Microsoft Word - unit 13_Trigonometry_F.docx

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1 wî KvYwgwZ Tignmety f~wgkv wî KvYwgwZ kãwu G m Q `ybwu MÖxK kã Tignn Ges metn _ K Tignn k ãi A_ wzbwu KvY hv wîfzr Ges metn k ãi A_ cwigvc A_ v wî KvYwgwZ A_ wîfz Ri cwigvc MwY Zi h kvlvq wzbwu KvY I wzbwu evûi cwigvc Ges Z`m úk xq welq Av jvwpz nq Zv K wî KvYwgwZ ejv nq AwZ cövpxb Kv j wî KvYwgwZi cwiwa ïaygvî wîfz Ri KvY, evû I ÿîdj wby qi g a mxgve wqj ez gv b MwY Zi h Kvb kvlvq wkÿvjv fi Rb wî KvYwgwZi Ávb GKvšÍ Acwinvh wî KvYwgwZ `ybwu kvlvq wef³ mgzj wî KvYwgwZ (Plane Tignmety) Ges MvjKxq wî KvYwgwZ (Spheical Tignmety) Avgv `i Av jvpbv ïaygvî mgzj wî KvYwgwZ Z mxgve _vk e BDwb Ui D Ïk GB BDwbU k l Avcwb wî KvYwgwZK KvY m ú K ej Z cvi eb, wî KvYwgwZK Kv Yi wewfbœ cwigvc Ki Z cvi eb, m~² Kv Yi wî KvYwgwZK AbycvZ wby q Ki Z cvi eb, mshy³ Kv Yi wî KvYwgwZK AbycvZ wby q Ki Z cvi eb, wewfbœ mgm vi mgvavb Ki Z cvi eb BDwbU BDwbU mgvwßi mgq BDwbU mgvwßi m ev P mgq 0 w`b GB BDwb Ui cvvmg~n cvv.: R vwgwzk KvY I wî KvYwgwZK KvY cvv.: KvY cwigv ci GKK cvv.: KvY cwigv ci wewfbœ cöwzávmg~n cvv.4: KvY cwigv ci wewfbœ GK Ki g a m úk cvv.5: wî KvYwgwZK AbycvZmg~n cvv.6: wî KvYwgwZK AbycvZ jvi cvi úwik m úk cvv.7: wî KvYwgwZK Abycv Zi aªæezv I mxgve Zv cvv.8: wî KvYwgwZK AbycvZ: mgm v I mgvavb cvv.9: wba vwiz Kv Yi wî KvYwgwZK Abycv Zi gvb cvv.0: mshy³ Kv Yi wî KvYwgwZK AbycvZ cvv.: mshy³ Kv Yi wî KvYwgwZK AbycvZ: mgm v I mgvavb

2 I cb zj GmGmwm cövmövg cvv. R vwgwzk KvY I wî KvYwgwZK KvY cvvwfwëk D Ïk GB cvv k l Avcwb R vwgwzk I wî KvYwgwZK KvY ey bv Ki Z cvi eb, abvz K KvY I FYvZ K Kv Yi msáv ej Z cvi eb, PŠKY ev PZzf vm e vl v Ki Z cvi eb, wî KvYwgwZK Kv Yi Ae vb Kvb PZzf v M _vk e Zv e vl v Ki Z cvi eb gyl kã R vwgwzk KvY, wî KvYwgwZK KvY, abvz K, FYvZ K, PŠKY ev PZyf vm g~jcvv R vwgwzk KvY: R vwgwz Z mvaviyz `ybwu iwk i wgj b KvY Dcbœ nq Ges Kv Yi cwigvy 0 n Z 60 Gi g a mxgve _v K Ges Zv memgqb abvz K nq, KLbI FYvZ K nq bv wî KvYwgwZK KvY: wî KvYwgwZ Z GKwU N~Y vqgvb iwk Aci GKwU w i iwk i cöwÿ Z Ny i wbw` ó Ae v b Avm Z h cwigvy AvewZ Z nq, ZvB Y B H iwk Øviv m ó Kv Yi cwigvc A g b Kiæb XX Ges YY `ybwu w i iwk j ^fv e Aew Z GLb hw` GKwU N~Y vqgvb iwk Avw` Ae &vb X n Z Nwi KuvUvi wecixz w` K N~Y b ïiæ K i A Ae v b G m w i nq Zvn j wî KvYwgwZK Kv Yi X X mávbymv i N~Y vqgvb iwk Øviv m ó Kv Yi cwigvy XA GLb hw` GB iwk wu GKB w` K Nyi Z Nyi Z Avw` Ae vb X cvi n q B C Ae v b G m w i nq Zvn j m ó Kv Yi cwigvy n e XB Ges Zv Y Pvi mg KvY A c v e nëi Avevi hw` N~Y vqgvb iwk wu Nwi KuvUv hw` K Ny i Avw` Ae vb n Z mw` K Ny i C Ae v b G m w i nq Zvn j m ó Kv Yi cwigvy n e XC wbw` ó cwigvy KvY wpwýz Kivi ci N~Y vqgvb ilvwu h cöv ší Ae vb K i mb cövší Ae vb K cövwšík ilv ev e vmva f±i (Radius Vect) ejv nq wî KvYwgwZK KvY jv K mvaviyz A, B, C, (Avjdv), (weuv), (Mvgv), (w_uv) BZ vw` A i Øviv m~wpz Kiv nq abvz K I FYvZ K KvY: Dc iv³ Av jvpbvq Avgiv A iwk Ges B iwk K Nwi KuvUv hw` K N~ i Zvi wecixz w` K N~i Z ` LwQ Ges C iwk K Nwi KuvUv hw` K N~ i mw` K N~i Z ` LwQ myzivs Kv bv iwk K Nwi KuvUvi wecixz w` K (Anticlckwise) N~iv j Dcbœ KvY K abvz K (Psitive) KvY ejv nq Ges Kv bv iwk K Nwi KuvUvi w` K (Clckwise) N~iv j Dcbœ KvY K FYvZ K (Negative) KvY ejv nq myzivs wî KvYwgwZK Kv Yi cwigvy 0 0 n Z ïiæ K i h Kvb gv bi n Z cv i Ges Zv abvz K I FYvZ K h Kvb gv bi n Z cv i Z e R vwgwzk Kv Yi cwigvc ïay abvz K, KLbI FYvZ K nq bv wî KvYwgwZK Kv Yi cwigvc Rvb Z n j PŠKY ev PZzf vm m ú K Rvb Z n e 4 evsjv `k Dš y³ wek we` vjq

3 D PZi MwYZ BDwbU PŠKY ev PZzf vm (Quadant): wpî j Ki j `Lv hv e, XX Ges YY mij ilv `ybwu ci úi K j ^fv e Q` K i mgzj ÎwU K PviwU As k wef³ K i Q GB PviwU As ki cö Z KwU K PŠKY ev PZzf vm (Quadant) e j wp Î XY, YX, XY I YX Ask K h_vμ g cö_g, wøzxq, Z Zxq I PZz_ PŠKY ejv nq wî KvYwgwZK Kv Yi cwigvy hvb nvk bv Kb Zv h Kvb GKwU PŠK Yi g a Ae vb Ki e cöwz PŠK Y Kv Yi cwigvy 90 0 ZvB, Dc ii Av jvpbv _ K ejv hvq GKwU abvz K Kv Yi cwigvc 90 A c v Kg n j g PZzf v M _vk e Avevi 60 I 450 g a _vk ji KvYwU g PZzf v MB _vk e GKBfv e Kv bv abvz K Kv Yi gvb 90 _ K 80 Gi g a _vk j q PZzf v M, 80 I 70 g a _vk j KvYwU q PZzf v M, Ges 70 I 60 Gi g a _vk j 4_ PZzf v M _vk e Abyiƒcfv e GKwU FYvZ K Kv Yi cwigvc 90 _ K 0 g a _vk j 4_ PZzf v M, 80 _ K 90 Gi g a q PZzf v M, 70 _ K 80 Gi g a q PZzf v M I 60 _ K 70 Gi g a _vk j g PZzf v M _vk e 0 I 60 ev Gi h Kv bv c~y mvswl K wyzk XX ilvi Ges 90 I 70 G `i h Kv bv c~y mvswl K wyzk YY ilvi Dci Ae vb Ki e GLb Avmyb Avgiv D`vni Yi gva g eys Z Póv Kwi wî KvYwgwZK KvY wkfv e Kvb& PŠK Y Ae vb K i D`vniY : 80 KvY Kvb& PŠK Y Ae vb K i? mgvavb: 80 KvYwU PZz_ PŠK Y Ae vb Ki e h nzz 80 = 90+0 e vl v: h nzz 80 KvYwU abvz K AZGe KvYwU Dcbœ n e KvY m wókvix N~Y vqgvb iwk wu Nwi KuvUvi wecixz w` K N~Y bi d j GLv b N~Y vqgvb ilvwu Nwi KuvUvi wecixz w` K Nyi Z Nyi Z wzbwu PŠK Y 70 AwZμg K i Ges Aewkó 0 Ny i 4_ PŠK Y A Ae v b _vk e myzivs 80 KvYwU 4_ PŠK Y Ae vb Ki e D`vniY : 80 KvYwU Kvb& PŠK Y Ae vb Ki e mgvavb: h nzz 80 = ( 90+0) 80 KvYwU g PŠK Y Ae vb Ki e e vl v: h nzz 80 KvbwU FYvZ K AZGe KvbwU Dcbœ n e KvY m wókvix N~Y vqgvb iwk wu Nwi KuvUvi w` K NyY bi d j GLv b N~Y vqgvb iwk wu Avw` Ae vb X n Z Nwi KuvUv hw` K Nv i mw` K Nyi Z Nyi Z wzbwu PŠK Y 70 AwZμg Kivi ci Aewkó 0 AwZμg K i g PŠK Y A Ae v b _vk e myzivs 80 KvYwU g PŠK Y Ae vb Ki e X X X q PŠKY q PŠKY Y 80 Y 80 Y Y Y Y A g PŠKY 4_ PŠKY A X X X mvims ÿc R vwgwz Z mvaviyz `ybwu iwk i wgj b KvY Dcbœ nq Ges Kv Yi cwigvy 0 n Z 60 Gi g a mxgve _v K Ges Zv memgqb abvz K nq, KLbI FYvZ K nq bv wî KvYwgwZ Z GKwU N~Y vqgvb iwk Aci GKwU w i iwk i cöwÿ Z Ny i wbw` ó Ae v b Avm Z h cwigvy AvewZ Z nq, ZvB H iwk Øviv mó Kv Yi cwigvc wî KvYwgwZ 5

