Trget Publictions Pvt. Ltd. Chpter 0: Trigonometric Functions 0 Trigonometric Functions. ( ) cos cos cos cos (cos + cos ) Given, cos cos + 0 cos (cos + cos ) + ( ) 0 cos cos cos + 0 + cos + (cos cos + ) 0 cos + cos 0 cos cos + cos 0 cos 0 cos 0 (n + ). (n + ) cos cos. cos Cse I: If cos > 0, cos 9,,,.[ (0, ), (0, )] 9,,, But cos > 0 ( must be in st or th Qudrnt) the possible vlues re,. Cse II: If cos < 0, ( cos ), 7 7, The vlues of stisfying the given eqution 7 between 0 nd re,,,.. These re in A.P. with common difference. cos + 0 + 0 + 0 Let t t + 0 t t + 0t t 0t + 0 (t ) (t ) 0 t or t or or or or
Trget Publictions Pvt. Ltd.. or n ± or n ± 7,,, or,,, There re solutions in [0, ]. θ, cos θ, tn θ re in G.P. cos θ θ tn θ cos θ θ. θ cosθ cos θ ( cos θ ) cos θ + cos θ 0 ( cos θ ) ( cos θ + cos θ + ) 0 cos θ 0.[ The eqution cos θ + cos θ + 0 hs no rel root] cos θ cos θ n ±. 0 cos 0 0 + cos 0 Two sides re equl only if nd cos 0. n +. Given, cos, cos nd cos re in G.P. cos cos ( cos ) cos cos + cos 0 cos ( cos + cos ) 0 cos 0 or cos + cos 0 cos 0 or cos + cos cos 0 cos 0 or ( cos + ) ( cos ) 0 cos 0,, The two smllest positive vlues of re nd. α β 7. We hve, cos Std. XII : Triumph Mths + cos + cos + cos +...to.[ce, infinite G.P.] cos cos cos Squring on both sides, we get cos cos cos n ± If cos θ cos. θ n ± α α. The mimum vlue of + b cos is + b. Mimum vlue of + cos is nd the mimum vlue of + is. The given eqution will be true only when + cos nd + If + cos cos + cos cos + cos n, n +.(i) + n + ( ) n. n + ( ) n..(ii) The vlue of [, ] which stisfies both (i) nd (ii) is.
Trget Publictions Pvt. Ltd. 9. + cos cos ( + cos ) cos cos ( cos ). cos + 0 ( + ) ( ) 0.[ ] n + ( ) n n + ( ) n The vlue of in [0, ] re nd. There re solutions. 0. tn sec + 0 tn ( + tn ) + 0 tn tn + 0 tn tn + + 0 (tn ) 0. In ABC, (A + B) A + B n + ( ) n.(i) Chpter 0: Trigonometric Functions. cos + tn tn + + tn + tn Let tn t t + t + t t t + 0 t t + 0 (t ) 0 t tn tn α n + α n + α. The equtions re cos + cos, ( ) y Let cos nd y b + b.(i) nd b.(ii) ( ).[From (i) nd (ii)] + 0 + 0 ( ) ( + ) 0 or But > 0, A + B + C nd B A + C B B.[ A, B, C re in A.P.] From eqution (i) if n, A + B A nd C A, B, C If, then b 9 cos y 9 cos y cos y cos y y
Trget Publictions Pvt. Ltd.. tn θ + tn θ+ + tnθ+ tn θ+ tn θ + tnθ + tn θ + tnθ. tn θ( tn θ ) + (tn θ+ )(+ tn θ) + (tn θ ) ( tn θ) tn θ 9 tn θ tn θ tn θ tnθ tn θ tn θ tn θ tn θ tn θ n + θ (n + ) 0 > 7 tn A > tn B A B > A > B A B tn A B tn + tn + A B tn tn 0 tn A + B + 7 0 7 C + 0 tn 00 C tn 0 cot C 0 tn C 0 Since, 0 7 > 0 tn B > tn C B > C B > C A > B > C > b > c Std. XII : Triumph Mths. A( ABC) 9 bca 9 bc 9. A bc b + c cos A bc cos (b c) + bc bc ( ) + 7. 7 + 7 + + 9 cm B 0 Let B 0, C A 0 A B C b c 0 0 + b c b ( ) c ( ) 0 A + + 0 + 0 0 + 0 C + 0
