TRIGONOMETRIC RATIOS & IDENTITY AND EQUATION Contents Toic Pge No. Theory 0-08 Exercise - 09-9 Exercise - 0-8 Exercise - 9 - Exercise - - Answer Key - 7 Syllbus Trigonometric functions, their eriodicity nd grhs, ddition nd subtrction formule, formule involving multile nd submultile ngles, generl solution of trigonometric equtions. Nme : Contct No. ARRIDE LEARNING ONLINE E-LEARNING ACADEMY A-79 indr Vihr, Kot Rjsthn 00 Contct No. 80007
TRIGONOMETRIC RATIOS & IDETITIE AND EQUATION v Bsic Trigonometric Identities: KEY CONCEPTS () sin² q + cos² q = ; - sin q ; - cos q " q Î R (b) sec² q - tn² q = ; ½sec q½ ³ " q Î R ( n ) ì í î ü +, n Î Iý þ n, n Î I (c) cosec² q - cot² q = ; ½cosec q½ ³ " q Î R { } v Circulr Definition Of Trigonometric Functions: PM sin q = OP OM cos q = OP tn q = sin q cos q, cos q ¹ 0 cos q cot q = sin q, sin q ¹ 0 v sec q =, cos q ¹ 0 cosec q =, sin q ¹ 0 cosq sin q Trigonometric Functions Of Allied Angles: If q is ny ngle, then - q, 90 ± q, 80 ± q, 70 ± q, 60 ± q etc. re clled ALLIED ANGLES. () sin (- q) = - sin q ; cos (- q) = cos q (b) sin (90 - q) = cos q ; cos (90 - q) = sin q (c) sin (90 + q) = cos q ; cos (90 + q) = - sin q (d) sin (80 - q) = sin q ; cos (80 - q) = - cos q (e) sin (80 + q) = - sin q ; cos (80 + q) = - cos q (f) sin (70 - q) = - cos q ; cos (70 - q) = - sin q (g) sin (70 + q) = - cos q ; cos (70 + q) = sin q (h) tn (90 - q) = cot q ; cot (90 - q) = tn q v Grhs of Trigonometric functions: () y = sin x x Î R; y Î [, ] A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
(b) y = cos x x Î R; y Î [, ] (c) y = tn x x Î R (n + ) /, n Î I ; y Î R (d) y = cot x x Î R n, n Î I; y Î R (e) y = cosec x x Î R n, n Î I ; y Î (-, - ] È [, ) (f) y = sec x x Î R (n + ) /, n Î I ; y Î (-, - ] È [, ) A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
v Trigonometric Functions of Sum or Difference of Two Angles: () (b) sin (A ± B) = sina cosb ± cosa sinb cos (A ± B) = cosa cosb m sina sinb (c) sin²a - sin²b = cos²b - cos²a = sin (A+B). sin (A- B) (d) cos²a - sin²b = cos²b - sin²a = cos (A+B). cos (A - B) (e) tn (A ± B) = tn A ± tnb m tn A tnb cot A cot B m (f) cot (A ± B) = cotb ± cot A tn A + tnb + tnc-tn A tnb tnc (g) tn (A + B + C) = - tna tnb - tnb tnc- tnc tn A. v Fctoristion of the Sum or Difference of Two Sines or Cosines: C+ D () sinc + sind = sin cos C-D C+ D (b) sinc - sind = cos sin C-D v C+ D (c) cosc + cosd = cos cos C-D (d) cosc - cosd = - C+ D sin sin C-D Trnsformtion of Products into Sum or Difference of Sines & Cosines: () sina cosb = sin(a+b) + sin(a-b) (b) cosa sinb = sin(a+b) - sin(a-b) (c) cosa cosb = cos(a+b) + cos(a-b) (d) sina sinb = cos(a-b) - cos(a+b) v Multile nd Sub-multile Angles : () sin A = sina cosa ; sin q = sin q cos q (b) cos A = cos²a - sin²a = cos²a - = - sin²a; cos² q = + cos q, sin² q = - cos q. tna (c) tn A = - tn A tn ; tn q = - tn q q tn A (d) sin A = + tn A -tn A, cos A = + tn A (e) (f) sin A = sina - sin A cos A = cos A - cosa (g) tn A = tn A - tn - tn A A A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
v Imortnt Trigonometric Rtios: () sin n = 0 ; cos n = (-) n ; tn n = 0, where n Î I (b) sin or sin cos or cos = = - = cos 7 or cos + = sin 7 or sin ; ; tn = - = - = cot 7 ; tn 7 = + + = + = cot - - (c) sin or sin 8 = 0 + & cos 6 or cos = v Conditionl Identities: If A + B + C = then : (i) sina + sinb + sinc = sina sinb sinc (ii) (iii) (iv) (v) sina + sinb + sinc = cos A cos B cos C cos A + cos B + cos C = - - cos A cos B cos C cos A + cos B + cos C = + sin A sin B sin C tna + tnb + tnc = tna tnb tnc (vi) tn A tn B + tn B tn C + tn C tn A = (vii) cot A + cot B + cot C = cot A. cot B. cot C (viii) cot A cot B + cot B cot C + cot C cot A = (ix) A + B + C = then tn A tn B + tn B tn C + tn C tn A = v Rnge of Trigonometric Exression: E = sin q + b cos q b E = + b sin (q + ), where tn = = + b cos (q - b), where tn b = b v Hence for ny rel vlue of q, - + b E + b Sine nd Cosine Series: sin + sin ( + b) + sin ( + b ) +... + sin ( + - b ) nb sin n = b sin æ sin ç + è n - ö b ø cos + cos ( + b) + cos ( + b ) +... + cos ( + - b ) nb sin n = b sin æ n - ö cos ç + b è ø A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