4 I cb zj GmGmwm cövmövg Kv bv iwk K Nwi KuvUvi wecixz w` K N~iv j Dcbœ KvY K abvz K KvY ejv nq Ges Kv bv iwk K Nwi KuvUvi w` K N~iv j Dcbœ KvY K FYvZ K KvY ejv nq cv VvËi g~j vqb.. wb œ DwjøwLZ KvY jv Kvb& PZzf v M Ae vb Ki e wby q Kiæb (i) 440 (ii) 50 (iii) 55 (iv) 740 cvv. KvY cwigv ci GKK cvvwfwëk D Ïk GB cvv k l Avcwb KvY cwigv ci wewfbœ c wz jv ey bv I e vl v Ki Z cvi eb gyl kã lvug~jk, kzg~jk, e Ëxq, wwwmö, MÖW, iwwqvb g~jcvv wî KvYwgwZK Kv Yi cwigvc msávbymv i mg Kv Yi cwigvy nj w i ev aªæe (Cnstant) mg KvY K g~j GKK a i wî KvYwgwZK KvY cwigv ci Rb wzb cökvi c wz e envi Kiv nq GB c wz jv njñ (i) lvug~jk c wz (Sexagesimal System) (ii) kzg~jk c wz (Centesimal System) (iii) e Ëxq c wz (Cicula System) (i) lvug~jk c wz: lvug~jk c wz Z mg KvY K KvY cwigv ci GKK aiv nq GB c wz Z GK mg KvY K ev 90 0 KvY K mgvb 90 fv M fvm K i cöwzwu fvm K GK wwmöx ejv nq GK wwwmö K mgvb 60 fvm K i cöwzfvm K GK wgwbu (=ne Minute) Ges GK wgwbu K mgvb 60 fvm K i cöwz fvm K GK m KÛ (= ne Secnd) aiv nq cöwzwu wwwmö K 60 wgwb U Ges cöwzwu wgwbu K 60 m K Û fvm Kiv nq A_ v mg KvY = 90 (beÿb wwmöx) = 60' (lvu wgwbu) ' = 60" (lvu m KÛ) z`ªzg fvm jv 60 e j Gi bvgkiy n q Q lvug~jk KvY cwigv ci GB c wz K mvaviy (Cmmn) ev e wuk (Bitish) c wzi ejv nq (ii) kzg~jk c wz: GB c wz Z GK mg KvY K mgvb 00 fv M fvm K i cöwzwu fvm K MÖW ejv nq cöwzwu MÖW K GK kzg~jk wgwbu Ges cöwzwu GK kzg~jk wgwbu K GKkZg~jK m KÛ fvm Kiv nq A_ v mg KvY = 00 g (GKkZ MÖW) = 00 (GKkZ kzg~jk wgwbu) = 00 (GKkZ kzg~jk m KÛ) 6 evsjv `k Dš y³ wek we` vjq

5 D PZi MwYZ BDwbU z`ªzg fvm jv 00 e j Gi bvgkiy n q Q kzg~jk GB c wz K divwm c wzi e j (iii) e Ëxq c wz: GB c wz Z g~j GKK nj iwwqvb (Radian) G K c wpý Øviv cökvk Kiv nq h Kvb e Ëi e vmv a i mgvb e ËPvc H e Ëi K `ª h KvY Dcbœ K i Zv K ejv nq GK iwwqvb KvY iwwqvb GKwU w i (Cnstant) KvY Q wp Î PQR e Ëi K `ª, e Ëi e vmva P = Ges e vmv a i mgvb Pvc PQ PQ Pvc R K `ª Z PQ Dcbœ K i Q D³ Kv Yi cwigvyb GK iwwqvb A_ v PQ P GK iwwqvb mvims ÿclvug~jk c wz Z mg KvY K KvY cwigv ci GKK aiv nq mg KvY K ev 90 0 KvY K mgvb 90 fv M fvm K i cöwzwu fvm K GK wwmöx ejv nq GB c wz Z GK mg KvY K mgvb 00 fv M fvm K i cöwzwu fvm K MÖW ejv nq GB c wz Z g~j GKK nj iwwqvb (Radian) G K c wpý Øviv cökvk Kiv nq h Kvb e Ëi e vmv a i mgvb e ËPvc H e Ëi K `ª h KvY Dcbœ K i Zv K ejv nq GK iwwqvb KvY iwwqvb GKwU w i (Cnstant) KvY cv VvËi g~j vqb.-. KvY cwigv ci wewfbœ c wz jv ey bv I e vl v Kiæb cvv. KvY cwigv ci wewfbœ cöwzávmg~n cvvwfwëk D Ïk GB cvv k l Avcwb Kv Yi wwmöx cwigvc I iwwqvb cwigv ci m úk wby qi Rb wb œv³ cöwzávmg~n ey bv I e vl v Ki Z cvi eb gyl kã e Ë, cwiwa, e vm AbycvZ g~jcvv Kv Yi wwmöx cwigvc I iwwqvb cwigv ci m úk wby qi Rb wb œv³ cöwzávmg~n Ges Kv Yi e Ëxq cwigvc m ú K Rvbv cö qvrb cöwzáv : h Kvb `ybwu e Ëi ^ ^ cwiwa I e v mi AbycvZ mgvb cögvy: g b Kiæb `ybwu e Ëi mvaviy K `ª e e ËwU Z n msl K mgvb evû wewkó ABCD... eûfzr AsKb Kiæb, A;, B;, C;, D;... hvm Kiæb GB ilv wj QvU e ËwU K A', B', C', D',...we `y Z Q` K i GLb wî KvYwgwZ 7 D C C B A B A

6 I cb zj GmGmwm cövmövg A',B'; B', C'; C', D';... hvm Kiæb Zvn j A'B'C'D'... ÿîwu QvU e Ë AšÍwj wlz n msl K mgvb evû wewkó eûfzr n e GLb A = B (e e Ëi `ybwu e vmva ) Ges A = B ( QvU e Ëi `ybwu e vmva ) A B myzivs, A B AZGe, AB, AB Ges AB Gi mvaviy KvY AZGe, AB Ges AB m` k KvYx wîfzr AB A AB A n. AB A ev, n. AB A n n A B ev,. AB. A A e e Ëi AšÍwj wlz eûfz Ri cwimxgv QvU e Ëi AšÍwj wlz eûfz Ri cwimxgv A_ v e e Ëi e vmva = QvU e Ëi e vmva...(i) GLb eûfz Ri evû msl v n hz ekx n e AB, AB... Ges Ab vb evû wji ˆ`N ZZ QvU n e hw` n Gi gvb Amxg nq Zvn j Dfq eûfz R evû wj e Ëi cwiwai mv _ cövq wg k _v K myzivs (i) bs mgxkiy wb œiƒc AvKvi aviy Ki e, e e Ëi cwiwa QvU e Ëi cwiwa e e Ëi e vmva = QvU e Ëi e vmva e e Ëi cwiwa QvU e Ëi cwiwa ev, e e Ëi e vm = QvU e Ëi e vm... (ii) Gfv e h Kvb msjmœ e Ë A¼b K i (ii) bs mgxki Yi mz Zv cögvy Kiv hvq myzivs h Kvb `ybwu e Ëi ^ ^ cwiwa I e v mi AbycvZ mgvb gšíe : h Kv bv e Ëi cwiwa I e v mi AbycvZ memgq mgvb I GKB aªæe msl v h Kv bv e Ëi cwiwa mb e Ëi e vm = aªæe msl v... (iii) G aªæe msl vwu K wmök ey π (cvb) Øviv cökvk Kiv nq π GKwU Ag~j` msl v Ges `kwg K cökvk Ki j GwU GKwU AšÍnxb A cšbtcywbk msl v (π ) gšíe : mvaviyz Pvi `kwgk vb ch ší π Gi Avmbœ gvb π = 46 e envi Kiv nq Kw úduv ii mvnv h π Gi gvb mnmª j vwak `kwgk vb ch ší wby xz n q Q h nzz π Gi Avmbœ gvb e envi Kiv nq m nzz DËiI n e Avmbœ ZvB DË ii cv k ÔcÖvqÕ jlv Aek KZ e ciez x mg Í Kv R Ab Kv bviƒc ejv bv _vk j Pvi `kwgk vb ch ší π Gi Avmbœ gvb 46 e envi Kiv n e Amywm vší: e Ëi e vmva ÔÕ n j, cwiwa n e π Kvb e Ëi e vmva K Ges e vm K d aiv n j (iii) bs m~î Abymv i, cwiwa d ev, cwiwa = d = = e Ëi cwiwa = 8 evsjv `k Dš y³ wek we` vjq

7 D PZi MwYZ BDwbU cöwzáv : e Ëi Kv bv Pvc Øviv Dcbœ K `ª KvY H e ËPv ci mgvbycvwzk cögvy: g b Kiæb, PQR e Ëi K `ª Ges e vmva P Q e Ëi Dci Ab GKwU we `y d j QP e Ëi GKwU Pvc Ges PQ e Ëi GKwU K `ª KvY Q Zvn j, K `ª PQ, Pvc QP Gi mgvbycvwzk n e A_ v, K `ª PQ Pvc QP cöwzáv : iwwqvb KvY GKwU aªæe KvY cögvy: g b Kiæb K `ª wewkó GKwU e Ëi e vmva GLb e Ëi e vmv a i mgvb K i e Ëi cwiwa n Z AB e ËPvc A¼b Kiæb A, B hvm Kiæb Zvn j mávbymv i AB = c GLb A mij ilvi Dci C j ^ A¼b Kiæb Zvn j AC = GK mg KvY Ges e ËPvc AC = e Ëi cwiwai GK PZz_ vsk = = 4 Agiv Rvwb e ËPvc Øviv m ó K `ª KvY e ËPvcwUi mgvbycvwzk C B AB AC AZGe = e ËPvc AB e ËPvc AC AB e ËPvc AB A ev, AC e ËPvc AC ev, c GK mg KvY ev, c = GK mg KvY h nzz Ges mg Kv Yi gvb aªæe, AZGe iwwqvb GKwU aªæe KvY Kv Yi e Ëxq cwigvc e Ëxq GKK A_ v iwwqvb GK K Kv bv Kv Yi cwigvc Zvi e Ëxq cwigvc (Cicula Measue) ejv nq cöwzáv 4: e Ëi h Kvb Pvc I Zvi e vmv a i AbycvZ H e Ëi K `ª mb Pvc Øviv Dcbœ Kv Yi mgvb cögvy: g b Kiæb, PQ GKwU wbw` ó KvY K K `ª K i Gi mgvb e vmva wb q GKwU e Ë AsKb Kiæb hv P Ges Q K h_vμ g A I B we `y Z Q` K i GLb e Ëi e vmv a i mgvb GKwU e ËPvc AC wbb Zvn j, Pvc AC = e vmva A Ges AC = GK iwwqvb Avgiv Rvwb, GKwU e ËPvc Øviv m ó K `ª KvY mb e ËPvcwUi mgvbycvwzk myzivs AB Pvc AB = AC Pvc AC ev, AB AC = Pvc Pvc AC AB ev, GK iwwqvb = Pvc ev, AB = Pvc GK iwwqvb ev, AB = s iwwqvb, hlv b Pvc AB = s Q B C A P myzivs e Ëi h Kvb Pvc I Zvi e vmv a i AbycvZ H e Ëi K `ª mb Pvc Øviv Dcbœ Kv Yi mgvb R P wî KvYwgwZ 9

