CLSS XI ur I CHPTER.6. Proofs d Simpl pplictios of si d cosi formul Lt C b trigl. y gl w m t gl btw t sids d C wic lis btw 0 d 80. T gls d C r similrly dfid. T sids, C d C opposit to t vrtics C, d will b dotd by c, d b, rspctivly (s Fig..5). c b Torm (si formul) I y trigl, sids r proportiol to t sis of t opposit gls. Tt is, i trigl C si si si C b c Proof Lt C b itr of t trigls s sow i Fig..6 (i) d (ii). Fig..5 C c c b C D b D C (i) (ii) Fig..6 T ltitud is drw from t vrt to mt t sid C i poit D [i (i) C is producd to mt t ltitud i D]. From t rigt gld trigl D i Fig..6(i), w v si, i.., c si () c d si (80 C) sic From () d (), w gt c si si C, i.., si si C c () ()
Similrly, w c prov tt si si b From () d (4), w gt si si si C b c For trigl C i Fig..6 (ii), qutios () d (4) follow similrly. Torm (Cosi formul) Lt, d C b gls of trigl d, b d c b lgts of sids opposit to gls, d C, rspctivly, t b + c bccos b c + ccos c + b bcosc Proof Lt C b trigl s giv i Fig..7 (i) d (ii) (4) c c b C D D b C (i) Fig..7 Rfrrig to Fig..7 (ii), w v C D + DC D + (C D) D + D + C C.D + or b + c bc cos Similrly, w c obti C C cos b c + c d cos c + b bcosc Sm qutios c b obtid for Fig..7 (i), wr C is obtus. covit form of t cosi formul, w gls r to b foud r s follows: (ii) b + c cos bc c + b cos c + b c cos C b
Empl 5 I trigl C, prov tt C b c t cot b + c C c t cot c + b C t cot + b Proof y si formul, w v Trfor, b c ksy ( ). si si si C b c k(si si C) b+ c k(si + si C) + C C cos si + C C si cos (+C) ( C) cot t π C cot t C t cot C b c Trfor, t cot b + c Similrly, w c prov otr rsults. Ts rsults r wll kow s Npir s logis. Empl 6 I y trigl C, prov tt si ( C) + b si (C ) + C si ( ) 0 Solutio Cosidr si ( C) [si cosc cos sic] () si si si C Now k(sy) b c Trfor, si k, si bk, si C ck Substitutig t vlus of si d sic i () d usig cosi formul, w gt + b c c + b si( C) bk ck b c
k k( b ( + b c c + b ) c ) Similrly, b si (C ) k (c ) d csi ( ) k ( b ) Hc L.H.S k (b c + c + b ) 0 R.H.S. Empl 7 T gl of lvtio of t top poit P of t vrticl towr PQ of igt from poit is 45 d from poit, t gl of lvtio is 60, wr is poit t distc d from t poit msurd log t li wic mks gl 0 wit Q. Prov tt d ( ) Proof From t Fig..8, w v PQ 45, Q 0, PH 60 P 5 45 d 0 60 H Q Fig..8 Clrly PQ 45, PH 0, givig P 5 gi P 5 P 50 From trigl PQ, w v P + (Wy?) or P pplyig si formul i Δ P, w gt P d si5 si50 si5 si50 i.., d si5 si 0 ( ) (wy?) Empl 8 lmp post is situtd t t middl poit M of t sid C of trigulr plot C wit C 7m, C 8m d 9 m. Lmp post subtds gl 5 t t poit. Dtrmi t igt of t lmp post. Solutio From t Fig..9, w v 9 c, C 7 d C 8 b. 4
Fig..9 M is t mid poit of t sid C t wic lmp post MP of igt (sy) is loctd. gi, it is giv tt lmp post subtds gl θ (sy) t wic is 5. pplyig cosi formul i ΔC, w v + b c 49 + 64 8 cos C () b 7 8 7 Similrly usig cosi formul i ΔMC, w gt M C + CM C CM cos C. Hr CM C4, sic M is t mid poit of C. Trfor, usig (), w gt M 49 + 6 7 4 7 49 or M 7 Tus, from ΔMP rigt gld t M, w v PM t θ M 7 or t(5 ) 7 (wy?) or 7( )m. EXERCISE.5 I y trigl C, if 8, b 4, c 0, fid 4. cos, cos, cosc (s. 5, 5, 0). si, si, sic (s. 5, 4 5, ) For y trigl C, prov tt. cos + b c C si 4. si b c C cos 5
