wa fags ADB i, murna mama c, £4, AD ste 0% fat are a &, - cos AD cos $A=sin AD cot c—sin 3A cot? ... @ wit ware faye ADCH, 5, 44, AD ote (n—0) naaat® fam aiténe qa &, cos AD cos 34=sin AD cot b—sin A cot (7-8) =sin AD cot b-sin 44 cot 6 os (ii)ateta fags H Prevafadt date 107 {i) aie (ii) A ASAT, 2cos AD cos $A=sin AD (cot c+cot 5) __ cot c+-cot b cot AD=~TcGs FA 2s @ _sin b cos ¢-+cos 6 sin ¢ “YF sin 6 sin ¢ cos $4 _ sin (bc) “sin b sin ¢ cos $4 22+ waren 2. un ddza air DE, aia Pegs ABC at AB aR AG gavait at anfemiar seat 3 ate fag P saat aa f1 ate farg P al Seige atat art mam: B,D, E ate C fargat & freer aa at fag wet fir LBPC=2/ DPE [aaargz, °60; atrz, °53; ara, 63] amafa 51 fet &@—(a) fieg D ote E, got AB aie AC % wen fry ZI aa: AD=BD sit AE=CE108 aici fraratifes (@) fag P, ata DE at aa 2 | sata, PD=PE=7)2 faz won g— ZBPC=2/ DPE vant—P faeg at detaa ai are firg A & frat santa —ana at, frg P ae at ater avait ¥ autit agar 94, By, Os IR 94 &, aie ate at LADP =6 . ZLBDP =7—-0 na, frat ADP # az anima TAT O,, PD, 0 sit AD, % awleare gt aay Tz, cos PD cos @=sin PD cot AD—sin 0 cot Og aT, cos 7/2 cos O=sin 7/2 cot AD—sin 6 cot O°. (@) 7 cot AD=sin 4 cot 6, se. (1) ait sar, fas BDP # a aaa saat 9, PD, (7—@) aR BD & #téare ga.8, cot BD =sin (7—0) cot 4, 733, AD=BD - @) ate, sin (7—0)=sin 0 oo cot AD=sin @ cot 0, see (2) (1) a (2) 4, cot 6, =cot 6, “ 6,= 0, » s+ (3) wat gan, Prat AEP ite CEP aw fag ae ane 2 fs, 3) she (4) a foots 6 @ Ome 8+ 84= bat Oy a 9+ Oat Oy+84= 20.405) ™, ZBPC=2/ DEPter forget Paererficter efate . 109 fecqoht—ea yar at Faq Aleta sarfafe at aerrat & franif ea ea RI wart img 4 1. ateta frye ABC & afe AE ary BAC #1 ave a4- fraraa & at fea wt cot b—cot¢ cot AE= “Jsin id * soe |) _ sin (c—b) “Ysnbsnesnga 1 2. arta frys ABC % oraz BC 1 afe D ae Be farg 2, at fre eet fr sin A cot AD=cot b sin BAD-4-cot Csin DAC aris ofeuts & at A & avafer ste ate anfemaat a (até A ate areare BC % ata at) seats sta FT | [aea—sea ¥ gat arr % fae, (i) afe AD aiafes anfrrrars 8 at, “LBAD= / DAC=3A sam ant & ofegrr Far way 7, cot AD sin A=cot b sin $4-+ cot ¢ sin $A =sin 4A (cot b-++cotc) _ cot b+cote cot AD=—y cord * tat sereca 11] 3, afe atta fryt ABC Horn: styt & aafearat F ae aie aan & ata at aang mam: 0,4, p a at faz we fe, cot 6 cos 34-+-cot ¢ cos $B+-cot # cos 3C =cota+cot b+cotc. [atrz, 1966]110 mista Pratatatr 4, af om atetta Pros & atait & aver anfgaraal a (ait ate arae & ata) arareat mum: 5, 5), 5, 2 at fs wat fie cot 6, sin $4-}-cot 8, sin $B+-cot 8, sin }C=2 [Atrgz, 1957] 5. tteitr fags ABC / A anata 2 | ana & arate ate aval fests stare BC at mam: X ate (aert wa BC a) Y fargst at ated g at fae at fe cot? AX+4 cot? AY = cot? b-+-cot® ¢ [fawa, 1963; atrz, 1963] 6. mits fryst ABC H, / C= 90° sik b= 30 2 | afe ate C& atafee anew a (ate ate ora % ate) arate 5 a at fas at far tan §= } (tan a+-tan 2a). 7. iter frys ABC at gut BC at neq farg afe fra ait frgsat A, Book CB ama gd ox & a fre wa fe, cot B cot C= cos*® fa, 8. meta faga ABC # af ata C, gota ® aera at at fra att fr, sin (B—6)=sin b sin B cot ¢ cos C, [aa C=a sin C=sin a, xarfe | see & sin (B—6)=sin B cos b—cos B sin b =sin bsin B(cotb—cot B) . . . (ii) aa, Pays ABC # atére qa sik (i) 8, cos a cos C=sin a cot b—sin C cot B =sin a (cot b—cot B) am, , cot b—cot B= cot C cos C sin (B—4)=sin 6 sin B cot ¢ cos C].aera Pryor # Preteatictar ar iL 9. ateita fags ABC # faa aa fr, tan a cos C+-tan¢ cosA ton b= tana tan ¢.cos d cos C [fecrot—ea yaa al Hae saat GA a ee Het fae wear amg 2 at sea 9 at |] [eta—aite Be gat AC ax BD wad | AABD att CBD % waa aa, cos AD cos A=sin AD cot AB—sin A cot 90° am, tan AD=cos A tan AB= cos A tane at THR, tan DC=cos C tan AB= cos C tan tan AD+tan DC 1—tan AD tan