8 I cb zj GmGmwm cövmövg GLb AB Gi e Ëxq cwigvc iwwqvb n j Zv K AB = iwwqvb AvKv i wjlv hvq Zvn j AB = s iwwqvb ev, iwwqvb = s iwwqvb ev, = s s mvims ÿc- h Kv bv e Ëi cwiwa I e v mi AbycvZ memgq mgvb I GKB aªæe msl v G aªæe msl vwu K wmök ey π (cvb) Øviv cökvk Kiv nq mvaviyz Pvi `kwgk vb ch ší π Gi Avmbœ gvb π = 46 e envi Kiv nq e Ëi Kv bv Pvc Øviv Dcbœ K `ª KvY H e ËPv ci mgvbycvwzk iwwqvb KvY GKwU aªæe KvY e Ëi h Kvb Pvc I Zvi e vmv a i AbycvZ H e Ëi K `ª mb Pvc Øviv Dcbœ Kv Yi mgvb e Ëi h Kvb Pvc s, Zvi e vmva Ges H e Ëi K `ª mb Pvc Øviv Dcbœ KvY n j s cv VvËi g~j vqb.-. cögvy Kiæb, iwwqvb GKwU aªæe KvY. cögvy Kiæb = s, hlv b s = e Ëi Pvc Ges = e Ëi e vmva cvv.4 KvY cwigv ci wewfbœ GK Ki g a m úk cvvwfwëk D Ïk GB cvv k l Avcwb- KvY cwigv ci wewfbœ GK Ki g a m úk wby q Ki Z cvi eb, m úk jvi mvnv h mgm v mgvavb Ki Z cvi eb g~jcvv- KvY cwigv ci wewfbœ GK Ki g a m úk : (K) lvug~jk I kzg~jk GK Ki g a m úk lvug~jk c wz Z mg KvY = 90Ges kzg~jk c wz Z mg KvY = 00 g myzivs 90 = 00 g 90 = 00 g 9 = 0 g g = 0 9 =. g (cövq) 0 evsjv `k Dš y³ wek we` vjq

9 D PZi MwYZ BDwbU Avevi, 0 g =9 g = 0 9 = 0.9 (cövq) (L) lvug~jk I e Ëxq GK Ki g a m úk cöwzáv Gi iwwqvb Kv Yi ey bvq Avgiv cvb, iwwqvb = mg KvY myzivs c = mg KvY c = c = c = Avevi, =.46 = 577'44.8".46 iwwqvb = 80 wkš Aci `ybwu c wz Abymv i mg KvY = 80 = 00 g mg KvY = 80 = 00 g = c iwwqvb = iwwqvb mg KvY = 90 = 00 g = c (M) g b Kiæb, GKwU wbw` ó Kv Yi cwigvy K lvug~jk kzg~jk I e Ëxq c wz Z h_vμ g D wwmöx, G MÖW Ges R iwwqvb wb ` k Kiv nj Zvn j, 80 = iwwqvb = iwwqvb D = D iwwqvb Avevi h nzz 00 g = iwwqvb G g = G iwwqvb myzivs = D = G = R D = G = R Dc iv³ Av jvpbv _ K eûj e eüz KvYmg~ ni wwwmö I iwwqv bi m úk `Iqv n jv: (i) = c (iii) 45 = c c 45 = (v) 90 = c c 90 = (ii)0 = c c 0 = (iv) 60 = c c 60 = (vi) 80 = c 80 = c (vii) 60 = c 60 = c e envwik Î iwwqvb cözxk (c) mvaviyz wjlv nq bv iwwqvb cözxk Dn i L m úk wj K Avgiv wb œwjwlzfv e c ev =, 0 =, 45 =, 60 =, 90 =, 80 = 60 = BZ vw` wî KvYwgwZ

10 I cb zj GmGmwm cövmövg D`vniY : 05'0" K iwwqv b cökvk Kiæb mgvavb: 0" = 0 60 ' = ' 5'0" = 5 ' = ' = ' = 60 = 0 AZGe 05'0" = 0 0 = 4 0 = = c = 0.6 c (cövq) mg KvY = 0800 c D`vniY : 0 g 5'0" K lvug~jk c wz Z cökvk Kiæb mgvavb: 0 g 5'0" = 0 g +0.5 g g =0.5 g = mg KvY = 0.05 mg KvY = = 8.58 = 8 ( )' = 8.608' = 8' ( )" = 8' 6.48" D`vniY : Kvb wîfz Ri KvY wji AbycvZ :4:5 KvY wj K wwmöx I iwwqv b cökvk Kiæb mgvavb: aiæb, GKwU Kv Yi cwigvy = x AZGe, kz g Z wîfz Ri Aci KvY `ybwu 4x Ges 5x. wîfz Ri wzb Kv Yi mgwó = 80 A_ v x+4x+5x = 80 x = 80 x = 80 = 5 cö_g KvYwU = x = 5 = 45 wøzxq KvYwU = 4x = 4 5 = 60 Z Zxq KvYwU = 5x = 5 5 = 75 Avevi, = c 45 = c 45 = 60 = c 60 = Ges 75 = 5c 75 = wîfz Ri KvY wzbwu 45, 60, 75 A_ev c, c 5c, Ges D`vniY 4: GKwU Mvxi PvKvi e vm 4 dzu Ges PvKvwU cöwz m K Û 8 evi Nv i MvwwUi MwZ em wby q Kiæb mgvavb: PvKvi e vm d = 4 dzu PvKvi e vmva = d = 4 dzu = dzu PvKvi cwiwa = =.46 dzu =.5664 dzu A_ v PvKvwU evi Ny i.5664 dzu `~iz AwZμg K i h nzz PvKvwU cöwz m K Û 8 evi Nv i AZGe m K Û MvwwUi AwZμg `~iz = dzu evsjv `k Dš y³ wek we` vjq

11 D PZi MwYZ BDwbU N Uv ev 60 60,,,,,,,,,, = dzu = MvwwUi MwZ em N Uvq 69 gvbj (cövq) = MR gvbj = gvbj = 69 gvbj (cövq) D`vniY 5: GKwU e &ËKvi c _ N Uvq 6 wk.wg. e M `Š GK e w³ 6 m K Û Ggb GKwU e ËPvc AwZμg K i hv K `ª 56 KvY Dcbœ K i e Ëi e vm wby q Kiæb mgvavb: N Uvq ev m K Û AwZμg K i = 6 wk.wg. = wg. m K Û Ó Ó Ó = wg. 6 m K Û Ó Ó Ó = wg. = 60 wg. e Ëi Pvc = 60 wg. Avgiv Rvwb s = s h nzz = e Ëi e vmva, s = e ËPvc, Kv Yi cwigvb GLv b s = 60 wg., c 80 AZGe gvb ewm q cvb = wg. = wg. = 6.8 wg e Ëi e vm = = 6.8 =.74 wg. = wg. (cövq) 60 D`vniY 6: 0 wguvi DuPz GKwU LyuwU KZ `~ i 5 KvY Dcbœ K i mgvavb: g b Kiæb AB LuywUwU f wgi Dci we `y Z 5 KvY Dcbœ K i aiæb A = h nzz AB Lye z`ª myzivs AB Gi ˆ`N A Gi ˆ` N i Zzjbvq Lye z`ª n e myzivs AB K GKwU e Ëi Pvc aiv h Z cv i hvi K `ª D Ges e vmva =. B GLb 5 = 5 60 = = iwwqvb Ges s = 0 wguvi GLb s = m úk n Z cvb 5 0 = ev, = wg. = = 7.5 wk.wg. (cövq) AZGe LuywUwU 7.5 wk.wg `y i 5 KvY Dcbœ K i wg. = wk.wg. = wk.wg. (cövq) D`vniY 7: c w_ex _ K m~ h i M `~iz 9,5,00,000 gvbj aiv nj hw` m~h `k Ki Pv Li Ae v bi mv _ KvY Dcbœ K i, Zvn j m~ h i e vm KZ? mgvavb: g b Kiæb m~ h i e vm D gvbj h nzz `k Ki Pv Li Ae v bi mv _ m~ h i A_ v m~ h i e vm Øviv Dcbœ Kv Yi cwigvb Lye QvU, myzivs `k K A_ v c w_ex n Z m~ h i `~i Z i Zzjbvq m~ h i e v mi cwigvy AwZ z`ª n e 5 wg. A wî KvYwgwZ

12 I cb zj GmGmwm cövmövg myzivs m~ h i e vm K mb e Ëi Pvc Kíbv Kiv h Z cv i hvi e vmva = 9,5,00,000 gvbj Ges K `ª nj `k Ki Ae vb GLb = 60 = iwwqvb GLb s = hlv b s = e Ëi Pvc, = e Ëi e vmv_ Ges = wbw` ó KvY, m úk n Z cvb D = 9,5,00, = = 8,6,0 gvbj (cövq) cv VvËi g~j vqb.4-. iwwqv b cökvk Ki: (i) 05'6" (ii) 405'6" (iii) 45'.". lvu g~jk c wz Z cökvk Kiæb (i) g 5'0" (ii) 5 g '5" iii) 5. GKwU PvKvi e vmva wg. n j Zvi cwiwa KZ? 4. GKwU wîfz Ri KvYmg~n h_vμ g x, 5 g Ges x Gi gvb wby q Kiæb 6 5. GKwU Mvwi PvKv 00 evi AveZ b K i 800 wguvi AwZμg K i PvKvi e vmva wby q Kiæb 6. GKwU ijmvxi PvKvi e vm.5 wguvi Ges PvKvwU cöwz m K Û 5 evi Nv i ijmvxi MwZ em N Uvq KZ? 7. `ybwu Kv Yi mgwó I AšÍi h_vμ g iwwqvb I n j z`ªzg KvYwUi gvb iwwqv b cökvk Kiæb 8. GKwU wîfz Ri wzbwu Kv Yi AbycvZ :: KvY wzbwu K wwmöx I iwwqv b cökvk Kiæb 9. GKwU Kv Yi wwmöx Z cwigvc D I iwwqv b cwigvc R c n j `Lvb h, D 80 = R. 0. lvug~jk I kzg~jk c wz Z GKwU Kv Yi cwigvc h_vμ g x I y n j `Lvb h, 50x = 8y. GK e w³ N Uvq 0 gvbj e M `Š 6 m K Û Ggb GKwU e ËPvc AwZμg K i hv e Ëi K `ª 56 KvY Dcbœ K i e Ëi e vm wby q Kiæb. hw` GKwU e ËPvc 40 dzu `xn e vmva wewkó e Ëi K `ª 60 g KvY Dcbœ K i, Zvn j e ËPvcwUi ˆ`N wby q Kiæb. wb œwjwlz mg q e ËvKvi Nwi N Uvi KuvUv I wgwb Ui KuvUvi ga ez x Kv Yi cwigvc K iwwqv b, wwmöx I MÖ W cökvk Kiæb (i) 0 Uv 5 wgwbu (ii) Uv (iii) 4 Uv 40 wgwbu 4. c w_ex _ K Puv `i `~iz 7600 gvbj Puv `i e vm c w_exi Kvb we `y Z 6 KvY Dcbœ K i Puv `i e vm wby q Kiæb 5. c w_ex _ K m~ h i `~iz wk.wg. Ges c w_exi K `ªwe `y Z m~ h i e vm KvY Dcbœ K i m~ h i e vm KZ? 6. c w_exi e vmva 6440 wk.wg. n j c w_exi Dci h `ywu vb K `ª KvY Dcbœ K i Zv `i `~iz wby q Kiæb 7. Puv `i e vm `k Ki Pv Li mv _ 0 KvY Dcbœ K i Ges m~ h i hw` m~h, Puv `i _ K 675 Y `~ i Aew Z nq, Zvn j Zv `i e v mi AbycvZ wby q Kiæb 8. GKwU Mv Qi D PZv 50 Ges MvQwUi `k Ki Pv Li Ae v b 9 KvY Dcbœ K i Mv Qi Mvv n Z `k Ki `~iz KZ? 4 evsjv `k Dš y³ wek we` vjq