5. C b c si cos 6. (b cos C c cos ) b c 7. (cos C cos ) (b c) cos 8. si( C) si( + C) b c 9. C C ( b + + c)cos cos 0. cos + b cos + c cos C si si C.. cos cos cos C + b + c + +. (b c ) cot + (c ) cot + ( b ) cotc 0 b c bc b c c b si+ si+ sic 0 b c 4. tr stds vrticlly o ill sid wic mks gl of 5 wit t orizotl. From poit o t groud 5m dow t ill from t bs of t tr, t gl of lvtio of t top of t tr is 60. Fid t igt of t tr. (s. 5 m) 5. Two sips lv port t t sm tim. O gos 4 km pr our i t dirctio N45 E d otr trvls km pr our i t dirctio S75 E. Fid t distc btw t sips t t d of ours. (s. 86.4 km (ppro.)) 6. Two trs, d r o t sm sid of rivr. From poit C i t rivr t distc of t trs d is 50m d 00m, rspctivly. If t gl C is 45, fid t distc btw t trs (us.44). (s. 5.5 m) 5.7. Squr-root of Compl Numbr CHPTER 5 W v discussd solvig of qudrtic qutios ivolvig compl roots o pg 08-09 of ttbook. Hr w pli t prticulr procdur for fidig squr root of compl umbr prssd i t stdrd form. W illustrt t sm by mpl. Empl Fid t squr root of 7 4i Solutio Lt + iy 7 4i T ( + iy) 7 4i or y + yi 7 4i Equtig rl d imgiry prts, w v y 7 () y 4 W kow t idtity ( ) ( ) + y y + ( y) 49 + 576 65 Tus, + y 5 () 6
From () d (), 9 d y 6 or ± d y ±4 Sic t product y is gtiv, w v, y 4 or,, y 4 Tus, t squr roots of 7 4i r 4i d + 4i EXERCISE 5.4 Fid t squr roots of t followig:. 5 8i ( s. 4i, + 4i). 8 6i (s. i, + i). i (s. + ± μ i ) 4. i (s. ± m i ) 5. i (s. i + ± ± ) 6. + i (s. ± ± i ) 9.7. Ifiit G.P. d its Sum CHPTER 9 G. P. of t form, r, r, r,... is clld ifiit G. P. Now, to fid t formul for fidig sum to ifiity of G. P., w bgi wit mpl. Lt us cosidr t G. P., 4,,,... 9 Hr, r. W v S Lt us study t bviour of s bcoms lrgr d lrgr: 5 0 0 0.6667 0.68748 0.074599 0.000007866 W obsrvr tt s bcoms lrgr d lrgr, bcoms closr d closr to zro. Mtmti- clly, w sy tt s bcoms sufficitly lrg, bcoms sufficitly smll. I otr words s 7
, 0. Cosqutly, w fid tt t sum of ifiitly my trms is giv by S. Now, for gomtric progrssio,, r, r,..., if umricl vlu of commo rtio r is lss t, t ( r ) r S ( r) r r I tis cs s, r 0 sic r <. Trfor S r Symboliclly sum to ifiity is dotd by S or S. Tus, w v S r. For mpls (i) +.... + + + (ii) + +... + EXERCISE 9.4 Fid t sum to ifiity i c of t followig Gomtric Progrssio..,,,... (s..5). 6,.,.4,... (s. 7.5) 9. 5, 0 80,,... 7 49 (s. 5 ) 4.,,,... 4 6 64 (s. ) 5 5. Prov tt 4 8... 6. Lt + + +... d y + b + b +..., wr < d b <. Prov tt y + b + b +... + y CHPTER 0 0.6 Equtio of fmily of lis pssig troug t poit of itrsctio of two lis Lt t two itrsctig lis l d l b giv by + y + C 0 () d + y + C 0 () From t qutios () d (), w c form qutio ( ) + y+ C+ k + y + C 0 () 8
wr k is rbitrry costt clld prmtr. For y vlu of k, t qutio () is of first dgr i d y. Hc it rprsts fmily of lis. prticulr mmbr of tis fmily c b obtid for som vlu of k. Tis vlu of k my b obtid from otr coditios. Empl 0 Fid t qutio of li prlll to t y-is d drw troug t poit of itrsctio of 7y + 5 0 d + y 7 0 Soluio T qutio of y li troug t poit of itrsctio of t giv lis is of t form 7 y + 5 + k( + y 7) 0 () i.., ( + k) + ( k 7) y + 5 7k 0 If tis li is prlll to y-is, t t cofficit of y sould b zro, i.., k 7 0 wic givs k 7. Substitutig tis vlu of k i t qutio (), w gt 44 0, i.., 0, wic is t rquird qutio. EXERCISE 0.4. Fid t qutio of t li troug t itrsctio of lis + 4y 7 d y + 0 d wos slop is 5. (s. 5 7y + 8 0 ). Fid t qutio of t li troug t itrsctio of lis + y 0 d 4 y + 7 0 d wic is prlll to 5 + 4y 0 0 (s. 5 + y 7 0). Fid t qutio of t li troug t itrsctio of t lis + y 4 0 d 5y 7 tt s its -itrcpt qul to 4. (s. 0 + 9y + 40 0. ) 4. Fid t qutio of t li troug t itrsctio of 5 y d + y 0 d prpdiculr to t li 5 y 0. (s. 6 + 05y 78 0.) 