Trget Publictions Pvt. Ltd. A( ABC) bc A + + 0 0 0 ( + ) (0 + ) ( + ) + ( + ) + +. Let b + c c + + b ( + b + c) k + b + c.(by property of equl rtio) b + c k, c + k, + b k, + b + c k 7k, b k, c k b + c cos A bc k + k 9k (k)(k) k 0k cos A 9. n + A B n + C Let AC n, AB n +, BC n + Lrgest ngle is A nd smllest ngle is B. A B Since, A + B + C 0 B + C 0 C 0 B n Chpter 0: Trigonometric Functions C (0 B) B A B C n+ n n+ B B B n+ n n+ BcosB B B B n+ n n+ cosb n+ n B n+ cos B n +, B n + n n ( cos B) n + n n+ + n + n n n + n+ + n + n n n + n + n + n + n n n 0 (n + ) (n ) 0 n or n But n cnnot be negtive. n The sides of the re,,. 0. According to e Rule, A b B c C R R A, b R B, c R C Now, b c R B C A R B R C.R A.R cos B cos C A (cos C cos B) A (B + C) (B C) A ( A) (B C) A A (B C) A (B C)
Trget Publictions Pvt. Ltd.. D C In ODC, OD OC r, DOC 0 7 A( ODC) r.r. 7 r 7 A Are of pentgon r 7 A Are of circle r A A r r 7 cos sec sec 0. Let k, b k, c k Now, s + b + c k + k + k k s(s )(s b)(s c) k k k k k k k E A O r 7 k 7k k k 7 k By e Rule, A R A R bca bc R bc R R bc k.k.k 7k 7 k Also rs, where r Rdius of incircle of ABC 7 k r s k R r R r 7 r 7 k 7k 7 B 7 k b + c. cos A bc + cos 0 7 7.[ ] Std. XII : Triumph Mths bca 0 s + b + c + + rs r s + ( ) 9 +. Let ABC be the tringle in which A < B < C A + C B.[ A, B, C re in A.P.] Also, A + B + C 0 B 0 B 0 c 0, b 9 (given) c + b cos B c 00 + cos 0 0 9 + 0 9 + 0 0 + 9 0 0 ± 00 7 0 ± ±. + b + c (b + c ) + b + c b c 0 + b + c b + b c c b c ( b c ) + ( ) b + c bc b + c cos A bc A bc bc bc
Trget Publictions Pvt. Ltd.. Are of p bc A p bc p bc bc p By e rule, b c.(i) b c From (i),. p p B C Let length of ltitude p Since, A + B + C A + + A A cos cos + P Chpter 0: Trigonometric Functions cos cos 0 7. tn A nd tn B re the roots of the qudrtic eqution + 0 tn A + tn B, tn A. tn B A B tn A tn + tn Β + A B tn tn tn A + B A+ B A + B C ABC is right ngled tringle.. Let ABC be the tringle nd nd b be the roots of the qudrtic eqution + 0. + b, b, C cos C + b c b ( + b) b c cos b ( ) c c c c, c Perimeter + b + c + 7
Trget Publictions Pvt. Ltd. 9. By e Rule, A b B c C R R A, b R B, c R C cosa+ bcosb+ ccosc + b+ c R A cos A + R Bcos B+ R CcosC + b+ c R ( A + B + C) s R [ (A + B)cos(A B) + CcosC ] s R [ ( C)cos(A B) + CcosC ] s R [ Ccos(A B) + CcosC ] s R C [ cos(a B) cos( (A B)) ] s + + R. C[cos (A B) cos(a + B)] s R. C. A B s R.. b. c s R R R bc sr bc A bc. R bc R bc R, sr, s r required vlue R. R r c B 0. r s ( + b + c) r R c + b+ c.[ B 90 ] c r + c+ b + c b + c b c( + c b) c( + c b) ( + c) b + c + c b + c b.[ + c b ] Dimeter + c b Std. XII : Triumph Mths. A, B, C 0 b c 0 k k, b k, c k 0 c k 0 k k ( 0 + ) ( 0 ) k cos cos k k (k ) (k ) b. A, B, C re in A.P. A + C B Also, A + B + C 0 B 0 A B C k b c A k, B bk, C ck c C + c A c ( C cos C) + c ( AcosA) c ( ck cos C) + c (k cos A) k cos C + kc cos A k( cos C + c cos A) kb.[ b cos C + c cos A] B.[ B 0 ]. cot tn tn + tn + tn tn + 9 tn tn