13 D PZi MwYZ BDwbU cvv.5 wî KvYwgwZK AbycvZmg~n cvvwfwëk D Ïk GB cvv k l Avcwb- m~² Kv Yi wî KvYwgwZK AbycvZmg~n wby q Ki Z cvi eb, h Kv bv Kv Yi wî KvYwgwZK AbycvZ wby q Ki Z cvi eb, wewfbœ PZzf v Mi wî KvYwgwZK Abycv Zi wpý wba viy Ki Z cvi eb gyl kã m~² KvY, wî KvYwgwZK AbycvZ, vbvsk, e vmva f±i, wî KvYwgwZK Abycv Zi wpý g~jcvv- wî KvYwgwZi GB As k cö_ g m~² Kv Yi Î wî KvYwgwZK AbycvZmg~n (sine, csine, tangent, secant, csecant, ctangent) m ú K Av jvpbv Kiv n e m~² Kv Yi AbycvZmg~ ni gva g h Kv bv Kv Yi wî KvYwgwZK AbycvZmg~n wby qi KŠkj Av jvpbv Kiv n e AbycvZ mg~ ni cvi úwik m úk Ges wewfbœ PZzf v M G `i wpý wk n e m m ú K e vl v Kiv n e wî KvYwgwZK AbycvZ msμvší KwZcq A f` m ú K aviyv `Iqv n e GQvvI Av`k KvYmg~ ni (0, 0, 45, 60, 90) wî KvYwgwZK AbycvZ Ges AbycvZmg~ ni m ev P ev me wb œgvb A_ v gv bi cwiwa m ú K Av jvpbvi GB As k AšÍf y³ _vk e m~² Kv Yi wî KvYwgwZK AbycvZmg~n wp Î XA GKwU m~² KvY aiæb KvYwU A evû Z h Kvb GKwU we `y P wbb P we `y n Z X Gi Dci PM j ^ Uvbyb Zvn j GKwU mg KvYx wîfyr PM Dcbœ nj PM Gi mv c P AwZfzR P j ^ A wîfz Ri AwZfzR (Hyptenuse), M fywg (adjecent side), PM j ^ (ppsite side) Ges PQ =(m~² KvY) GLb PM Gi wzbwu evû P, M I PM wb q QqwU AbycvZ MVb Kiv hvq PM M PM P P M AbycvZ jv,,,,, P P M M PM PM GB AbycvZ jv K Kv Yi wî KvYwgwZK AbycvZ ejv nq f~wg M X Zvn j Kv Yi wî KvYwgwZK AbycvZ K wb œv³fv e AwfwnZ Kiv nqt PM j ^ = sine ( Kv Yi mvbb) ev ms c sin P AwZfzR M f wg = csine ( Kv Yi KvmvBb) ev ms c cs P AwZfzR wî KvYwgwZ 5

14 I cb zj GmGmwm cövmövg PM j ^ tangent ( Kv Yi U vb R U) ev ms c tan M f wg M f wg ctangent ( Kv Yi KvU vb R U) ev ms c ct PM j ^ P AwZfzR secant ( Kv Yi m K U) ev ms c sec M f wg P AwZfzR csecant ( Kv Yi Kv m K U) ev ms c csec PM j ^ h Kv bv Kv Yi wî KvYwgwZK AbycvZ g b Kiæb GKwU N~Y vqgvb iwk Avw` Ae vb X n Z N~Y b ïiæ K i P Ae v b G m XP KvY Dcbœ K i GLb P we `y n Z XX ilvi Dci PM j ^ A¼b Kiæb g~jwe `y n Z P we `yi `~iz P K P we `yi e vmva f±i e j g b Kiæb, M x Ges PM y GLb P we `yi vbvsk (x,y) Ges e vmva f±i P n j Kv Yi wî KvYwgwZK AbycvZ jv wb œwjwlzfv e msávwqz nq: PM P we `yi y- vbvsk y sin P P we `yi e vmva f±i M cs P PM tan M M ct PM P sec M P csec PM P we `yi x- vbvsk P we `yi e vmva f±i P we `yi y- vbvsk P we `yi x- vbvsk P we `yi x- vbvsk P we `yi y- vbvsk P we `yi e vmva f±i P we `yi x- vbvsk P we `yi e vmva f±i P we `yi y- vbvsk x y (hw` x 0 x x (hw` y 0 y nq) nq) (hw` x 0 nq) x (hw` y 0 y Dc ii ey bvq P we `y Ges we `y wfbœ we `y e j P 0 mbrb sin Ges cs memgqb A_ en hw` cövwší evû P, x A ÿi Dci Aew Z nq Zvn j y0, KviY x A ÿi Dci Aew Z h Kv bv we `yi y Gi vbvsk k~b Giƒc Kv Yi Rb ct, csec msávwqz bq Avevi hw` cövwší evû P, y A ÿi Dci Aew Z nq Zvn j x 0, KviY y A ÿi Dci Aew Z h Kv bv we `yi x Gi vbvsk k~b Giƒc Kv Yi Rb tan, sec msávwqz bq wî KvYwgwZK Abycv Zi wpý g b Kiæb GKwU N~Y vqgvb iwk Avw` Ae vb X n Z N~Y b ïiæ K i P Ae v b G m XP KvY Dcbœ K i GLv b P we `yi Ae vb XY, YX, XY A_ev YX GB PviwU PZzf v Mi h Kv bv GKwU Z n Z cv i GLb P we `y n Z XX ilvi Dci PM j ^ A¼b Kiæb g~jwe `y n Z P we `yi `~iz P K P we `yi e vmva f±i e j g b Kiæb, M x Ges PM y nq) X Y Y P (x,y) y x M X 6 evsjv `k Dš y³ wek we` vjq

15 D PZi MwYZ BDwbU X Y P (x,y) y x M X P (x,y) y X M x Y X Y g wpî Y Y q wpî Y X M x y P (x,y) Y q wpî X Y 4_ wpî N~Y vqgvb iwk Avw` Ae vb X n Z N~Y b ïiæ K i cwigvy KvY Dcbœ Kivi ci PviwU PZzf v Mi h Kv bv GKwU Z Ae vb Ki Z cv i h nzz e vmva f±i P ( ) me `v abvz K AZGe Kv Yi wewfbœ Abycv Zi wpý x I y Gi wpý A_ v M I PM evûi cwigv Yi Dci wbf i K i (wpî) GLb hw` N~Y vqgvb iwk i kl Ae vb A_ v e vmva f±i cö_g PZzf v M Ae vb K i (g wpî) Zvn j x, y, cö Z KB abvz K n e myzivs Zv `i mkj AbycvZ abvz K n e myzivs cö_g PZzf v M Aew Z mkj wî KvYwgwZK AbycvZ abvz K n e Y hw` e vmva f±i wøzxq PZzf v M Ae vb K i (q wpî) Z e x FYvZ K Ges y I abvz K myzivs wøzxq PZzf v M x ewr Z AbycvZ q PZzf vm g PZzf vm A_ v sin I csec AbycvZ `ybwu abvz K Ges Ab vb AbycvZmg~n sin I csec mkj AbycvZ FYvZ K AbycvZ abvz K abvz K hw` e vmva f±i Z Zxq PZzf v M Ae vb K i (q wpî) Z e x I y X X DfqB FYvZ K Ges abvz K myzivs Z Zxq PZzf v M x I y m ^wjz q PZzf vm 4_ PZzf vm AbycvZ A_ v tan I ct AbycvZ `ybwu abvz K Ges Ab vb tan I ct cs I sec AbycvZ abvz K AbycvZ abvz K AbycvZmg~n FYvZ K hw` e vmva f±i PZz_ PZzf v M Ae vb K i (4_ wpî) Z e y FYvZ K Y Ges x I abvz K myzivs PZz_ PZzf v M y ewr Z AbycvZ A_ v cs I sec AbycvZ `ybwu abvz K Ges Ab vb AbycvZmg~n FYvZ K wbw` ó KvY Dcbœ Kivi ci N~Y vqgvb iwk Kvb& PZzf v M Ae vb Ki e Zv Rvb Z cvi j cv ki wp Îi mvnv h Avgiv wî KvYwgwZK Abycv Zi wpý wby q Ki Z cvwi X x M y X P (x,y) wî KvYwgwZ 7

16 I cb zj GmGmwm cövmövg cvv.6 wî KvYwgwZK AbycvZ jvi cvi úwik m úk cvvwfwëk D Ïk GB cvv k l Avcwb- wî KvYwgwZK AbycvZ jvi cvi úwik m úk wby q Ki Z cvi eb g~jcvv- cv ki wp Î PM = AZGe msávbyhvqx PM P (i) sin, csec P PM PM sin, P P csec PM M P (ii) cs, sec P M cs, sec sec cs PM M (iii) tan, ct M PM PM tan, M M ct PM (iv) tan PM M PM M P P cs sin cs Abyiƒcfv e, ct sin (v) wpî n Z cx_v Mviv mi cöwzáv Abyhvqx PM M P PM M P ev, [Dfq c K P P P ev, ev, PM P M P sin cs ev, sin cs... () (vi) wpî n Z cx_v Mviv mi cöwzáv Abyhvqx csec sin ct tan P Øviv fvm K i] P M 8 evsjv `k Dš y³ wek we` vjq

17 D PZi MwYZ BDwbU PM M P PM M P ev, [Dfq c K M M M PM P ev, M M ev, tan sec M Øviv fvm K i] ev, sec tan... () (vii) wpî n Z cx_v Mviv mi cöwzáv Abyhvqx PM M P PM M P ev, [Dfq c K PM Øviv fvm K i] PM PM PM M P ev, PM PM ev, ct csec ev, csec ct... () (), () I () bs m~î jv K wb œwjwlz AvKv ii wjlv hvq sin cs ; cs sin sec tan ; sec tan csec ct ; csec ct cvv.7 wî KvYwgwZK Abycv Zi aªæezv I mxgve Zv cvvwfwëk D Ïk GB cvv k l Avcwb- wî KvYwgwZK Abycv Zi aªæezv wby q Ki Z cvi eb, wî KvYwgwZK Abycv Zi mxgv wby q Ki Z cvi eb, wî KvYwgwZK Abycv Zi mxgve Zv wby q Ki Z cvi eb gyl kã wî KvYwgwZK AbycvZ, aªæezv, mxgve Zv g~jcvv- wî KvYwgwZK Abycv Zi aªæezv GKwU wbw` ó cwigvy Kv Yi Rb h Kvb wbw` ó wî KvYwgwZK Abycv Zi gvb me mgqb aªæe wî KvYwgwZ 9