DC tan AD a tan DC at are eats at at oie ahr ser emi] 10, We % ges ot feat ata adys ABCD at ama gaat AB otk BC at ararat mae: a ste be wat / ABD=0 @ | fag att fe wa, tan b= tan (AD+DC)= cos asin b—sin a (cos b cos B+-cot C sin By —cot A sin b+ sin a (cos 6 sin B—cot Ccos B) [atrz, 1950] Rat qa 8 fanaa go fase qa tan0= 46. AT BT aa (Half Angle Formulae) ateita Pays % adatat & Pratatrcte agar (area, ttatge ate dare) A garat & fastofadta agardt & oat Hat wet | feo—atater qa 2, cos A= °° a—cosb cose — sindsine112 ater Prato fate ai aa faatofifs &, gat caTz, 2cos*$d=1+cosA 2141.1 COS a—cos b cose 2 cost 4a ising _ sin} sin c+-cos a—cos b cos ¢ ~ sin b sin ¢ __ COS a—cos (b+¢) sin dsine «(2tb+e\ . (b-peta’ _2sin( EF) sin ("= 9 ) sin d sin a+btc=2s —a+b+ce=2s—2a=%s—a) a—b-+}-c=2s—2b = 2(s—b), ware A_2sins sin (s—a) 2 cost 5 = sin } sin¢ A J@ ssin (s—a)) cos 5= a 2 sindsine / 2 sin? 4 l—cos A _ sin 6 sin c—cos a--cos cos ¢ sin b sin ¢ __ cos (b—c)—cos a ~ sin b sine » fa—bte\.. (atb—c _ 2sin( z ) sin (=) sin b sin ¢aici Pargor 3 faatteafiette eae li: _ sin 2 sin (s—4): sin (s—e) sin dsinc . A sin (s—b) sin (s—e) - sin'g= J ( sin b sine ) arte sfeuni 3, ES sin NI 4_ tan5= nol bs cos 5 - Jee (s—8) si sin —) sin s sin(s—a) JO) « /( (=9) sin 5 sin (s—a) sin (a) 1 sin (s—a) sin (sb) sin (s—c) ~ sin axa. ( sin 5 ). a tt, I (sa) sin (s—b) sin a) =tany sins a, _tanr tan3dA= sin (s—a) [weara 7, aver (7°71) % ar 2a fis r, fags ABC & areata aa a fasat cater’ 1] ster afer at ite F fer care & eat wea 21 cos Jam [(SB ssn) nC) sindsine } sin n (s—6) sin (s—c)\) tan j4= JC sin s sin (s—a) ) . . (10) i t b. = an _ ~ Sin (s—a) J 8 ato xo114 aera Pratrerfife 4°61. qa (8) se (9) & agate a, sin A=2sin 44 cos 4A sin (s—b) sin(s—o)\ |/sin s sin (s—a) a sin b sinc ( sin 5 sine ) =a: a/ {sin ssin (s—a)sin (s—6)sin (s—o)} sin A 2 sy /{sin s sin (s— 6 Sina sinasinb sinc © {sin s sin (s—a) sin(s—9) sin(s—c)} site aret (44) & atest a 3, sin A_sinB_sinC_ 2n sina smb since sinasindsine _ V0 cos? a— cos* b—cos? ¢-+-2 Cos a cos b cose) sin asin 6 sine sags afeatai &, n= / {sin 5 sin (s—a) sin (s—b) sin (s—c)} = V/(1—cos? a—cos? b—cos* c-2 cos a cos b cose) 4:7. waysT aa (Half side Formulae) ateta frye at at gare & Prarafadta aged (wer, staat site e3ez) at ara & Paateafincttr agra H set H ara HAT | agen starter a7 (4), cos az 22° A-+cos BceosCG sin Bsin C siz ava faatafafa &, 2 cos? 4a=14-cosa z cos A--cos B cos =1+~“an Bsin Cate finest # Preafindter aber 115 _ Sin B sin C+-cos A-+-cos B cos C ~ sin B sinC cos A+-cos (B—C) ~ sin B sin C A+B—C A—B+C ron (APO) au ( B20 = sin B sin € ’ aT at, A+B4C=25 a, —A+B+C=28—24=2 (S—A); gerfe | aa, 2 cos? 4a= 2 #008 C—P) cos (SO) © corde- (SO GBeas-0) wait THR, 2 sin? 4a=1—cos @ = 1698 A-+cos B cos C ~ sin Bsin C sin B sin C—cos A— cosBcosC “sin Bin cos (B+-C)+cos A sin Bsin@ (A+B4-C) B+C—A) 2cos( 3 )eos (2 EE— ) sin Bain @ singe |(Snesno |) aude ofr &, _ sin nie tan wie 2 g Nis116 airetrr farraferit = (Ac sees 1) (ag Reo) = (ss —cos $ cos (S—A) 2) : cos (S—B) cos (S—C _ _ —cos § = 605 (5 4), (a (S—A) cos (S—B) cos a=) wa HT at, . —cos 5) _ A (a (S—A) cos (S—B) cos (S— a) =tanR a, tang =cos S—A) tan R aaa H, 5 5A) 1 @ —cos § cos (S—. . sin $=, |(—“Ga Bain G ) - ++ Gb ‘COS cos (S-—B) cos (S— ©) cos (S—C)' cos $= (sine sin Bsin C \ .. + (12) a_ —cos § cos (S—A) tan d= | Se ew S—O) ... (3) = tan R cos (S—A). [acura 7, ater (73) 3 a@r fa R, fags ABC % afera aa # fasat eufat & 1] 4&7. ar la (12) a, sin a=2 sinScoss 272 w2 (= SaP ae) = gaFantV¥ {—cos S cos (S—A) cos(S—B) cos (S—C)}airetra fanrst & Prarie eae 117 site omer (4°4.2) & aeges ater GT A, sina sind sinc oN snA~sinB sinC sin AsinBsinC _ i —cos?A—cos® B— cos? C—2 cos A cos B cos C) ~ sinAsinBsinG _ arden ofan 2, N=-+/(1—cos? A—cos* B—cos? C—2 cos A cos B cos C) =/{—cos$ cos (§—A) cos (S$ —B) cos (S—C)} 47.2. fecauit (1) ae ad stat sie ad gait & daa qa art (4°6) ate (4°7) & gat # a, b cst A,B, Cal aaa ma FH aaa A sea feat aT aaa 21 aa, sin b sine ~ A sin(s—b) sin(s—<)\ sing = JC ~ a, ware a ama wT aay Tz, sin : sin (s—c) sin sin (s—¢) sin (s—a) - ("tee ) sin ¢ sina —cos S cos($—A) ame, sin 5= ( sin B sin C a Baac) a, waaat at aale wa T aaa Tz, « 5 j¢—cos § cos (S—B) sing =, ( sin C sind ) seanfa | (2) aver (4°6) te (47) H weet Pe wea ae Ba aTiee atife fat & agar att frye % erg ak gat 180° a aa et & aie eatfere sete oe Stay site ae BT Uw aaa A HH ste & | aa: sak area, Paver ate dere Pratafida agar at ae aan gaz |118 aia fratafafer (3) cava Pratafafe & gare gt ater fratatate F ae eter aie ad gor qa Praga & fra & fate er B scat fas ade) aa fargo at gore ara di ste stat at ara ara aeaT a; waaT At fet & ate goed at art ata eer A ae Hater Gat at we aT gor ate ater gat at ageitt TTT Ft age acer aa Bat Z| [wearer 6, arr (6°3) # ag ere at ara |] (4) are (4°7) & aati & at . a 2 a Sin 5, cos, tang mathe arecfe art srt at | vale aw0 os Gi) ait sar, cos (S—B)>0 Lo. Gill) att cos (S—C)>0 woe. (iv)aes fare % faarofdta ate 119 (i), Gi), Gi) atk (iv) & ara ta? oz, —cos S cos (S—A) sin Bsin C >0 +: sin B>0, sin C > 0; aaife are & art oF aH AT 2 | sin Ja= (aveafar tft) sin? 4a= gat vate gn faa ae aad z [fe cos 4a sie tan Ja weal ara witzareafaa7e | 473, sae 1. deta fags ABC A gard at a a agate % azaz a, wats a+-b+c=7 | faa ea fe sin?14+sin?1B+sin?4C=1 [arrz, 1962] ad ata a (8) 2, , sin?}A= sin (s—b)sin (s—c) sin b sine reg feat 3 fr atbto=n sam] sin (7,2—6)sin(n/2—c) sin 4 sin ¢ =cot bcote see (i) sin? LA= zal TTR, sin? 4B=cot acotc ~ +. (ii) ai, sin? iC = cot a cot b . . . (iii) @), Gi) ate Gi) &, sin? $4+sin? $B+sin? $C = cot b cot ¢4+-cot¢ cot a +cotacoth . . . (iv) wa, ava faatgfafe 4, ,_ _tana-+tanb-+tan c—tana tanb tane tan(atb-+e)= + 1 3Eeeent120 after fraterfafer at, 2 tanatand=tanatanb+tantanc-+tanctana Tq, a+b4c=nr a tan (a+-6-++c)= tan 7=0 tan a+-tan 6-+tan c= tan a tan 6 tane ai ax tan atand tane #1 aM Bo, cot b cot ¢-+cot ¢ cot a+cot acotb=1.. . (v) aa: (iv) ote (v) & sites ofagrr area a oat 21 waren 2. avila frys ABC # fre wa fr, —cos § tan Jatan B= oe a, 28 =A4B+C aed got qa (13) a, tan e= /(saais=men(S=0)) —cos § cos (S—B) tan p= d (a (S—Ajcos 0) . ; —cos S$ cos(S—A) tan jatandb= | Se osGeO)) «(Se eo) __—cos S ~ cos (S—C) aah ger H age are frat sais ara ta > 0 ab 0; eater wet vat aT ATT TATE HT H fore Healt HT aM UGH ear ae |aiteter farasr # faprenfirctr date lat sareet 3, ates Pras ABC # afe A=B=2C2 at fas we fe, 8 sin? 3C (cos s-}sin 4C) cos $C =cos a [arr, 1952] saafa— A=B fat’! ve a=b eee G) ak, © A=2C feared A ~C ¢ sin xasinC=2sing cosy aed ater qt (8) 4, sin (s—6) sin (s—e) C jfsinssin(s—e) Jt sin} sin ¢ )= asin g, al Sear) Asin? C, sins sin (0) sin (s—b) sin (s—e) sin’ 2 “sinasin b sin b sine C , sins _sin(s—b) m 4sints 2 sina” “sing | at 4 sin? $ sin s sin c= sin asin (s—6) =sin a sin (s—a) “s @) _e¢ « pfa+b+e =sin asin (Ay*)-4} =sina sin§ " @® ~4C. . 6 co. +o 4sin? Z sins 2sing cos 5=sin asing 2 272 2 oC. co eas, at, 8sin’ 5 sins cosy=sina . . + (iii)129 | attr Praeife aa, ; cos s-+sin 37 cos s+ [cose ean sin asin b sin (s—a) Sina = COS 5-++ ana “ @ _ Cos 5 sin asin (s—a) ~ sins _ d{sin (s-a)—sin (s—a)}+-sin (s—a) -_ sin a _ sin (s+-a)-+sin (s—a) 2Qsina 2 sin s cosa =o 2sina =sin scota » 4. (iv) iii) ste (iv) IM RA Tz, «oC ¢ . 6 . . 