Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

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1 MT TRIGONOMETRIC FUNCTIONS AND TRIGONOMETRIC EQUATIONS C Trigonometric Functions : Bsic Trigonometric Identities : + cos = ; ; cos R sec tn = ; sec R (n ),n cosec cot = ; cosec R {n, n I} Circulr Definition of Trigonometric Functions : I PM OM cos,cos, tn,cos, cot,, sec,cos OP OP cos cos cosec, C Grphs of Trigonometric functions : y = x, x R; y [, ] y = cos x, x R; y [, ] y = tn x, x R (n + ) /, n I; y R Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

2 MT y = cot x, x R n, n I; y R (e) y = cosec x, x R n, n I; y (, ] [, ) (f) y = sec x, x R (n + )/, n I, y (, ] [, ) Prctice Problems :. The vlue of cos 5 + cos +... cos 5 + cos 9 will be 7 9. If sec tn = k then the vlue of cos will be k k k k k k k k. If cosec A + cot A = then tn A equls [Answers : () () () c] Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

3 MT C Trigonometric Functions of Allied Angles : If is ny ngle, then, 9 ±, ±, 7 ±, ± etc. re clled Allied Angles. ( ) = ; cos ( ) = cos (9 ) = cos ; cos (9 ) = (9 + ) = cos ; cos (9 + ) = ( ) = ; cos ( ) = cos (e) ( + ) = ; cos ( + ) = cos (f) (7 ) = cos ; cos (7 ) = (g) (7 + ) = cos ; cos (7 + ) = (h) tn (9 ) = cot ; cot (9 ) = tn Prctice Problems :. If nd is obtuse then cot equls. The vlue of is none of these. The vlue of cos. cos. cos...cos will be cnnot be found. The vlue of cos + cos + cos...cos will be none of these 5. The vlue of log [cos + cos + cos... cos ] will be not defined. The vlue of log [cot. cot. cot... cot 9 ] will be not defined [Answers : () d () () () b (5) d () ] C Trigonometric Functions of Sum nd Difference of Two Angles : (A ± B) = A cosb ± cosa B cos (A ± B) = cosa cosb A B A B = cos B cos A = (A + B). (A B) cos A B = cos B A = cos (A + B). cos (A B) (e) (f) (g) tn A tn B tn(a B) tn A tn B cot Acot B cot(a B) cot B cot A tn A tn B tnc tn A tnb tnc tn(a B C) tn A tnb tnbtnc tnctn A Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

4 MT Prctice Problems :. If A + cos B = nd B + cos A = b then the vlue of (A + B) will be + b b b b. If tn ( + ) = nd tn ( ) = then tn will be none of these. If tn = nd tn = then will be. The vlue of cos equls to The vlue of cosec sec is equl to. The vlue of tn 5x. tn x. tn x is equl to tn 5x tn x tn x 5x x x cos 5x cos x cos x none of these 7. The vlue of cos9 cos9 9 9 equl to tn 5 tn tn none of these. If A + B + C = (A, B, C > ) nd the ngle C is obtuse then tn A. tn B > tn A. tn B < tn A. tn B = none of these [Answers : () c () b () b () c (5) c () (7) () b] C5 Fctoristion of the Sum nd Difference of Two Sines nd Coes : C D C D C D cos C D C D C D cos C D C D cosc cos D cos cos C D C D cosc cos D Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

5 MT 5 C Trnsformtion of Products into Sum nd Difference of Sines & Coes : A cosb = (A + B) + (A B) cosa B = (A + B) (A B) cos A cos B = cos (A + B) + cos (A B) A B = cos (A B) cos (A + B) C7 Multiple nd Sub-multiple Angles : A = A cosa; = cos cosa = cos A A = cos A = A; cos = + cos, = cos. (e) (f) tn tn A tn A ;tn tn A tn tn A A tn ;cos A A A = A A cos A = cos A cosa tn tn A A (g) tn A tn A tna tn A C Importnt Trigonometric Rtios : n = ; cos n = ( ) n ; tn n =, where n I 5 or cos75 5 orcos cos5 or cos 75 5 or tn5 cot 75 ;tn 75 cot5 or 5 & cos or cos 5 5 Prctice Problems :. If A + B = nd cos A + cos B = b then the vlue of (A + B) will be b b b b b b b b. The vlue of 7 cos7 cos equl to Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