18 I cb zj GmGmwm cövmövg g b Kiæb XP wî KvYwgwZK Abycv Zi gvb A evû Z P we `yi Ae v bi Dci wbf i K i bv A evû Z Aci GKwU we `y P wbb GLb P I P we `y n Z X Gi Dci h_vμ g PM I PM j ^ A¼b Kiæb AZGe PM Ges PM `ybwu m` k mg KvYx wîfyr h nzz PM Ges PM m` k, P PM P PM PM P P AZGe A_ v ev P PM P P P PM PM M P M M P P Abyiƒcfv e, A_ v ev M P P P M M PM M PM PM M M X Ges A_ v ev M M PM M M M PM PM myzivs XA = n j PM PM P P sin, csec P P PM PM M M P P cs, sec P P M M PM PM M M tan, ct M M PM PM N~Y vqgvb iwk i kl Ae vb h Kv bv PZzf v M Ae vb KiæK bv Kb, wîfzr `ybwu m` k KvYx niqv QvvI Abyiƒc evûi wpî GKB cök wzi nq myzivs h Kv bv wîfzr n Z wî KvYwgwZK AbycvZ wby q Ki j Zv `i gvb I wpý GKB n e myzivs wbw` ó cwigvy Kv Yi Rb h Kvb wbw` ó wî KvYwgwZK Abycv Zi gvb aªæek wî KvYwgwZK mxgv wî KvYwgwZK Abycv Zi wbw` ó mxgv Av Q GB mxgvi evb i wî KvYwgwZK AbycvZ Ae vb K i bv GLb Avgiv wî KvYwgwZK Abycv Zi mxgv m ú K Rvb ev g b Kiæb, PM= GKwU m~² KvY Ges PM M (i) Kv Yi cö Z K wî KvYwgwZK AbycvZ PM Gi `ybwu evûi AbycvZ myzivs Giƒc cö Z K AbycvZ GKwU abvz K ev Íe msl v P (ii) PM G AwZfyR P e nëg evû myzivs PM P Ges M P PM M ev, Ges [P Øviv Dfq c K fvm K i] P P A_ v sin Ges cs P P Avevi Ges M PM M A_ v csec Ges sec (iii) PM G h Kvb `yb evûi mgwó Z Zxq evû A c v e nëi myzivs PM + M > P PM M ev, [P Øviv Dfq c K fvm K i] P P A_ v sin cs 40 evsjv `k Dš y³ wek we` vjq

19 D PZi MwYZ BDwbU wî KvYwgwZK Abycv Zi gv bi mxgve Zv Avcbviv Rv bb sin cs h nzz ev Íe msl vi em me `v abvz K, myzvivs sin Ges cs cö Z KwUi gvb abvz K n e Avevi G `i hvmdj e j KvbwUi gvb + A cÿv e nëi n Z cv i bv A_ v sin ev cs Gi gvb + A cÿv e nëi ev A cÿv ÿz`ªzi n Z cv i bv A_ v sin Ges cs h nzz csec sin Ges sec myzivs csec Ges sec Gi gvb A_ev wkš tan I ct Gi gv bi cs Kv bv mxgv wba viy Kiv hvq bv cvv.8 wî KvYwgwZK AbycvZ: mgm v I mgvavb cvvwfwëk D Ïk GB cvv k l Avcwb- wî KvYwgwZK AbycvZ m úwk Z mgm vi mgvavb Ki Z cvi eb g~jcvv- c~e ez x cvvmg~ n wî KvYwgwZK AbycvZ, wewfbœ PZzf v M wî KvYwgwZK Abycv Zi wpý, wî KvYwgwZK AbycvZmg~ ni cvi úwik m úk, wî KvYwgwZK Abycv Zi aªæezv I mxgve Zv BZ vw` m ú K Av jvpbv Kiv n q Q ez gvb cv V wî KvYwgwZK Abycv Zi cö qvm A_ v wewfbœ mgm v I mgvavb wb q Av jvpbv Kiv n e D`vniY : sin AbycvZ K ct Gi gva g cökvk Kiæb mgvavb: Avgiv Rvwb, sin csec Avevi, csec ct csec ct sin ct D`vniY : hw` sin nq, Z e cögvy Kiæb sec tan, hlb cö_g PZzf v M Aew Z 9 mgvavb: sin 9 GLb sin cs cs sin wî KvYwgwZ 4

20 I cb zj GmGmwm cövmövg cs 84 9 h nzz cö_g PZzf v M Aew Z, AZGe sin GLb sec tan cs cs D`vniY : cögvy Kiæb, mgvavb: sin sin sin sin cs Gi gvb abvz K n e sin sec sin sin sin sin sin sin sin sin sin sin sin sin sin sin sin sin sin cs cs sin sin sin sin sec cs cs cs cs tan sec sin D`vniY 4: cögvy Kiæb, tan sec cs sin sin cs tan sec mgvavb: cs cs tan sec sin sin cs cs cs cs cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin cs sin sin cs cs sin cs sin sin cs sin sin cs cs cs cs cs sin cs sin sin cs sin cs cs cs cs cs D`vniY 5: cögvy Kiæb, mgvavb: sec sec A sec A ct A cseca sec A A sec A sec A sec A sec A sec A sec A sec A tan A 4 evsjv `k Dš y³ wek we` vjq

21 D PZi MwYZ BDwbU sec A sec A tan A tan A tan A cseca ct A cs sin cs A A A ct A A sin A ct D`vniY 6: cögvy Kiæb, sin sin cs mgvavb: sin sin sin sin sin sin sin sin sin cs cs cs D`vniY 7: cögvy Kiæb, sin csec cs sec tan ct mgvavb: sin csec cs sec sin csec sin csec cs sec cs sec sin cs csec sec csec sec 4 csec sec csec sec ct tan tan ct a D`vniY 8: m~² KvY Ges tan n j cögvy Kiæb, sin cs b a sin a sin a sin a mgvavb: tan b cs b cs b sin b b sin a sin b sin a a sin b sin a sin a sin a b a a sin a b a a b a b Avevi cs sin a b a b a b a b a b GLb sin cs a b a b a b D`vniY 9: hw` sin A cs A a Ges sec A cseca b nq, Z e cögvy Kiæb, ba a mgvavb: b a sec A cseca sin A cs A sec A csecasin A cs A sin Acs A sec A cseca sin Acs A sec A cseca.sin Acs A sin Acs Asec A sin Acs AcsecA sin A cs A a wî KvYwgwZ 4

22 I cb zj GmGmwm cövmövg cv VvËi g~j vqb.8- wb œwjwlz A f` jv cögvy Kiæb ( 5) sin A cs A. cseca cs A sin A 4 4. sec A sec A tan A tan A cs. (i) csec ct cs sin (ii) sec tan sin cs cs 4. ct cs cs cs cs sin sin 5. sin sin cs cs cs A sin A 6. cseca sec A sin Acs A cs Asin A 7. sec A tan A cs A cs A sec A tan A sin A sin B cs A cs B 8. 0 cs A csb sin A sin B sin cs sin cs 9. csec sin cs sin cs tan tan 0. ct ct sin tan cs ct. sec csec 4 4. sec A tan A sec Atan A sin. tan sec sin mcs msin m 4. nct n tan n sin sin 5. sec sec sin sin hw` sin sin nq, Z e `Lvb h tan tan 7. (i) sin n j cs I tan Abycv Zi gvb KZ n e? 9 (ii) cs n j tan I csec Abycv Zi gvb KZ n e? 4 8. x acs bsin Ges y asin bcs n j `Lvb h, x y a b 44 evsjv `k Dš y³ wek we` vjq

23 D PZi MwYZ BDwbU sin k k 9. k tan tank n j `Lvb h, sin k sin sin cs 0. tan n j cögvy Kiæb, cs sin cs sin cs cvv.9 wba vwiz Kv Yi wî KvYwgwZK Abycv Zi gvb cvvwfwëk D Ïk GB cvv k l Avcwb- wba vwiz Kv Yi wî KvYwgwZK AbycvZmg~ ni gvb wby q Ki Z cvi eb, mgm v mgvav bi AbycvZmg~ ni gvb cö qv M Ki Z cvi eb g~jcvv- wewfbœ Kv Yi hgb 0 0, 0 0, 45 0, 60 0 Ges 90 Kv Yi Rb AbycvZ jvi gvb wfbœ nq GLb Avmyb Kvb& Kv Yi Rb wî KvYwgwZK Abycv Zi gvb wk nq m m ú K Rvwb 45 0 Kv Yi wî KvYwgwZK AbycvZ cv ki wp Î XA = 45 0 A evû Z P h Kvb we `y Ges P n Z X evûi Dci PM j ^ GLb PM = 45 0, PM = 90, AZGe PM = 45 myzivs M = PM aiæb M = PM = a P P M PM a GLb, sin 45 P a M a cs45 P a PM a tan 45 M a M a ct 45 PM a sec45 csec45 P M PM a a a a P PM a a a a a 45 0 a M a A X wî KvYwgwZ 45

24 I cb zj GmGmwm cövmövg 0 I 60 Kv Yi wî KvYwgwZK AbycvZ 0 A_ev 60 Kv Yi Rb Abycv Zi gvb wfbœ n e GLb Avgiv `wl GB `ybwu Kv Yi wî KvYwgwZK Abycv Zi wk gvb cviqv hvq wp Î XA = 0 A evû Z P h Kvb we `y Ges P n Z X A P evû Z PM j ^ GLb PM = 0, PM = 90 a 60 AZGe PM = 60 a PM K P ch ší Ggbfv e ewa Z Ki hb PM = PM nq P hvm Kiæb 0 a 0 M X GLb PM PM AZGe PM = PM = 0 a 60 a Ges PM = PM = 60 P AZGe PP GKwU mgevû wîfyr A aiæb PM = PM = a AZGe PP = PM + P M = PM + PM= PM = a P = P = a GLb M P PM a a AZGe sin0 tan 0 sec0 4a a a PM a P a PM a M a P a M a a cs0 ct 0 csec0 M a P a M a PM a P a PM a Avevi, sin 60 tan 60 sec60 M P M PM P PM a a a a a a cs60 ct 60 csec60 PM a P a PM a M a P a M a k~b (0) Kv Yi wî KvYwgwZK AbycvZ k~b Kv Yi Avw`evû I cövwšík ilv Awfbœ G Î P(x,y) we `y XX A g~jwe `yi Wvbcv k Aew Z AZGe x =, y = 0 GLb 46 evsjv `k Dš y³ wek we` vjq