8 sin’ 7 Sin s cos 5 \coss-tsin 5) =sin a sin s cota 8 sin® $ (cos s-sin 5) cos 5 cos a wart ave 5 1. afeca ater frye & ata at att are gaat} acaz i at fra at fr, cos* 5tcos? Frcostf = 1 [arrer, ’54] 2. diet fry ABC # fag wa fr, « (a+b+e sin (“5 ) coe B on & sina = COS F COS 7meta fogs # faatyfndts date 123 S, 5, N ait n & ae wat & fre fag wet fr, A n 3. tan 5 Sin sin Ga) A B_sin (s=6) 4. tang tan5 o-—sins A B Cc 2 5. tang tan 5 tang= saints 6. tang -—__ 2° cos($S—B) cos (S—C) @ 4 ¢ cos S 7. tang tan 5 tan 5= ‘ N 8. sin(s—a) = a Rt 2sin 7 Cosy COST 9. —cos S=— n 2cos% cos b cos & 207° 2 2 10. cos(S—A) =" ;§ — 2 cos5 sing sin 5 ind sins sing Sing _sin (s—<) sin ¢ 11. @ C sin 5 ab 608 FOOD _ cos (S—C) @) ~~ o = Sin @ cos 5 . A. B_. Cc 12, sing tang —sin btan 3 =sin(a—) cots124 ateita Prerofifer 13, aitetta Pgs ABC H afe A= B=2C & at fra wa fe, 8sin (2+5) «sin? 5 . cos 5 = sin® a 14, ateta Fras ABC # af, . cos G=—tan § tan’, @ a fae ad fr, C=A+4B 15. aafrarg ater Prgs ABC & fire ae Fr, a. A 2cosssin 5 =1. 48 fafer gersco vem 1 (@) aia Pras ABC H fae wa fe, sin (A+B) _ cos a-+cos b “sind ~~ THeose ‘Pte, 1957] (@) aieia Pays ABC # afe C= A+B & at fra wet fe, 1—cos a—cos b-+-cos c=0 [amg, 1957] aevfe—(a) sin(4+B) _ sin A cos B-cos A sin B sinc sin C -(¢ sin B A sin 2) cos B +(ee cos area ae stargt at F ata waa gz, sin @\ cos b—cos a cosc¢ , /sin 8 cos a—cos b cos ¢ = (24) sin a sin ¢ (2 c) sindbsine (cos b-+-cos @)—cos a cos c—cos b cos ¢ ~ sin? ¢aivetta Paras # faretegfrctte eteter 125 (cos a+-cos b) (l—cos c) = 1—cos* ¢ _ C08 4-+Cos 4 ~"T--cos ¢ @ ofa (a) ¥# C= A+B war or, sin C_ cos a+cos b snC 1-+-cos¢ 1—cos a—cos b-++-ccsc=0 wargem 2. asta Pays ABC H faz at fs, sin ¢ 1—cos a cos b cos ¢ @ ee (; Taos d cos B cot ) [arrz, 1964] @ sin?a-}sin?b+sin?¢ _1—cos a cos b cos¢ sin? A+ sin? B+ sin? C™ 1--cos A cos B cos C [arrar, 1948] aanfa (=) Fraet YF F cos b cos ¢ FAM WAT Tz, 1—cos a cos b cos c= 1—cos a(cos a—sin b sin ¢ x cos A) =(1—cos?a)-++-cosa sind sine x cos A = K? {sin?A-}-cosa sin B sin Cx cos A} sina sind sine 7a, K= 3 d7 sin B sin C aa, aye Hater Br (4) cos A+-cos B cos C cos a= : = sin B sin C 1—cos a cos b cos c= K# {sin?.A+(cos A cos B cos C) x cos A}126 atteiter Preatertin fet = K? {(sin? 4+ cos?.4)-4-cos A cos B cos Cy} = K? (1+ cos A cos B cos C) K= ( 1—cos a cos} cos ¢ ) I+-cos A cos B cos C TH, cams “So ReTAT a, sina __1—cosacosb cose - sin A~ 1+-cos A cos B cos O @) erie aert k= Sina _sing sine , sin A sin B sin G ‘? Ssinta —Ssin?a DsintA~ 1K? Dsin? a =k? afar (a) 8K at art wat oz, Zsin?a _1—cos a cos b cos ¢ 2sin?A~ 1+-cos A cos B cos C saremt 3. tite faye ABC Raat Bais ant ear farg & afte favgail at ga mam: 8, 6, YF 1 fre wa fs cos, 6 sin (b—c)-+-cos ¢ sin (c—a)+-cos w sin (a—b)=0 [sera, 1954] argfr 52 am at ttt fryer ABC # atu & mis AD, BE sik CF, TRA UH Fat a O fang ac Het |afettr Brgse 8 Breteafattar ate 197 fet: 40=6, BO=¢ te CO=y fags AOB % sara are warat (6, 4, AB=«, B)2) ® atdare qa aT cos ccosf=sinc cot sin $ cot B soe (i) gat sare fay AOC % @tere a a, cos b cos #=sin beot 0—sin 4 cot $ we. fi) [@ xcosb .. Gi) x cos c] frat ae a, O=(sin 4 cos c—cos b sin ¢) cotd— sin 4 cos ¢cot s cosh cot a) cot @ sin (b—c)=sin $A (cos ¢ cot 4C—cos 6 cot 3B) at, cos @ sin (6—c)=sin 0 sin $ (cos c cot £ cos b cot) z 3 alga ga &, +o... Git} . sin@ sing A A0B%, Ea sing sing sin 0 sin 4=sin sin 2 - + + (iv) . sin sing Adocs, cA sing sins sin @sin $= singin § . .128 attetrr faratenfartt (iv) a (v) 3, sin 0 sin = sing sing =sin ¢ sin pak (aa at) ii) Fata wat oz cos @ sin (—o=k (cos ¢ cot 5 cos b cot 3 ) (vi) eat sare an firs ae ane g fs (sata aH gaaal at aay 8), cos ¢ sin (¢— a= K( cos acot 4 —cos¢ cot 5) (vii) 2 2 ak, cos ys sin (a—b)=K (cos b cot F—cos acot 5 )ositn aie cits oferay a) sieht &, cos 6 sin (b-+c)+-cos ¢ sin'(c—a)-+-cos # sin (a—b) = 0 varemt 4. atta fags ABC & af, A=B=2C a ot fre at fe, COS @ COS 3708 (45) -saafa—arfer Ta & aes Ze Tz, a a * - a COS a COS 5 COS ¢ COS 5 —Sin ¢ sin 5 2 2 2 : a r .. cos a= cos c—sin ¢ tan Dre qd) agra at ater qa (4) a, cos A--cos B cos C sin B sin C _cos 2C(1+cosC) ., =—mnICsn CO feat2?, A= B=2C cosa = =cot 2C cot 4C - . . (ii)ahaa Pays # faatfrdte aa 129 aa (i) & aet cet 8, args Atavet ga (4) COS ¢ FI, TAH AAT aa (6)8 sin site wdaty ga (13) & tan} at wa twat a, cos ¢—sin ¢ tan? = ©% C+-cos A cos B 2N o—sin ¢ tang = sinAsinB sin Asin B x eos S' cos (S—A) N — £08 G-+-cos A cos B+-cos ($-+$—A)+ cos (S—S+A) sin Asin BO — __ cos C+-cos A cos B-+-cos (B+-C)+-eos A 7 sin A sin B ‘os C+-cos? 2C-+-cos 3C+-cos 2C = “fear, A=B=2C _ (cos C-+cos 3C)+-cos 2C (1+-cos 2C) - sin? 2C _ 2 cos 2C cos C-+-2 cos 2C cos? C — sin? 2C a2 cos 2C' cos C(1+-cos C) Cc. . 4sin 2C cos 5 608 ysin gy z 4 cos 2C cos C. cost 4sin 2C cos C' cos Sain 5 =cot 2Ceot$ - +. (iii) Gi) ate Gil) & wae aaa Ta ame 1 We fro 9130 meta Parrott sat tag 6 1, ateta frat ABC H faz at fr, ‘ @) sin(a+b) cos A+cos B sin ¢ 1—cosC (@) afe atb=nte a a faa st fe, 1-+-cos A-+-cos B—cos C=0 2. atefa frat ABC & fae wet far, sin (B— C) _ cos c—cos b @) ‘sm A ~~ I—cosa ¢ sin (b—c) _ COs C—cos B 7) sin a 1-+-cos A [feces ser 1 atk 2, cera arqeratel & aaa & aa a aA EI zat art (4.11), (4.14)—satexa 2 | 3. atette Page ABC & gat b att c wm gat at aaqes Z| fee amet fr, (sin ¢ + cos ¢) sin A=2 cos? § sin (B+-C) 4) ater fry ABC at got AB ste AC® nem fargstt at faart ara dtdaa arr, Be C at aie agg Th amt BC at D farg 1% frerat 2 | fare aa fe, (@) BD+CD=2 (a) cos AD= sind? sin. —b cosec® 2 2 2 5. ateta Prat ABC % faz at fa, sin b sin c+-cos b cos ¢ cos A =sin B sin C—cos B cos C'cos a [sera, 1955] [ aea—atatest a 8,sitar Brags 8 ferateafircter ie 131 am qa =sin d sin ¢+-(cos a—sin b sin ¢ cos A) cos A =sin 5 sin ¢ (1—cos? 4)-Lcos a cos A =sin B sin C sin? a-+-cos a cos A =sin B sin C (1—cos? a)-+-cos a cos A =sin B sin C—cos a (cos A~sin B sin C cos a) =sin B sin C4-cos a (—cos B cos C) =sin B sin C—cos B cos C cos a] 6. ater Frys ABC ¥ fire at fr, cot a sin c= cot A sin B-L-cos ¢ cos B [fawr, 1962] [ata—arfet cer at aver ait stave & get aaa ae ret TAT Sherr GAT aT agTAtT FA |] 7. sitet frga ABC ¥ afa, C= ALB & fre 5a fr, () gst AB at meq fag faye Batt gta wart et REI at Gi Et Bo, HTH eT fig st aay gitee & front ay are at gett tet & | 8. ateia fay ABC at gone at aut aang a gonit arar a ait Pago ABC’ are aa | fare we Fe, cos A= cos A’—} tang tan 3 sin? A’ [arrerer, 1958] [ea cos $ — cos 2 cos Cos 4=a——_+, 3 2 sin 5 sin 5 sin A VU—Zcos* a+2 x cos} a) sing 6 sind ¢ No132 10. 11. 12, 13, ateter frarafafer ange cat 3 ga a aT waa aee HVA Tz, cos a—cos b cos ¢ atfant tt = din Bsince ~ 70° A meta fags ABC # fra ait fr, sin (b+-s) sin (b—c) sin? A=sin (B+-C) sin (B—C) sin? a [ata—acee 1 atk 2 & sfeqrat ar agai eet 1] ates frat ABC # afe gat AB at wea fag DQ at fae art fir, in? A—sin? cot BCD—cot ACD = 2 Aas’ B sin A sin B sin C _ sin (ad) sin (a—6) ~~ sina sin 6 sin ¢ idea fags ABC 3 faa A fa, cos a=cos (b-4-¢)+2 sin b sine cost $ [da—2 cost 4 =1-4c0s 4] afe ateita fags ABC at gat BC om agate & at fee wet fr, cos A+ cos B cos C=0 alee fags ABC tafe sie Aste BR aist at areit negra tard aera g at fag at fe, at at a=b in2£ = cos? 2 4 vos 2 2b saat, sin® 5 = cos’ gq TeO8s COS 5 + cos? 5 [ate—arer (4.2.3) F sargem 2% afearr ar vgaT aa i]aera Pras H fretted eat 133 14, ata frga ABC Hon ate N & are wat & fae fas wa fr, (@) 4nN =sin asin 6 sinc sin A sin B sin C n?