0.7. Siftig of origi qutio corrspodig to st of poits wit rfrc to systm of coordit s my b simplifid by tkig t st of poits i som otr suitbl coordit systm suc tt ll gomtric proprtis rmi ucgd. O suc trsformtio is tt i wic t w s r trsformd prlll to t origil s d origi is siftd to w poit. trsformtio of tis kid is clld trsltio of s. T coordits of c poit of t pl r cgd udr trsltio of s. y kowig t rltiosip btw t old coordits d t w coordits of poits, w c study t lyticl problm i trms of w systm of coordit s. Fig. 0. To s ow t coordits of poit of t pl cgd udr trsltio of s, lt us tk poit P (, y) rfrrd to t s OX d OY. Lt O X d O Y b w s prlll to OX d OY rspctivly, wr O is t w origi. Lt (, k) b t coordits of O rfrrd to t old s, i.., OL d LO k. lso, OM d MP y (s Fig.0.) 9 Y O L Y' 0' M' k P{(, y) ( ', y')} M X' X
Lt O M d M P y b rspctivly, t bsciss d ordits of poit P rfrrd to t w s O X d O Y. From Fig.0., it is sily s tt OM OL + LM, i.., + d MP MM + M P, i.., y k + y Hc, +, y y + k Ts formul giv t rltios btw t old d w coordits. Empl Fid t w coordits of poit (, 4) if t origi is siftd to (, ) by trsltio. Solutio T coordits of t w origi r, k, d t origil coordits r giv to b, y 4. T trsformtio rltio btw t old coordits (, y) d t w coordits (, y ) r giv by + i.., d y y + k i.., y y k Substitutig t vlus, w v d y 4 6 Hc, t coordits of t poit (, 4) i t w systm r (, 6). Empl Fid t trsformd qutio of t strigt li y + 5 0, w t origi is siftd to t poit (, ) ftr trsltio of s. Solutio Lt coordits of poit P cgs from (, y) to (, y ) i w coordit s wos origi s t coordits, k. Trfor, w c writ t trsformtio formul s + d y y. Substitutig, ts vlus i t giv qutio of t strigt li, w gt ( + ) (y ) + 5 0 or y + 4 0 Trfor, t qutio of t strigt li i w systm is y + 4 0 EXERCISE 0.5. Fid t w coordits of t poits i c of t followig css if t origi is siftd to t poit (, ) by trsltio of s. (i) (, ) (s (4, )) (ii) (0, ) (s. (, )) (iii) (5, 0) (s. (8, ) ) (iv) (, ) (s. (, 0)) (v) (, 5) (s. (6, )). Fid wt t followig qutios bcom w t origi is siftd to t poit (, ) (i) + y y y + 0 (s. y + y + 6y+ 0) (ii) y y + y 0 (iii) y y + 0 (s. y y 0 ) (s. y 0 ) CHPTER.5. Limits ivolvig potil d logritmic fuctios for discussig vlutio of limits of t prssios ivolvig potil d logritmic fuctios, w itroduc ts two fuctios sttig tir domi, rg d lso sktc tir grps rougly. 0
Lord Eulr (707D 78D), t grt Swiss mtmtici itroducd t umbr wos vlu lis btw d. Tis umbr is usful i dfiig potil fuctio d is dfid s f (), R. Its domi is R, rg is t st of positiv rl umbrs. T grp of potil fuctio, i.., y is s giv i Fig... Y O grp of y X Fig.. Similrly, t logritmic fuctio prssd s log : R + Ris giv by log y, if d oly if y. Its domi is R + wic is t st of ll positiv rl umbrs d rg is R. T grp of logritmic fuctio y log is sow i Fig... Y O X grp of y log Fig.. I ordr to prov t rsult lim, w mk us of iqulity ivolvig t prssio 0 wic rus s follows: + + ( ) olds for ll i [, ] ~ {0}.
Torm 6 Prov tt lim 0 Proof Usig bov iqulity, w gt lso + lim 0 + + lim + ( ), [, ] ~ {0} 0 + 0 d lim[ + ( ) ] + ( )lim + ( )0 0 0 Trfor, by Sdwic torm, w gt lim 0 log( + ) Torm 7 Prov tt lim 0 Proof Lt log (+ ) log ( + ) y + y y y or. y y y. T y lim lim y (sic 0 givs y 0) y 0 y 0 y lim y s lim 0 y 0 y log ( + ) lim 0 Empl 5 Comput lim 0 Solutio W v lim lim 0 0 y lim, wry y 0 y.
Empl 6 Comput lim 0 si si si Solutio W v lim lim 0 0 si lim lim 0 0 0 log Empl 7 Evlut lim Solutio Put +, t s 0. Trfor, log log ( + ) log ( + ) lim lim sic lim 0 0 EXERCISE. Evlut t followig limits, if ist 4 +. lim (s. 4). lim 0 0 5 si. lim (s. 5 ) 4. lim 5 5 0 5. 7. lim log( + ) lim 0 (s. ) 6. (s. ) 8. ( ) lim 0 cos log ( + ) lim 0 si (s. ) (s. ) (s. ) (s. )