25 D PZi MwYZ BDwbU y 0 sin0 0 x cs0 y 0 tan0 0 x sec0 x GLv b ct 0 I csec 0 mávwqz bq, KviY y = 0 mg Kv Yi (90 ) wî KvYwgwZK AbycvZ 90 Kv Yi Î cövšíevû Y eivei Ges P(x,y) we `y YY A g~jwe `yi Dc ii w` K Aew Z d j x = 0 Ges y = Y myzivs sin90 y x x y 0 cs ct90 0 csec90 y GLv b tan 90 I sec 90 mávwqz bq, KviY x = 0 e env ii myweav _ wewfbœ Kv Yi wî KvYwgwZK Abycv Zi gvbmg~n QK AvKv i wb œ `Lv bv n jv KvY AbycvZ sin 0 cs tan ct AmsÁvwqZ sec csec AmsÁvwqZ jÿbxq: wî KvYwgwZK Abycv Zi gvbmg~n K g b ivlvi mnr wbqg X X AmsÁvwqZ AmsÁvwqZ Y Y P(x,y) P(x,y) Y 0 0 X X wî KvYwgwZ 47

26 I cb zj GmGmwm cövmövg (K) 0,,, Ges 4 msl v jvi cö Z KwU K 4 Øviv fvm K i fvmd ji em g~j wb j h_vμ g sin 0, sin 0, sin 45, sin 60 Ges sin 90 Gi gvb cviqv hvq (L) 4,,, Ges 0 msl v jvi cö Z KwU K 4 Øviv fvm K i fvmdj jvi em g~j wb j h_vμ g cs 0, cs 0, cs 45, cs 60 Ges cs 90 Gi gvb cviqv hvq (M) 0,, Ges 9 msl v jvi cö Z KwU K Øviv fvm K i fvmdj jvi em g~j wb j h_vμ g tan 0, tan 0, tan 45 Ges tan 60 Gi gvb cviqv hvq (D jl h, tan 90 msávwqz bq) (N) 9,, Ges 0 msl v jvi cö Z KwU K Øviv fvm K i fvmdj jvi em g~j wb j h_vμ g ct 0, ct 45, ct 60, ct 90 Gi gvb cviqv hvq (D jl h, ct 0 msávwqz bq) () secant Gi gvb csine Gi gv bi YvZ K wecixz Ges csecant Gi gvb sine Gi gv bi YvZ K wecixz n 90 Kv Yi wî KvYwgwZK AbycvZ g b Kiæb n GKwU c~y msl v GLb XX K x Aÿ Ges YY Y K y Aÿ we epbv K i n Gi wewfbœ gv bi Rb n 90 Kv Yi P(0, ) wî KvYwgwZK Abycv Zi gvb wb œwjwlzfv e cviqv hvq (i) hlb n 4k, hlv b k GKwU c~y msl v A_ v hlb n 0, 4, 8,,..., 4, 8,,... BZ vw` ZLb cövwšík P(, 0) P(, 0) evû P abvz K x Aÿ X Gi Dci c myzivs P we `yi X X vbvsk n e, 0, hlv b P 0. 0 myzivs sin 0, cs P(0, ) Y (ii) hlb n 4k, hlv b k GKwU c~y msl v A_ v hlb n, 7,,...,, 5, 9,,... BZ vw` ZLb cövwšík evû P abvz K y Aÿ Y Gi Dci c myzivs P we `yi vbvsk n e 0,, hlv b P 0. 0 myzivs sin, cs 0 (iii) hlb n 4k, hlv b k GKwU c~y msl v A_ v hlb n, 6, 0,...,, 6, 0,... BZ vw` ZLb cövwšík evû P FYvZ K x Aÿ X Gi Dci c myzivs P we `yi vbvsk n e, 0, hlv b P 0. 0 myzivs sin 0, cs (iv) hlb n 4k, hlv b k GKwU c~y msl v A_ v hlb n, 5, 9,...,, 7,,... BZ vw` ZLb cövwšík evû P FYvZ K y Aÿ Y Gi Dci c myzivs P we `yi vbvsk n e 0,, hlv b P 0. 0 myzivs sin, cs 0 D`vniY : gvb wby q Kiæb, sin 45 sin 60 cs 45 cs60 mgvavb: sin 45 sin 60 cs 45 cs evsjv `k Dš y³ wek we` vjq

27 D PZi MwYZ BDwbU D`vniY : gvb wby q Kiæb, mgvavb: D`vniY : hw` tan 45 sin 60 ct 0 tan 45 sin 60 ct 0 sec A 0 nq Z e cögvy Kiæb, sin mgvavb: sin A sin.0 sin 60. tan A tan 0 Avevi tan A tan tan A sin A tan A 45 tan A tan A A sin 60 cs60 sin 60 cs60 D`vniY 4: `Lvb h, sin 60 cs60 sin 60 cs60 sin 60 cs60 sin 60 cs60 mgvavb: evgcÿ = sin 60 cs60 sin 60 cs sec cs0 wî KvYwgwZ 49

28 I cb zj GmGmwm cövmövg Wvbcÿ = cs 0. evgcÿ = Wvbcÿ D`vniY 5: mgvavb Kiæb, sin cs, hlb abvz K my² KvY mgvavb: sin cs cs cs cs cs cs cs 0 cs 4cs cs 0 cs cs cs cs cs 0 nq cs 0 A_ev cs 0 cs cs hv m e bq cs cs D`vniY 6: mgvavb Kiæb, tan ct 4 mgvavb: tan ct 4 hlb tan 4 tan tan 4 tan tan 4 tan 0 90 tan 4tan 0 tan tan tan 0 tan tan 0 tan tan 0 tan nq tan 0 A_ev, tan 0 tan tan 60 tan 60 h nzz , 60 tan 0 tan0 mvims ÿc 0,,, Ges 4 msl v jvi cö Z KwU K 4 Øviv fvm K i fvmd ji em g~j wb j h_vμ g sin 0, sin 0, sin 45, sin 60 Ges sin 90 Gi gvb cviqv hvq 4,,, Ges 0 msl v jvi cö Z KwU K 4 Øviv fvm K i fvmdj jvi em g~j wb j h_vμ g cs 0, cs 0, cs 45, cs 60 Ges cs 90 Gi gvb cviqv hvq 0,, Ges 9 msl v jvi cö Z KwU K Øviv fvm K i fvmdj jvi em g~j wb j h_vμ g tan 0, tan 0, tan 45 Ges tan 60 Gi gvb cviqv hvq (D jl h, tan 90 msávwqz bq) 9,, Ges 0 msl v jvi cö Z KwU K Øviv fvm K i fvmdj jvi em g~j wb j h_vμ g ct 0, ct 45, ct 60, ct 90 Gi gvb cviqv hvq (D jl h, ct 0 msávwqz bq) 50 evsjv `k Dš y³ wek we` vjq

29 D PZi MwYZ BDwbU secant Gi gvb csine Gi gv bi YvZ K wecixz csecant Gi gvb sine Gi gv bi YvZ K wecixz sin 90 cs ; cs90 sin tan 90 ct90 tan ; ct ; ; sec90 csec ; csec90 sec cv VvËi g~j vqb.9- gvb wby q Kiæb ( 4). sin 0 4sin 0. 5sin 90 cs0 6 tan 45 sec 45. ct cs sec 4sin tan.sin.tan.tan 4 6 cögvy Kiæb (5 9) cs0 5. sec60 tan 60 cs0 6. cs cs sin sin sin cs cs sin sin 0 cs 0 cs 60 tan 0 8. tan 0 9. sin csec cs ct sec tan cs ct sec tan cs xsin cs tan n j x Gi gvb wby q Kiæb tan sin x ct sec n j x Gi gvb wby q Kiæb A 60, B 0 n j wb œwjwlz m~î jvi mz Zv hvpvb Kiæb (i) sina B sin AcsB cs Asin B (ii) csa B cs AcsB sin Asin B tan A tan B (iii) tana B tan Atan B tan A (iv) tana tan A (v) csa 4cs A cs A (vi) sin B sin B 4sin B mgvavb Kiæb ( 6). cs sin, abvz K my² KvY 4 wî KvYwgwZ 5

30 I cb zj GmGmwm cövmövg 4. sec tan, abvz K my² KvY 5. cs ec ct, abvz K my² KvY 6. sin cs, abvz K my² KvY cvv.0 mshy³ Kv Yi wî KvYwgwZK AbycvZ cvvwfwëk D Ïk GB cvv k l Avcwb- mshy³ Kv Yi wî KvYwgwZK AbycvZ wby q Ki Z cvi eb gyl kã mshy³ KvY, wî KvYwgwZK AbycvZ g~jcvv-, 90, Giƒc KvY K Kv Yi mshy³ KvY ejv nq mshy³ Kv Yi wî KvYwgwZK AbycvZ wby qi mgq `~i Z i wpý jv Aek B we epbv Ki Z n e Z e iy ivl Z n e h e vmva f±i me mgqb abvz K Giƒc Kv Yi wî KvYwgwZK AbycvZ K Kv Yi wî KvYwgwZK Abycv Zi gva g cökvk Kiv nq ( ) Kv Yi wî KvYwgwZK AbycvZ g b Kiæb, GKwU N~Yv qgvb iwk Avw` Ae vb X n Z Nwi KuvUvi wecixz w` K Ny i XP = Ges Nwi KuvUvi w` K Ny i XQ = KvY Dcbœ K i GLb P we `y _ K X Gi Dci PN j ^ AsKb Kiæb PN K ewa Z Ki j Zv Q ilv K Q we `y Z Q` K i Y g b Kiæb, P I Q we `yi vbvsk h_vμ g (x, y) I x, y Zvn j N = x, PN = y Ges N x, QN y GLb PN I QN mg KvYx wîfzrøq n Z cvb, PN = QN, NP = NQ Ges N mvaviy evû myzivs, ΔPN I ΔQN me mg PN= QN; P = Q = (e vmva f±i) vbv ¼i wpý we epbv K i cvb, x x, y y y y myzivs, sin () = sin QN = sin PN = sin x x cs () = cs QN = cs PN = cs y y tan () = tan QN = tan PN = tan x x Abyiƒcfv e, csec () = csec sec () = sec ct () = ct X Y P Q N X 5 evsjv `k Dš y³ wek we` vjq