Trget Publictions Pvt. Ltd. cot cot tn tn + tn tn tn tn tn tn + + 7. Let cosec 0 θ cosec θ 0 cot θ cosec θ 0 9 cot θ, tn θ cosec 0 θ tn tn + () tn tn Let cosec 0 θ cosec θ 0 cot θ cosec θ 0 9 cot θ 7, tn θ 7 cosec 0 θ tn 7 tn + () tn tn Let cosec ( n )( n n ) cosec θ n ( n + )( n + n+ ) + + + θ n cot θ n cosec θ n (n + ) (n + n + ) (n + ) (n + + n + ) (n + ) + n (n + ) + n + (n + ) + n (n + ) + n (n + n + ) cot θ n n + n + tn θ n n + n+ θ n tn n + n+ tn n+ n + ( n+ ) n tn (n + ) tn n Chpter 0: Trigonometric Functions cosec ( n + )( n + n+ ) tn (n + ) tn n Given series tn tn + tn tn +. +. + tn (n + ) tn n tn (n + ) tn tn (n + ). Let cos b θ cos b θ cos θ b tn + cos b + tn cos b tn + θ + tn θ + tnθ tnθ + tnθ + tnθ ( + tn θ ) + ( tn θ ) tn θ ( + tn θ) tn θ tn θ cos θ b + tn θ b. cos α cos β cos αβ + α β Given, cos cos y α cos y y + ( ) α 9
Trget Publictions Pvt. Ltd. cos α ( ) ( ) 0 y y + y y cos α y ( ) cos α y Squring on both sides, we get ( y ) cos α y cos α + y y + y cos α y cos α + y + y y cos α cos α + y y cos α α 7. +... +... + cos... cos +... +... +.... θ cosθ. 0 + < < + + + + + 0 ( ) 0 0 or But 0 does not stisfy the given eqution.. + cos ( ) cos ( ) Std. XII : Triumph Mths cos ( ). (i) Let θ θ cos θ cos ( ).(ii) From (i) nd (ii), we get (squring both sides) 7 7 (From the given reltion it cn be seen tht is positive) 9. L.H.S. 7 + cos cos 7 + tn tn + 9 cot cot 7 + cos cos7 7 + tn tn + cot cot cos + cos 7 7 + tn tn + cot cot + + + 7 7.[ cos ( ) cos ] + 7 7 7 7 b, b 7 + b + 7 0
Trget Publictions Pvt. Ltd. 0. + + + + + + + + cos + cos + +..... >.[.7] tn ( ) > tn ( ).[ tn is n increg function] tn ( ) > A >.(i) θ θ θ θ ( θ θ) Put θ θ 7 7 (0.) Chpter 0: Trigonometric Functions.7 0., 0. < 0. (0.) < (0.).[ is lso n increg function] < <...(ii) (0.) < <.(iii) From (ii) nd (iii), we get B + < + B <.(iv) From (i) nd (iv), A > B. cot + cot y + cot z tn + tn y + tn z tn + tn y + tn z tn (tn + tn y + tn z) tn 0 Let A tn, B tn y, C tn z tn ( A + B) + tn C tn (A + B + C) tn(a+ B)tnC tn A + tn B + tn C tn A tn B tn A + tn B tnc tna tnb tn A + tn B+ tn C tn A tn Btn C tn A tn B tn B tn C tn C tn A tn (A + B + C) 0 tn A + tn B + tn C tn A tn B tn C tn (tn ) + tn(tn y) + tn(tn z) tn(tn ) tn(tn y) tn(tn z) + y + z yz
Trget Publictions Pvt. Ltd.. Let I (sec ) + (cosec ) (sec + cosec ) sec cosec sec sec + (sec ) sec I + ( sec ) sec + sec + sec +. sec 0 The minimum vlue of (sec ) + (cosec ) is. Std. XII : Triumph Mths. tn + tn + tn +.[ > ] + tn ( ) tn tn + tn + tn. tn + tn + + tn + tn + + tn + + + + + ( + )( + ) + + + + ( + ) ( + ) + 0 0 7 0 ( 7 ) 0 ( ) ( + ) 0. cos 9 9 cos 0 0 cos 9 9 cos cos 0 0 cos 9 cos + 0 cos + cos 0 cos cos 0 cos cos 0 cos 7 cos 0 nd 0 7 0 7 0 Principl vlue is 7. 0 0,, But > 0, 7. cot + tn + tn. tn + tn + tn + tn + tn