6 MT A C. If cot B, then A, B, C re in cosc cos A A.P. G.P. H.P. none of these cos A cosb. The vlue of A B n A B cos A cosb n equls n A B tn n A B cot both nd re correct 5. The vlue of... will be. The vlue of. 7.. will be 5 7. The vlue of cos cos cos 5 cos 7. The vlue of 5 7 cos cos cos cos will be 9. The vlue of equls to. The vlue of cos.cos.cos.cos will be. The vlue of 5 7 equl to. Prove the following sttements : cos A A + = cos A (A + cosa) ( A. cosa) = A + cos A A cos A cos A A coseca cos A + A = A. cos A Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

7 MT 7 (e) A A seca tna (f) cos eca cot A tn A cos A (g) ( + cota coseca) ( + tna + seca) = (h) cos eca cot A A A cos eca cot A (i) (j) cot A tnb cot B tn A sec cos cot A.tnB cos ec cos. cos cos.. [Answers : () () c () () d (5) b () d (7) c () c (9) c () d () c] C9 Rnge of Trigonometric Expression : E = + b cos E b ( + ), where tn b b cos( ),where tn b Hence for ny rel vlue of, b E b Prctice Problems :. Find the vlue of x for which following expression will hve mximum vlue : cos x + x cos x + x. Find the rnge of following trigonometric expression cos x + x cos x + x x + cos x C Sine nd Coe Series : ( ) ( )... n n n C (i) (ii) cos cos( ) cos( )... cos n n n cos Trigonometric Equtions : Eqution involving trigonometric functions of vrible re clled trigonometric equtions. The trigonometric eqution my hve infinite number of solutions nd cn be clssified s : Principl solution : The solution of trigonometric equtions for which x < re clled principl solution. Generl solution Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

8 MT Importnt Points :. = = n. cos = = (n + ). tn = = n. = = n + ( ) n, where, 5. cos = cos = n ±, where [, ]. tn = tn = n +, where, 7. =, cos = cos, tn = tn = n ±. = = (n + ) 9. cos = = n. cos = = (n + ). = nd cos = cos = n + Prctice Problems :. The most generl vlue of stisfying the eqution tn = nd cos = is 7 n 7 7 n n ( ) n none of these. The number of solution of the given eqution tn + sec = where < < is. If (sec + tn ) = 5 then the generl solution of is n n. If = then the generl vlue of is n n n,(n ) n,(n ) n,(n ) none of these 5. If tn m = tn n then then the consecutive vlue of will be in A.P. G.P. H.P. none of these. The number of solutions of the eqution tn x + sec x = cos x lying in the intervl [, ] is 7. The complete solution of the eqution 7cos x + xcosx = is given by n (n I) n (n I) n tn (n I). If + cos = cos then the generl vlue of is n,k tn (k, n I) n n n n 9. The generl solution of the eqution x cos x = is n,n I n,n I n,n I n,n I [Answers : () c () c () b () b (5) () d (7) d () b (9) b] Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

9 MT 9 ADDITIONAL PRACTICE PROBLEMS. Prove tht. If A + B + C =, then prove tht (i) (ii) (iii) (iv) (v) (vi) A + B + C = A B C A + B + C = cos A cos B cos C cos A + cos B + cos C = cos A cos B cos C cos A + cos B + cos C = + A B C tn A + tn B + tn C = tna tnb tnc A B B C C A tn tn tn tn tn tn (vii) A cot cot B cot C A B cot.cot.cot C (viii) cot A cot B + cot B cot C + cot C cot A = (e) (f) (A B) (B A) cos(a B) cos(b A) tn( A B) A A 5A 7A cos A cos A cos 5A cos 7A tna A A 5A A 5A 7A A 5A. If A + B + C = then prove tht tn A tn B + tn B tn C + tn C tn A =. If A + B = 5 show tht ( + tna) ( + tnb) = 5. Prove tht : 7 5 cos 7 cos 5 7A A A A tn cos A.sec 5A cos A cos A cos A cos A A A A A A cos A.cos A Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