31 D PZi MwYZ BDwbU (90 0 ) Kv Yi wî KvYwgwZK AbycvZ g b Kiæb, GKwU NyYv qgvb iwk X Ae vb n Z Nwi KuvUvi wecixz w` K Ny i XP = KvY Dcbœ K i Aci GKwU N~Yv qgvb iwk GKB Avw` Ae vb X n Z GKB w` K Ny i XY = 90 0 KvY Dcbœ Kivi ci Y Ae vb n Z Nwi KvuUvi w` K Ny i YQ = KvY Dcbœ K i Zvn j, XQ = 90 0 GLb P = Q ilv wbb Ges P I Q n Z X ilvi Dci h_vμ g PN I QM j ^ AsKb Kiæb g b Kiæb, P I Q we `yi vbvsk h_vμ g (x, y) I x, y Zvn j N = x, PN = y Ges M x, QM y GLb ΔPN I ΔQM mg KvYx wîf R n Z cvb, PN = QM Ges P = Q myzivs, wîfzrøq me mg AZGe, QM = N; M = PN; P = Q = (e vmva f±i) A_v, y x, x y myzivs, sin (90 0 y x ) = sin XQ = cs XP = cs cs (90 0 x y ) = cs XQ = sin XP = sin tan (90 0 y x ) = tan XQ = ct XP = ct x y Abyiƒcfv e, csec (90 0 ) = sec sec (90 0 ) = csec ct (90 0 ) = tan `ªóe : 90 0 Ges ci ú ii cwic~ik KvY `ybwu cwic~ik Kv Yi Rb GKwUi sine AciwUi csine, GKwUi tangent AciwUi ctangent Ges GKwUi csecant AciwUi secant Gi mgvb hgb, 0 0 I 60 0 KvY ci ú ii cwic~ik AZGe sin 0 0 = cs 60 0 ev sin 60 0 = cs 0 0, tan 0 0 = ct 60 0 ev tan 60 0 = ct 0 0, csec 0 0 = sec 60 0 ev csec 60 0 = sec 0 0 ( ) Kv Yi wî KvYwgwZK AbycvZ g b Kiæb, GKwU NyYv qgvb iwk Avw` Ae vb X n Z Nwi KuvUvi wecixz w` K Ny i XP = KvY Dcbœ K i H GKB ilv GKB w` K AviI Ny i PQ = 90 0 KvY Dcbœ K i Zvn j, XQ = GLb P = Q wbb Ges P I Q n Z x A ÿi Dci h_vμ g PN I QM j ^ A¼b Kiæb g b Kiæb, P I Q we `yi vbvsk h_vμ g (x, y) I x, y Zvn j N = x, PN = y Ges M x, QM y GLb ΔPN I ΔQM mg KvYx wîf R n Z cvb, PN = QY = QM Ges P = Q, myzivs wîfzrøq me mg AZGe, QM = N; M = PN; P = Q = (e vmva f±i) vbvs Ki wpý ch v jvpbv K i cvb, y x, x y myzivs, sin (90 0 y x + ) = sin XQ = cs XP = cs cs (90 0 x y + ) = cs XQ = sin XP = sin wî KvYwgwZ 5

32 I cb zj GmGmwm cövmövg tan ( ) = tan XQ Abyiƒcfv e, csec ( ) = sec sec ( ) = csec ct ( ) = tan y x = ct XP = ct x y (80 0 ) Kv Yi wî KvYwgwZK AbycvZ c~ e i wbqg AbymiY K i R vwgwzk c wz Z (80 0 ) Kv Yi wî KvYwgwZK AbycvZ wby q Kiv hvq wkš wb œi c wz AbymiY K i Avgiv mn R (80 0 ) Kv Yi wî KvYwgwZK AbycvZ wby q Ki Z cvwi myzivs, sin (80 0 ) = sin { (90 0 )} = cs (90 0 ) = sin cs (80 0 ) = cs { (90 0 )} = sin (90 0 ) = cs tan (80 0 ) = tan { (90 0 )} = ct (90 0 ) = tan Abyiƒcfv e, csec (80 0 ) = csec sec (80 0 ) = sec ct (80 0 ) = ct ( ) Kv Yi wî KvYwgwZK AbycvZ R vwgwzk c wz Z ( ) Kv Yi wî KvYwgwZK AbycvZ wby q Kiv hvq Avevi wb œi c wz AbymiY K i Avgiv mn RB ( ) Kv Yi wî KvYwgwZK AbycvZ wby q Kiv hvq myzivs, sin ( ) = sin { ( )} = cs ( ) = sin cs ( ) = cs { ( )} = sin ( ) = cs tan ( ) = tan { ( )} = ct ( ) = ( tan ) = tan Abyiƒcfv e, csec ( ) = csec sec ( ) = sec ct ( ) = ct (70 0 ) Kv Yi wî KvYwgwZK AbycvZ sin (70 0 ) = sin { (90 0 )} = sin (90 0 ) = cs cs (70 0 ) = cs { (90 0 )} = cs (90 0 ) = sin tan (70 0 ) = tan { (90 0 )} = tan (90 0 ) = ct Abyiƒcfv e, csec (70 0 ) = sec sec (70 0 ) = csec ct (70 0 ) = tan ( ) Kv Yi wî KvYwgwZK AbycvZ sin ( ) = sin { ( )} = sin ( ) = cs cs ( ) = cs { ( )} = cs ( ) = ( sin ) = sin ) tan ( ) = tan { ( )} = tan ( ) = ct Abyiƒcfv e, csec ( ) = sec sec ( ) = csec ct ( ) = tan 54 evsjv `k Dš y³ wek we` vjq

33 D PZi MwYZ BDwbU (60 0 ± ) Kv Yi wî KvYwgwZK AbycvZ cöwgz Ae v b (60 0 ) Ges ( ) KvY `ybwu h_vμ g ( ) I Kv Yi ms M wg j hvq myzivs, (60 0 ) Kv Yi wî KvYwgwZK AbycvZ ( ) Kv Yi mswkøó Abycv Zi mgvb Ges ( ) Kv Yi wî KvYwgwZK AbycvZ Kv Yi mswkøó Abycv Zi mgvb AZGe, sin (60 0 ) = sin ( ) = sin cs (60 0 ) = cs ( ) = cs tan (60 0 ) = tan ( ) = tan csec (60 0 ) = csec ( ) = csec sec (60 0 ) = sec ( ) = sec ct (60 0 ) = ct ( ) = ct Ges, sin ( ) = sin cs ( ) = cs tan ( ) = tan csec ( ) = csec sec ( ) = sec ct ( ) = ct mshy³ Kv Yi wî KvYwgwZK AbycvZ wby qi wbqg (i) hw` KvYwU Gi Rv wyz Ki mv _ hvm A_ev we qvm nq (80 0 ±, 60 0 ± ), Z e mb Kv Yi AbycvZ K Kv Yi wî KvYwgwZK Abycv Z cökvk Ki j g~j Abycv Zi Kvbiƒc cwiez b nq bv A_v sine, csine, tangent BZ vw` sine, csine, tangent-b _v K wkš Zvi wpý abvz K bv FYvZ K n e Zv wby q Kivi Rb KvY K abvz K m~² KvY a i KvYwU Kvb& PZzf v M Ae vb K i Zv wby q K i wp ýi PZzf vm wby q wbqg Abyhvqx mn RB wby q Kiv hvq (ii) hw` KvYwU Gi we Rv wyz Ki mv _ hvm A_ev we qvm nq (90 0 ±, 70 0 ±,... BZ vw`), Z e mb Kv Yi wî KvYwgwZK AbycvZ K Kv Yi wî KvYwgwZK Abycv Z cökvk Ki j sine, csine, tangent, ctangent, secant, csecant cwiewz Z n q h_vμ g csine, sine, ctangent, tangent, csecant, secant nq wkš Zvi wpý abvz K bv FYvZ K n e Zv wby q Kivi Rb KvY K abvz K m~ÿ KvY a i g~j KvYwU Kvb& PZzf v M Ae vb K i Zv wby q K i wp ýi PZyf vm wbqg Abyhvqx mn RB wby q Kiv hvq cv VvËi g~j vqb.0-. mshy³ Kv Yi wî KvYwgwZK AbycvZmg~ ni gvb wby q Kiæb wî KvYwgwZ 55

34 I cb zj cvv. mshy³ Kv Yi wî KvYwgwZK AbycvZ: mgm v I mgvavb GmGmwm cövmövg cvvwfwëk D Ïk GB cvv k l Avcwb- mgm v mgvav b mshy³ Kv Yi wî KvYwgwZK AbycvZ cö qvm Ki Z cvi eb g~jcvv- c~ e i cv V Avcbviv mshy³ Kv Yi wî KvYwgwZK AbycvZ m ú K Ávb AR b K i Qb ez gvb cv V wewfbœ mgm v mgvav b AbycvZmg~n cö qvm Ki Z cvi eb D`vniY : gvb wby q Kiæb, (i) sin 70 0 (ii) sec 50 0 (iii) tan( ) mgvavb: (i) sin (70 0 ) = sin ( ) = sin 0 0 = e vl v: GLv b N~Yv gvb iwk hw` Avw` Ae vb n Z Nwi KuvUvi wecixz w` K N~Y b ïiæ K i 70 0 KvY Dcbœ Kivi ci Z Zxq PŠK Y Ae vb Ki e h nzz KvYwU n (hlb n Rv msl v) AvKv ii sin AcwiewZ Z _vk e Ges h nzz Z Zxq PŠK Y sin FYvZ K d j wpý FYvZ K n e (ii) sec (50 0 ) = sec ( ) = csec 60 0 = (iii) tan (590 0 ) = tan (590 0 ) [ h nzz tan () = tan ] = tan ( ) = ( ct 60 0 ) = ct 60 0 = D`vniY : gvb wby q Kiæb, (i) cs sin tan tan 0 (ii) sin cs cs ( 00 0 ). sin ( 0 0 ) mgvavb: (i) cs sin tan tan 0 = cs ( ) + sin ( ) + tan ( ) + tan 0 = cs cs 8 0 tan 0 + tan 0 = 0 (ii) sin cs cs ( 00 0 ). sin ( 0 0 ) = sin ( ). cs ( ) cs ( ).sin ( ) = cs 0 0. cs 0 0 (sin 0 0 ). ( cs 60 0 ) [sin( )= sin, cs( )= cs] 4 =. +. = + = = D`vniY : gvb wby q Kiæb, (i) sin sin 5 8 (ii) sec 4 7 sec 9 7 mgvavb: (i) sin sin ct cs cs 8 ct 4 + cs cs 8 56 evsjv `k Dš y³ wek we` vjq

35 D PZi MwYZ BDwbU = {sin ( 8 )} + {sin ( 8 )} +{cs ( + 8 )} + cs 8 = sin 8 + sin 8 + cs 8 + cs 8 = ( sin 8 + cs 8 ) + ( sin 8 + cs 8 ) = + = (ii) sec 4 7 sec ct 4 4 ct 4 = {sec ( 7 )} + { sec ( )} +{ct ( )} {ct ( 4 )} = sec 7 = sec 7 = sec 7 = + tan sec 7 + ct 4 ct 4 sec {ct ( 5 7 )} {ct ( 7 )} 5 5 sec 7 + tan 7 tan 5 7 tan 7 + tan 5 7 tan 7 = 0 D`vniY 4: `Lvb h, sin ( + ) ct ( + ) cs ( ) = sin ( ) sin ( ) ct ( + ) mgvavb: evgcÿ = sin ( + ) ct ( + ) cs ( ) = sin ( + ) ct ( + ) cs ( ) = cs ( tan ) ( cs ) = cs. tan. cs = cs. sin cs = sin. cs cs Wvbcÿ = sin ( ) sin ( ) ct ( + ) = cs. ( cs ). ( tan ) = cs.( tan ).( cs ) = cs. tan. cs = cs. sin cs = sin. cs cs evgcÿ = Wvbcÿ D`vniY 5: hw` n Gi gvb ïb A_ev h Kvb ALÛ msl v nq Z e, sin {n + () n 4 } Gi gvb wby q Kiæb mgvavb: hlb n = 0, ZLb sin {n + () n 4 } = sin 4 = hw` n Rv msl v Ges m GKwU ALÛ msl v nq Z e, n = m sin {n + () n 4 } = sin {m + () m 4 } = sin (m + 4 ) = sin 4 = Avevi, n we Rv msl v n j Z e, n = m + n e sin {n + () n 4 } = sin {(m+) + () m+ 4 } wî KvYwgwZ 57