__ 1—cos a cos b cos ¢ @ = feos A cos B cos C* 15, atta Pegs ABC eat sare at gfe got AB at eater ae ae dat wat aga farg CHF stax stat 8 | fre Fat fs, cot A cot B=cos® a fadta ave gta ax o: waaat & daar az 49. oTe-atatet Ya feet Weta fags at da gamit aka Atal e ae wert eatfta SAT | x Nes wa 90* mata 53 ara at fags ABC H, gat a, b, cate at Brat CF att aaa cari TEATS | cat BC at D fargo Ga THTE AST A BD= 90°, AD ai faansit | CD=(90°—a) ~.. @134 aii Frat fahe safe. fast ABD ¥, araret qa (1) &, cos AD=cos AB cos BD-+sin AB sin BD cos B =Ccos ¢ cos 90°-+sin ¢ sin 90° cos B =sin c cosB » es Gi) ak frys ACD #, eit sar, cos AD=cos b sin a—sin b cosacosc. . . (iii) Gi) ate Gi) 3, cos AD % art aaat ee Tz, sin ¢ cos B=cos bsina—sinbcosacosC . (14) we aT §, frye & saaat A anita ma F aaat ox Areafafaa at aa ak oe fret ot ant & | sin a cos C=cosc¢sinb—sinecesbcosA . (15) stk, sin b cos A=cos asinc—sinacosccosB . . (16) 49.1. fecqot 1 Pras at dt gf ate yoreit atk at aati Ft af @) feat om sty a ara: ate TT C, Gi) fe a® sty HF ga ata Fs B, (ii) fae yar & aa Sr az RA gu ato fear @ 3% wre #% qt BC=a,aireita fartsr H Prevent eater 135 (iv) area: Bt aay are Ua ora aT AY Sts BT gag AST FT area ya, FF yat AC= |b, site (v) Ba yar a tae yoy, FH yt AB=o, ve Tt TTET-ATA qa @ fret fafa are ere tar aT aaa zg | AAA AT [sin (aa: YT). cos (ATTA FT) —cos (a Yt). sin (ataa TIT). cos (HT: BIT) ] = [sin @tadt yar). cos Zaz stT)] AM aad 7 at Me stat B— sin a cos 6—cos a sin b cos B=sin ¢ cos B, at aga-waTEA q7(14)z1 aan atafe Bat oar at ama /A= saa a et! WE T= 6, Mas T= a se Aad yaI=d sit HR AM wat a< gat frosted ga sta etm, sin ¢ cos a—cos ¢ sin a cos B=sin 6 cos A ot argrwtatet ar (16) 1 wa: fea gaat stat H fat a es ay arr: HUY A aL, Ba TET Ie Gat Maal HT ATTA TS, ar amit agar, didi amratamd ga azaat gia fae oT ata él ferret 2 area-wlarea BA at STATA i) Arete Pago & are gaat a ate afe are at at staat waa SLAM AT FT THT E | Gi) ga qa a fate sate faust % Frater , afer feafeat HF equ de ae aa HAH Aa 2 1 Fa, aia at, fret freer # at gored (a, 6) site wad ata at at C aa gaa fram at Paka act a asta oa yet c ake atg B136 aire Fahafa fe BT ATT ate eet F ferq ga HaTeT ET cos ¢=cosacos b+sinasinbcos.C . . . (i) ak aa aa aT, Gi) alagei eae) waver yr Fa, bate CR ma aM eats ae Shoat am ma date l vege saa ae HRY aK OAK C& ae (fet ga) ari at (ii) Henfa ae ah BRA at ama att | ga dfava feafe (B at ata ata ate 8) & gearer att & faa gn areratarer aa cos ¢ cos B=sin a cos b—cos asin b cos ¢ ar watt aad Be fea urs ft ae qa aera aa Bat afeearfer art ght | ara ge qa a dt Bear a ana at oat & reart 6 @faz) | aita fart (Astronomy) % ¥f arateret qa gt ad ¥ saat fae ata & 1 feet aateite fas ar eee (fagatar, Right ascension) aig Saeittaa (THA, Declination) #17 at az, sae erat Tate gat metier aree Hey wT eA net atte FT fratey & agar @, fraat at yond site sak ata ar at ae 2 | 410, AIH WERE Ta am at, fags 4'B'C" fags ABC a1 ata Prys 2 | a: a =n—A, b' =2—B, ¢ =1—-C a! Mana, Barb Can ep 1 aa qata fry A’B'C’ & area atares gar (14), sin ¢’ sin B’=sin a’ cos b’—cos a’ sin 6’ cos C’ @) & wart & av wat a, sin (7—C) cos (n—b)=sin (m—A) cos (7—B) —cos (m—a) sin (n—B) cos (nm—c)ateta Gras & Prettiest 137 —sin Ccos )=sin A cos B—cos A sin B cos ¢ sin C cos b=sin A cos B-++cos Asin Bcose (17) wandt @adta anh aah ox Be a st ga are fea aT ana &1 at, at, B: waaat & dats GT zara (Delambre’s) ate Afrax (Napier’s) atgaraTd (Analogies) 41, FATT ATGWMATE (Delambre’s Analogies) cos(4f) cos (Sy *) = 2. 8 sin § cos § d sin § sin 5 2 (A+ o3(*3 . sin (5 2) co cos (“) ———- 2 « (20) wo ost . fA— : — cin (4) sn (FS) ... @D Cc + 6 cos 5 sing Bay's aTgeTaTe HY aedeay aM ager (18) ears FV | we ary a (8) az (9) 8, cos A+B =cos4 cos BL sin 5 sin > 2 2 2 2 2 = |(Sinbsne) Pawasine) -|(* (s—d) sin ea) JF (s—a) sin (s—e) ) sin 6 sin ¢ sin @ sin ¢138 aiteirr Paratha fiefer (2 (s—a) sae) sin «¢ sin sin a __ sin snd sin (s—})sin sin (s—6)sin (s—a) “ine l( sin 5 sin a “nein =a) _ Sin s—sin (s—c) (= (s—a) sin (;—b) ) a sin ¢ sin a sin 2 cos (+*) sin G4 = AS * sin © . 