10 MT SINGLE CORRECT CHOICE TYPE. The vlue of... is equl to. The vlue of tn. tn. tn. tn is equl to. If A, B, C re cute positive ngles such tht A + B + C = nd cot A. cot B. cot C = k then k k 9 cos A cosb 9. The vlue of A B equls n k k A B cos A cosb n n A B tn A C. If cot B, then A, B, C re in cosc cos A A.P. G.P. H.P. none of these. If x cos y cos cos then the vlue of is equl to x y z cos 5. If x + x = then the vlue of expression. For cos x + cos x + cos x + cos x is equl to, if n x cos, n y, n n n n n z cos. then xyz = xz + y xyz = xy + z xyz = x + y + z both nd re correct 7. If A + B + C = (A, B, C > ) nd the ngle C is obtuse then tn A. tn B > tn A. tn B < tn A. tn B = none of these n A B cot both nd re correct. If 5x + x + x = then the vlue of x other thn lying between x is. If + cos = cos then the generl vlue of is n n n n. cos x + x = 7 posses solution for ll > < [, ]. The eqution x + b cos x = c where c > b hs one solution two solutions no solution infinite solutions Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

11 . The generl solution of the eqution x cos x = is n,n I n,n I n,n I n,n I 5. The solution of the eqution cos x x = re both nd re correct. The number of solutions of the eqution cosx = x in [, ] is 7. The generl solution of the eqution MT ANSWERS (SINGLE CORRECT CHOICE TYPE). b... c 5.. d 7. b. 9. d. c. b. d. c. b 5. d. d 7.. c 9. d. c cos x x. is n n + n none of these. The lest positive nonintegrl solution of (x + x) x = rtionl irrtionl of the form p irrtionl of the form n odd integer p, where p is irrtionl of the form n even integer p, where p is 9. If x x +, x, then the solution set for x is 5,, 5, 5,. The number of vlues of x [, ] stisfying cos x x is Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

12 MT EXERCISE BASED ON NEW PATTERN COMPREHENSION TYPE Comprehension- Consider the cubicl eqution qx px x p =. This eqution hs roots tn, tn nd tn equls to / / /. The vlue of.. is independent of p independent of q indepedent of both p nd q none. The vlue of cos. cos. cos is (A) (B) (C) (D) independent of p independent of q indepedent of both p nd q none MATRIX-MATCH TYPE Mtching- Column - A 9 r r r r Column - B r (p) / r cos (q) 5 (r ) cos (r) 9/ (r ) (s) /5 Mtching- Column - A Column - B (A) The number of solutions (p) of the eqution tn + 9 = sec in the intervl ( /, /) is (B) The number of solutions (q) of the eqution tn x + sec x = cos x lying in the intervl [, ] is (C) The number of vlues (r) x [, ] stysfying cos x x (D) The number of rel (s) solutions of (x, y) where y = x, y = cos (cos x), x, is Mtching- Column - A Column - B (A) cos x ( + ) cos x (p) ny irrtionl (B) ( + ) = hs rel number solution if is cos x cos x (q) hs rel solution if is (C) + x = cos x hs (r) exctly one solution if is (D) ( x+ cos x) x = (s) hs no solution if is Mtching- Consider tringle ABC Column - A Column - B (A) The mximum vlue of (p) / cosa + cosb + cosc (B) The mximum vlue of (q) / A B C (C) The minimum vlue of (r) / cos A + cos B + cos C (D) The mximum vlue of (s) / ( cos A )( cos B)( cos C) Mtching-5 Consider n cute ngle tringle ABC Column - A Column - B (A) The minimum vlue of (p) tn A tn B tn C (B) The minimum vlue of (q) tn A. tn B. tn C Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

13 MT (C) The minimum vlue of (r) 9 tn A + tn B + tn C (D) The mximum vlue of (s) / cos A + cos B + cos C MULTIPLE CORRECT CHOICE TYPE. If ( cos ) cos( ), then cos cos. Let A, B, C be three ngles of tringles such tht A nd tn B. tn C = p. The possible vlue of p is cos. If f(, ) = then cos f(, ) = f(, ) = f(, ) f(, ). Let f n ( ) tn. ( + sec ) ( + sec ) ( + sec )... ( + sec n ). Then f f n fn ( ) lim f n n = 5. cos x. x m mx is n identity in x. Then m, n, n 5, m n. If [, ] nd cos = (cos ) then the vlue of is Let [x] = the gretest integer less thn or equl to x. The eqution x = [ + x] + [ cos x] hs no solution in, no solution in, solution in, no solution for x R. If cos nd cos then ( + ) = ( + ) = 5 5 tn 7 tn 7 9. If + nd re solutions of for the eqution tn tn = where nd. Then. The possible rel vlues of x which stisfy tn x tnx Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