36 I cb zj GmGmwm cövmövg = sin (m + 4 ) = sin ( 4 ) = sin 4 = myzivs n Gi gvb ïb A_ev h Kvb ALÛ msl v n j Avgiv cvb, sin {n + ( ) n 4 } = D`vniY 6: mgvavb Kiæb tan cs sin + = 0, hlb 0 0 mgvavb: tan cs sin + = 0 ev, tan cs sin cs. cs + = 0 ev, tan cs tan. cs + = 0 ev, tan + tan. cs cs = 0 ev, (tan + ) cs (tan ) = 0 ev, (tan + ) ( cs ) = 0 nq, tan + = 0 tan = tan = tan tan = tan ( ) A_ev, tan ( ) tan = tan A_ev, tan 5 =, 5 A_ev, cs = 0 cs = cs = cs 0 0 A_ev, cs = 0 0, h nzz, 0 0 = 0 0,, 5, D`vniY 7: mgvavb Kiæb sin cs = cs, hlb mgvavb: sin cs = cs ev, cs cs = cs ev, cs = cs ev, cs + cs = 0 ev, cs + cs cs = 0 ev, cs (cs + ) (cs + ) = 0 ev, (cs + ) (cs ) = 0 nq, cs + = 0 cs = = cs 80 0 = 80 0 A_ev, cs = 0 cs = cs = cs = cs 60 0 = cs ( ) = 60 0, 00 0 h nzz, evsjv `k Dš y³ wek we` vjq

37 D PZi MwYZ BDwbU = 60 0, 80 0, 60 0 cv VvËi g~j vqb.- gvb wby q Kiæb (-4). (i) sin (ii) ct 90 0 (iii) csec (iv) sec (580 0 ) (v) ct (560 0 ). (i) sin ( ) (ii) csec (iii) tan ( 5 9 ). (i) sin sin cs cs (ii) cs 40 0 sin (00 0 ) sin cs (iii) tan 5 0 ct tan ct (iv) sin tan 90 0 ct cs (i) sin 7 (ii) cs 4 (iii) sin 5. `Lvb h, + sin sin cs sin + cs 4 + sin 5 + sin cs sin + sin + sin (i) sin ( 5 + ) cs ( ) ct (7 + ) = sin ( ) sin ( ) ct (5 + ) (ii) cs ( +) csec ( +) ct (5 ) = sin ( 6. n GKwU ALÛ c~y msl v n j gvb wby q Kiæb, ) sec (4 ) tan ( ) = sin (i) tan { n + ()n 4 } (ii) sin {n + ( )n 6 } (iii) csec {n + ( )n 6 } 7. 7 sin `Lvb h, tan cs csec 8. wb œi mgxkiy jv n Z Gi gvb wby q Kiæb, ct sec 5 (i) sec =, hlb (ii) tan =, hlb (iii) sin =, hlb mgvavb Kiæb: hlb (i) (sin cs + ) = cs + 4 sin (ii) sin + cs = 0 (iii) sin cs + ct = 0 (iv) tan + sec = (v) tan 4 sec + 7 = 0 (vi) ct + tan = sec (vii) ct + csec = wî KvYwgwZ 59

38 I cb zj GmGmwm cövmövg P~všÍ g~j vqb- eûwbe vpbx cökœ. lvug~jk c wz Z KvY cwigv ci GKK wb Pi KvbwU? (K) wwwmö (L) MÖW (M) iwwqvb (N) KvbwUB bq. Gi 60 fv Mi GK fvm mgvb KZ? (K) (L) 5 (M) 0 (N) 60. GKwU PvKvi e vmva 0 m.wg. PvKvwU GKevi Nyi j KZ m.wg. `~iz AwZμg Ki e? (K) 5 (L) 0 (M) 5 (N) 0 4. e vmv a i GKwU e Ëi cwiwa I e v mi AbycvZ KZ? (K) : (L) : (M) (N) GKK 5. lvug~jk c wz Z- (i) 60 (ii) 60 (iii) mg KvY 5400 Dc ii Z _ i wfwë Z wb Pi Kvb&wU mwvk? (K) (i) I (ii) (L) (i) I (iii) (M) (ii) I (iii) (N) (i), (ii) I (iii) 6. s m~ Î K Kvb& GK K cökvk Ki Z nq? (K) wwwmö (L) iwwqvb (M) MÖW (N) m KÛ 7. Kv bv Pvc e Ëi cwiwai mv _ iwwqvb KvY Dcbœ Ki j - (i) K `ª iwwqvb KvY Dcbœ Ki e (ii) DcPvc I AwaPvc mgvb n e (iii) PvcwU n e e Ëi e nëg R v Dc ii Z _ i wfwë Z wb Pi Kvb&wU mwvk? (K) (i) I (ii) (L) (i) I (iii) (M) (ii) I (iii) (N) (i), (ii) I (iii) wb Pi Z_ _ K 8-0 bs cö kœi DËi w`b: GKwU wîfz Ri wzbwu Kv Yi AbycvZ : 4 : 5 8. wîfz Ri e nëg KvYwUi e Ëxq gvb KZ? (K) (L) 6 4 (M) 5 (N) 9. wîfz Ri ÿz`ªzg KvYwUi gvb lvug~jk c wz Z KZ? (K) 0 (L) 45 (M) 60 (N) wîfzrwu Kvb ai bi wîfzr? (K) m~² KvYx (L) mg KvYx (M) ~j KvYx (N) KvbwUB bq. sine Gi wî KvYwgwZK AbycvZ wb Pi KvbwU? (M) f wg f wg (N) j ^ AwZfzR (K) j ^ j ^ (L) f wg AwZfzR. csine Gi wî KvYwgwZK AbycvZ wb Pi KvbwU? (K) j ^ j ^ (L) f wg AwZfzR (M) f wg j ^ (N) f wg AwZfzR 60 evsjv `k Dš y³ wek we` vjq

39 D PZi MwYZ BDwbU. ct n jv- sin cs (i) (ii) (iii) tan cs sin Dc ii Z _ i wfwë Z wb Pi Kvb&wU mwvk? (K) (i) I (ii) (L) (i) I (iii) (M) (ii) I (iii) (N) (i), (ii) I (iii) wb Pi Z _ i Av jv K 4-7 bs cö kœi DËi w`b- C cv ki wp Î ABC-G m~ÿ KvY 4. AC KZ? A (K) 4 (L) 5 (M) (N) 5 5. sin Gi gvb KZ? (K) (L) (M) (N) 6. cs Gi gvb KZ? (K) (L) (M) (N) 7. tan Gi gvb KZ? (K) (L) (M) (N) 8. sin cs n j, Gi gvb- (i) 0 (ii) 0 (iii) 90 Dc ii Z _ i wfwë Z wb Pi Kvb&wU mwvk? (K) (i) I (ii) (L) (i) I (iii) (M) (ii) I (iii) (N) (i), (ii) I (iii) 9. A 0, B 60 n j cs A B Gi gvb KZ? (K) 0 (L) (M) - (N) KvYwU Kvb PZzf v M _vk e? (K) cö_g (L) wøzxq (M) Z Zxq (N) PZz_. cs 0 Gi gvb KZ? (K) (L) (M) (N) wb Pi Z _ i Av jv K - bs cö kœi DËi w`b- A cs x. A Gi gvb KZ? (K) sin x (L) sin x (M) cs x (N) csx. x Gi gvb KZ iwwqvb? (K) 6 (L) 4 (M) (N) B wî KvYwgwZ 6

40 I cb zj GmGmwm cövmövg m Rbkxj cökœ 4. K I L vb `ybwu c w_exi K `ª KvY Dcbœ K i c w_exi e vmva R Ges vb `ybwui ga ez x `~iz S (K) `Lvb h, c w_exi cwiwa R [c w_ex K e Ë g b K i] (L) R Ges S Gi g a m úk vcb Kiæb (M) hw` c w_exi e vmva 6440 wk.wg. nq Ges K I L vb `ybwu c w_exi K `ª 57 KvY Dcbœ K i Z e vb `ybwui ga ez x `~iz KZ wby q Kiæb 5. GKwU PvKv 0.88 wk jvwguvi c_ h Z 0 evi Nv i (K) PvKvwU GKevi Nyi j AwZμvšÍ `~iz wby q Kiæb (L) PvKvwUi e vmva wby q Kiæb (M) PvKvwUi e vmv a i mgvb e vmva wewkó Kv bv e Ëi wguvi `xn Pv ci m yl Kv Yi cwigvy wby q Kiæb 6. hw` ct csec a nq Z e (K) csec ct Gi gvb wby q Kiæb a (L) cögvy Kiæb h, cs a (M) `Lvb h, a cs a sin a 7. A I B n j 6 (K) cs A B I cs A B Gi gvb wby q Kiæb cs A B cs A B cs Acs cs A B cs A B sin Asin (L) cögvy Kiæb h, B (M) cögvy Kiæb h, B 8. hw` cs sin sin nq Z e 5 (K) iwwqv bi Rb cs sin Gi gvb wby q Kiæb 6 (L) cögvy Kiæb h, cs sin cs (M) `Lvb h, csec cs 6 evsjv `k Dš y³ wek we` vjq

41 D PZi MwYZ BDwbU DËigvjv- cv VvËi g~j vqb.. (i) g PZzf v M (ii) q PZzf v M (iii) q PZzf v M (iv) 4_ PZzf v M cv VvËi g~j vqb.4. (i) 0. c (ii) 0.4 c (iii) 0.5 c. (i) 050'.5" (ii) 4'6.4" (iii). 9 wg wguvi wk.wg. (cövq) 7. ( c ) 8. 0,60,90, 80 6, c, c. 080 dzu. 7. dzu (cövq). (i) 9 4, 40', 58.g (ii), 90, 00g (iii) 00,. g gvbj (cövq) wk.wg. (cövq) 6. wk.wg. (cövq) 7. : gvbj (cövq) 5 9, cv VvËi g~j vqb.8 7. (i) 5, 40 4 (ii), cv VvËi g~j vqb cv VvËi g~j vqb.. (i). (i) (ii) (iii) (iv) (v) (ii) (iii). (i) (ii) (iii) 0 4. (i) (ii) (iii) 6. (i) (ii) (iii) ±, ± (iv) 8. (i) 0 0 (ii) 00 0 (iii) (i) 60 0, 0 0 (ii) 0 0, 40 0 (iii) 0 0, 5 0, 50 0 I 5 0 (iv) 80 0 (v) 0 0, 0 0 (vi) 0 0, 50 0 (vii) 60 0 wî KvYwgwZ 6

42 I cb zj GmGmwm cövmövg P~všÍ g~j vqb. K. K. N 4. N 5. N 6. L 7. N 8. N 9. L 0. K. L. N. L 4. M 5. L 6. M 7. K 8. L 9. K 0. K. L. N. K 4. (L) S R (M) 66.4 wk.wg. 5. (K) 44 wguvi (L) wguvi (M).57 iwwqvb 6. (K) a 7. (K) 0, 8. (K) 64 evsjv `k Dš y³ wek we` vjq