6 ¢ no cn (58)_ en (8) “sin C/2 ~~ cos «/2 are oteft aterrant it get ware ennfta at at aadt 21 wate fago % aaaat % dts canes areal & agin & ae aaah aura sre at tat & wife reward (18) site (21) sat (19) site 20) wat ot ree aR J Lats ard age erat afar gana (Delambre) % art WR AAT aera F ate S freast | 412, diaz atgwand (Napier’s Analogies) freafefae sarguia, gah atewet afteaa frre (Napier) & ae ot, afta ateeaarsit are & fers 2 | tan (442 ) cos (5) 2/7 ON 2! Le. (22) cot > 2ati Pras # faatafadte date 139 a5 nn coty sin (5) tan ats) A-B e )_ lz) (24) tan5 cos ( 3 ) tan (=) sin (452) tan§ “an (752) (25) arte avira a fate fatal ater entire feat ot aaa & | sarear — fafa—1 A Cc... B Cc tan At tan Sn tang tan 5+ tan gtang 2 2 1 A B —tan 5 fang aiaty qat arr en fas at ah g fe, (Ret ger 4, wer me 5) A B_ sin (s—c) tan 2 tan 27 sins sin (s—6) | sin (s—a) - A+B) Cc _ sins sin s tan + ) tan gy ain) =) sin s _ sin (s—4)+sin (s—a) ~~ sin s—sin (s—c) 2 sin (2-3-4) cos ca) ~~ be (2) ee FE)140 aftetter faratanfirfer fat 21 aret ga (5) 2, sin A_ sin B sina sind sin A+-sin B= K (sina+sin 6). . . (ii) ai sin A—sin B= K (sin a~sind). . iii) weaRT stare gr (3) &, cos A+-cos B cos C=sin BsinC cosa. . . (iv) =(K sin b)sinC cosa *5 (i) =K (am a) oe. @ aalt wae, cos B+-cos A cos C=(K sina) sin Ccosb. . . (v) (iv) ate (v) at sits? a, (cos A-+-cos.B) (1--cos C)= K sin C(sin b cos asin acos 1) . =K sin C sin (a+6) ss . (vi) Gi) a (vi) & ; . : & (sin a+sin 6) sin A+sin B = Sst) —> + & sin C sin (a+-6) cos A-++cos B ered } _K Gin asin 6) (1-+cos ©) = sin C sin (a+4) aa fratafafs &, A+B arr Tet =tan —>— so. + (Wii)aver Past # Prentice det 141 _ 2 sin (44) cos (54) 2eost $ - 2 sin (S +") cos (4) asin ¥ cos § a—b ol) cot Le. (wilt) cos (i) (vii) az (viii) 3, wn (Pe) lS) 2 a ) cot 5 cos. ( 5 gat sar, (iii) Hz (vi) B, sin A—sin B _ sin a—sinb 1-+cos C cos A--cos B~ sin (a+b) * sin@ sin 24 tan (45) = ae cot 5 tan “ary sin (3) a, oe ey (3) s+ + (23) a, ATERTAT (22) site (23) H gata Payor sramet & arr fa zara Beh He aTERTATT oq FT TH | fafa 31 cares arecraret & AAA (eT aren) Afaae steer - ma FT THT FL Fe, aarea areetat (20) sz (18), sn (72) (54 cos ¢ cos 2 4142 attire Prato firfar cos (+) cos (47) 2 ae sin cc os © x £98 5 AT at fpat &, tan 24-8 cos (7) — 7 Ne) » 22) cot C/2 cos +) (+ wil gare, Oy aftax grew wt cae erecta & ararear TTT aT famert aver Fefiner at ort eat #1 412.1. fequi carer sik afore arama a saragie tet & ga are # fe, (=) sa a wer ce 4, @ tant, in (=), tan (“$°) cea, a ante cet H are oH A 0) Se aw a ee Se co GH & a tan 458, tan a » keake, ot aft oer tae aver oat ATE I Serbs mecirer eT TERME aT ere ET H sate fe wat zt 413, sae 1 . afore arama & eaves aTqeraT aT rea att | fer —afirae aresmat (22) (A+B cl ccs (<5) | | |aireita Pergo faenrerfercitr eater 143 ava faatnfafa &, sec? @=1-+-tan? 6 a— by af ALB cos? cos* 5 (tg CE cost( 22) * sint sint $ sine’ S{1-+cos(a-+5)}-+cos § (1 +eos(a-+8)} cost (455) 2 cost (“F*) -. 2 cost = l+cos 6 aa, afer Tat FT, (sine +cos? S)+ sin? © 9 (Cos a.cos b—sina sin b) aw i 6 cos” 3 © (cos acos b+sin asin b) =1-+ cos a cos 6 (sin? C/2-+-cos? C/2)+- sina sin b (cos? C/2—s:n? C/2) =1-+cosa cos b+sin a sin b cos C =1+cos¢ *" states a7 (i) =2 cos? C/2 () ae catia ach ea TTT TR, cos (47) cos att) ~ C ‘os sin 9 c z ag cara argerat (18) 2 1 set sane, Ie agama sit Frafra # at aadt &1144 ait Prarrerferhet wager 2 aria fran ABC # fre wat fr, sin (A+B) _ cos a+cos b > snC ~ I+cos¢_ [arre, °57] aaafer | carey areqtat (18) atx (20), cos SS cost sin’ 7 coss 2 2 sin a cost? cos © _ os § / 2 “89 TM BWA Tz, _ A+B AB a+b a—b 2 sin —5~ cos _ 2608 gy COS ~~ +. CO 7 26 2 sing cos5 2 cos’ 2 ara Paatafata &, sin (A+B) _ cos a+cos b sinC 1 -F-cos ¢ feeqot—ag sea area ae FaTea Ga H aquatT F sarexy 1, “are (4.8) 3 ger Prem war & | Stat fafa at gear ae | sateen 3 meta Prat ABC # fre att fr, sin(a+6)tan (43) tan §=sin (a—B6) cot (437 )co$aetna aide 145 aaefer— afrere aveerat (22) site (23), tan aye cos ae CO = ath cot 3 cos A—B . a—b tan a) sin o cot sin @F2 2 2 qu ea fz, : A+B, A-B a—b . ab tan > tan 5— cos > sn rT Cc = a+b. a+b cot? 3 cos ~y—~ sin _ sin (a—b) “sin (@+6) sin (a+5) tan 4r8 tan : A-B._ OC =sin (a—b) cot zy cot T waren 4. aTeig frat ABC % fea vO fir, cos a tan B-+cos 6 tan A+tan C =cos a cos 6 tan A tan B tan C . [arret, 1962] fatal (@igratares aa % aqeatr 8) ateratater Ga (16), cos 6 sin a cos C-+-cos A sin ¢=cos a sin b area qa a, cos b (K sin A) cos C-+-cos A (K sin C) = cos a (K sin B) ate fro 10146 agtr Prerefite aet K at§ wax ufa 3 at, cos bsin A cos C-+-cos A sin C=cos a sin B =cos a tan B.cos B agen tater a (3) % cos B at wrt tat I, cos b sin A cos C-++cos A sin C =cos a tan B {cos A cos C+sin A sin C cos b} aii get ¥ cos A-cos Car att 34 Tz, cos b tan A+tan C=cos a tan B {—1-+-cosb fan A tan Cc} om, cos a tan B-+cos 6 tan A-+-tan C , : =cos a cos b tan A tan B tan C. fafa-2-caat—aind farg CR stare AB ox ‘CD wa a9 ait | ar at, : LACD=0 we /BCD=$ aefa 55 frys ACD & aeqes staret G7 (3) @, cos 90° = —cos A cos 6-+sin A sin 6 cos 6 aq, - cot @=tan A cos b oes @ater Pras 8 -faetregferctter ate 147 aat sare, frgt BCD 2,” . cot¢d=tan Beos@- '. . . (ii) 13a, *LC=04¢ ts eae : cot C=cot (6-4) ~ cot 6 cot d—1. cot cot ¢ @ se @ eae, cos a cos b tan A tan B—1 cot C= COS a t tan B-+-cos ‘btan A aT cosa tan B--cosb tan A tan C= cose cosh tan B tan saree 5. aafeatg atte frat ABG > att A & oH TT ator aa of stare BC at, Daeg re ga score seat @ fir BD = =a, DC=8 wh AD=8, fra 0 fe, tan 30 tan $B=tan 20-48) tan -) A ees << amet 56 am at anfeag Prag ABC H, bec eafac, ZB=20\. LADB=6 LADC=1—6 att ara ait,148 1 aRvetrer Pereira fags ABD # Ffrae atewaat (24) &, .tan $(b-+8) cos 0—B) ' “tan fa ~ cos 4(0-+B) & ait sar, frat ADC & Hfraz srewrat (25) a, tan tan $(5—8) b—8) _ sin (a—0—-C) tan “tan #8 * sin $7—0+C) LB=LC t tan?) cos (242) : tan B/2 cos cos (259) G) atk Gi) st ge aA a, tan a) tan (5°) _ cos (5") cos (*54) tan a/2 tan B/2 7 cos (3*) cos (5) : =1 nil wet aig 7 aieita fags ABC % faa #2 fr, sin(A—B) _sin’b—sin a. snC — 1—cose sin(a—b) _ cos B—cos A sinc 1-+cosC sin sin bre) _ “cos B--cosC sina _ l-cosG +C a B-C\.,4 cos? 5 +sin® ( 5 ) sint § 4. cos? ; =sin? Ptaieita Fangor. i Serene, ate * 149 5. sin? £ 5 =sint 4 cos? 4 tsin? (O55) sin . 4 4 6. ‘cos? Seccost (5) cos? At cost (A) sin? 7. afe aieta frye ABC at gard a, b,c aavat ait (Arith- metical Progresion) He at fee aa fe sin? 2B cos (54)= ‘A _ sin sin xy waa | a, b,c mame tat Hz batt? @ ak a+b+cr25 ba(~6), woes Gi) aaa arqerat (19) &, . C—A o+a cos (254 *) sin (F*) sin 2 sin 5/2 “ea ()) & (ii) ++ + (ii) are Frat ga (8) 3, sin B z sin fine nda fo" 2072150 S24 et Braden . B? ‘* |/sin (s—a) sin (s—e)\ - Sgn : df sin asin ¢: ) A. isin (s—b)sin(s—c) jsin (s— 6)sin(s—c) smysng J snbdsmce AN sinasinbd , __sind . 2, = sin vowee GY) (ii) st Gv) 3, : cos (4) sin 2 a2 7, Bb . A.C sin 5. sin > sing 8. dafgang aria fraw ABC & ote BC 4%, D om ee fry 2 | faa at fe, cos BD PEN 2 _ cos AD BC ~ cos AB cos =5* [eta—oareey 5 at apis Fagare, wear Preafefaea ST faa ast F— cos (=) cos 5 cos ‘cos (“FF cos b aie gaa atce afeura ersz 3 1] 9. afe fret ateita Pras % fame at ate at att 7 & ae Bt at faa wot fe ot stat at age ort ar ater ane & sr oe. aaa ara [aa— BiCecr ~ B+C 2 0 s BY. 0 . O0 “0 ¢ 90° ate b<90°1 arefa % BA sik BC at agrat at fare BY az faa & 1 Pras ABC, a)