. DEFINITION : TRIGONOMETRIC EQUATION An eqution contining trigonometric function of unknown ngles re known s trigonometric equtions.. PERIODIC FUNCTION : A function f(x) is sid to be eriodic if there exists T > 0 such tht f(x + T) = f(x) for ll x in the domin of definitions of f(x). If T is the smllest ositive rel numbers such tht f(x + T) = f(x), then it is clled the eriod of f(x). The eriod of sin x, cos x, sec x, cosec x is nd eriod of tn x nd cot x is.. GENERAL SOLUTION OF STANDARD TRIGONOMETRICAL EQUATIONS : Since Trigonometricl functions re eriodic functions, therefore, solutions of Trigonometricl equtions cn be generlised with the hel of eriodicity of Trigonometricl functions. The solution consisting of ll ossible solutions of Trigonometricl eqution is clled its generl solution.. Generl Solution of the eqution sin q = 0. By Grhicl roch, The bove grh of sin q clerly shows tht sinq = 0 t q = 0, ±, ±, ±... It follows tht when sin q = 0 is q = n : n Î I i.e. n = 0, ±, ±.... Generl solution of cos q = 0. By grhicl roch, + / / O / / The bove grh of cosq clerly shows tht cosq= 0 t q = ± /, ± /, ± /,... It follows tht when cosq = 0 q = (n + ) /, n Î I. i.e. n = 0, ±, ±... A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
. Generl solution of tn q = 0. Proof: If tnq = 0 or sinq cos q = 0 sin q = 0, it follows tht generl solution of tnq = 0, it sme s of sinq = 0 so tht, generl solution of tn q = 0 is q = n ; n Î I Note : Generl solution of secq = 0 nd cosecq =0 does not exist becuse secq nd cosecq cn never be equl to 0.. Generl solution of the eqution sin q = sin. Proof : If sinq = sin or, sinq sin = 0 F HG F HG or, sin q- I K J I K J or, sin q - or, q+ q - = (m + ) F HG cos q + I K J = 0 = 0 or cos q+ = m ; m Î I nd q = m + ; m Î I or q = (m + ) ; m Î I F HG ; m Î I q = (ny even multile of ) + or q = (ny odd multile of ) by combining these two results q = n + ( ) n ; n Î I.. Generl solution of the eqution sinq = k, where k. Note: I K J Let be the numericlly lest ngle such tht k = sin then, sinq = sin \ q = n + ( ) n, where n Î I nd = sin k = 0 The eqution cosec q = cosec is equivlent to sin q = sin so these two equtions hving the sme generl solution.. Generl solution of the eqution cos q = cos. Proof : If cos q = cos or, cos q cos = 0 or, sin q+ F HG F HG or, sin q+ I K J I K J F HG.sin q- I K J F HG = 0 = 0 nd sin q- q+ = n ; n Î I I K J = 0 A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 6
nd q- = n ; n Î I q = n ; n Î I nd q = n + ; n Î I for the generl solution of cos q = cos, combine these two result which gives. q = n ±, n Î I.. Generl solution of the eqution cosq = k, where k. Let be the numericlly lest ngle such tht k = cos, then cosq = cos \ q = n ±, where n Î I nd = cos k Note : The eqution secq = sec is equivlent to cosq = cos, so the generl solution of these two equtions re sme..6 Generl solution of the eqution tn q = tn. Proof : If tn q = tn or, sinq sin = cosq cos or, sinq.cos cosq sin = 0 or, sin(q ) = 0 or, q = n ; n Î I q = n + ; n Î I.6. Generl solution of the eqution tnq = k. Let be the numericlly lest ngle such tht k = tn, then tnq = tn \ q = n +, where n Î I nd = tn k Note : The eqution cotq = cot is equivlent to tnq = tn so these two equtions hving the sme generl solution.. GENERAL SOLUTION OF SQUARE OF THE TRIGONOMETRICAL EQUATIONS :. Generl solution of sin q = sin. Proof : If or, sin q = sin sin q = sin (Both the sides multile by ) or, or, cosq = cos cosq = cos q = n ± ; n Î I q = n ± ; n Î I. Generl solution of cos q = cos. Proof : If or, cos q = cos cos q = cos q (multily both the side by ) or, or, or, + cosq = + cos cosq = cos q = n ± q = n ± ; n Î I A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 7