14 . Consider the eqution sec + cosec = c. Then eqution hs two roots between nd if c < eqution hs four roots between nd if c < eqution hs two roots between nd if c > eqution hs four roots between nd if c >. If A = {/cos + } nd B /. Then A B is 5 Assertion-Reson Type 5 Ech question contins STATEMENT- (Assertion) nd STATEMENT- (Reson). Ech question hs choices (A), (B), (C) nd (D) out of which ONLY ONE is correct. (A) (B) (C) (D) Sttement- is True, Sttement- is True; Sttement- is correct explntion for Sttement- Sttement- is True, Sttement- is True; Sttement- is NOT correct explntion for Sttement- Sttement- is True, Sttement- is Flse Sttement- is Flse, Sttement- is True. STATEMENT- : The solution of this eqution x + cos x x. cos x = is m nd m where m STATEMENT- : If both sides of n eqution will be squred then we will lwys get the correct solution. MT x y. STATEMENT- : If cosec (x y) then x y hs no rel vlue if x nd y re non-zero rel. STATEMENT- : x + y > x y. STATEMENT- : The eqution sec + cosec = c hs two roots between nd if < c <. STATEMENT- :. STATEMENT- : If x + y cos = nd x y cos =, n /, then the locus of point (x, y) is stright line. STATEMENT- : Locus represents the reltion between x nd y. 5. STATEMENT- : There is no rel x for e x e x =. STATEMENT- : The minimum vlue of e x + e x is.. STATEMENT- : (cos x) = cos( x) does not possess ny rel solution for x. STATEMENT- : The mximum vlue of x + cos x is. 7. STATEMENT- : The eqution cos x + cos kx = hs only one solution if k is irrtionl. STATEMENT- : The eqution cos x + cos kx = hs mny solutions if k is rtionl.. STATEMENT- : The vlues of tht stisfy cos + cos re in AP. STATEMENT- : n + cos n for ll nd n. 9. STATEMENT- : In tringle ABC, + b + c STATEMENT- : AM GM. STATEMENT- : The hrmonic men of the exrdii of tringle is three times the inrdius. STATEMENT- : AM HM (Answers) EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE. d. b. MATRIX-MATCH TYPE. [A-q; B-r; C-p; D-p]. [A-p; B-q; C-s; D-r]. [A-q; B-r, s; C-p; D-q]. [A-p; B-q; C-r; D-s] 5. [A-p; B-q; C-r; D-s] MULTIPLE CORRECT CHOICE TYPE., b., b, c, d., b., b, d 5., c, d., d 7., b, d., c, d 9., c. c, d., d., b ASSERTION-REASON TYPE. C. A. A. A 5. B. A 7. B. B 9. B. B Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57

15 . If. cos. =. ( + ), then show tht tn, tn nd tn re in hrmonic progression. If cos b b cos b ( b). If A + B + C = then prove tht cos A + cos B cos C = + cos A. cos B. C A + B + C = + cosa. cosb. cosc then, show tht (B + C A) + (C + A B) + (A + B C) = A. B. C.. If A + B + C = s then prove tht : (s A) (s B) + (s). (s C) = A. B s. (s A). (s B). (s C) = cos A cos B cos C + cos A. cos B. cos C.. Prove tht tn + tn + tn n tn n + n cot n = cot.. If m ( + ) = cos ( ). Prove tht m m m. Show tht + + > If =, prove tht (i) cos cos cos cos cos.. MT 5 INITIAL STEP EXERCISE (SUBJECTIVE) cos s + cos (s A) + cos (s B) + cos (s C) =. + cos A. cos B. cos C Solve for x : sec x = ( ) tn x Find x (, ) if cosx cos x cos x...to. If ( cot ) = cos ( tn ). Prove tht either cosec or cot is equl to positive or negtive integer. n, where n is 5. If x + y + z = xyz, then show tht x y z xyz x y z ( x )( y )( z ) x x (ii) y y If xy + yz + zx =, then prove tht z z ( x )( y )( z ) FINAL STEP EXERCISE cos.cos + + cos. (SUBJECTIVE). cos ( ) = (iii) tn + tn + tn + tn = tn. tn. tn. tn (cot + cot + cot + cot). 5. If cos (x y), cos x, cos(x + y) re in hrmonic progression then evlute y cos x.sec. ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE) n,n where n, ANSWERS SUBJECTIVE (FINAL STEP EXERCISE) 5. ± Einstein Clsses, Unit No.,, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi, Ph. : 995, 57