. Generl solution of tn q = tn. Proof: If tn q = tn or, tn q tn = (using como. nd divid. rule) tn tn q + tn q - = + tn - or, + tn q + tn - tn q = - tn or, - tn q - tn + tn q = + tn or, cosq = cos Þ q = n ± ; n Î I. GENERAL SOLUTION OF TRIGONOMETRICAL EQUATION : cos q + bsin q = c Ste I Ste II Ste III Consider trigonometricl eqution cosq + bsinq = c, where, b, c Î R nd c b + To solve this tye of eqution, first we reduce them in the form cosq = cos or sinq = sin. Algorithm to solve eqution of the form cosq + bsinq = c. Obtin the eqution cosq + bsinq = c Put = r cos nd b = rsin, F I HG K J where r = + b nd tn = b i.e. = b tn Using the substitution in ste - II, the eqution reduces r cos(q ) = c Þ cos(q ) = c r Þ cos(q ) = cosb (sy) Ste IV Solve the eqution obtined in ste III by using the formul. Generl solution of Trigonometricl eqution cosq + bsinq = c. A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 8
PART - I : OBJECTIVE QUESTIONS * Mrked Questions re hving more thn one correct otion. Section : Bsic Formul nd Angle trnsformtion A-. The vlues of the exression (sinx + cosecx) + (cosx + secx) (tnx + cotx) wherever defined is equl to 0 7 (D) 9 A-. The vlue of tn º tn º tn º... tn 89º is 0 (D) A-. The vlue of sin ( + q) sin ( q)cosec q is equl to 0 sinq (D) none of these A-. æ ö æ ö æ7 ö ç ç ç è ø è ø è ø æ ö æ ö cosç x -.tnç + x è ø è ø tn x -.cos + x -sin -x when simlified reduces to: sin x cos x sin x sin x cos x (D) sin x é æ ö ù A-. The exression êsin ç - + sin ( + ) ú ë è ø û é 6 æ ö 6 ù êsin ç + + sin ( + ) ú ë è ø û is equl to 0 (D) sin + sin 6 A-6. cos (0 q) sin (60 q) is equl to 0 cos q sin q (D) sin q cos q A-7. If cos cos cos... cos79 = x +, then x is equl to 0 (D) none of these A-8. Given tn = tn b, then the vlue of sin( + b) is equl to sin( - b) 0 (D) A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 9
A-9. The vlue of the exression æ ö æ ö æ 7 ö æ 9 ö ç + cos ç + cos ç + cos ç + cos is è 0 ø è 0 ø è 0 ø è 0 ø 8 6 (D) 0 A-0. If (sec A + tn A) (sec B + tn B) (sec C + tn C) = (sec A tn A) (sec B tn B) (sec C tn C) then ech side is equl to 0 (D) none A-. In tringle ABC, ngle (, q) is : o A = 6, AB = AC = & BC = x. If + q x = then the ordered ir Section :Addition of Angles nd Multile Angles Formul B-. sin qcos q - cosqsin q when simlified reduces to sinq sin q cosq (D) cos q B-. If q = 9, then sinq - sinq is equl to sin6q + sinq - (D) - B-. sin cos 8 + cos8 cos98 Exct vlue of sin cos7 + cos7 cos97 is- 0 (D) none of these B-. The vlue of cos 90 + is sin 0 (D) none B-. If A nd B re comlimentry ngles, then : æ A ö æ B ö æ A ö æ B ö ç + tn ç + tn = ç + cot ç + cot = è ø è ø è ø è ø æ A ö æ B ö æ A ö æ B ö ç + sec ç + cosec = (D) ç - tn ç - tn = è ø è ø è ø è ø B-6. q & q re cute ngles such tht sinq = nd cosq =, then- 0 tn(q + q ) = tn(q + q ) = tn(q + q ) = (D) cot(q + q ) = 6 A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 0
B-7. cosq cos F H I F + q.cos q K H - K equls, I cos q sinq cosq (D) sinq æ cot A B-8. If A + B =, then the vlue of ö æ cot B ç. è+ cot A ö ç ø è+ cotb ø is (D) B-9. + b tn If sin = sinb, then is equl to - b tn (D) B-0. In tringle ABC if tn A < 0 then: tn B. tn C > tn B. tn C < tn B. tn C = (D) none B-. If tn º = x, then tnº -tnº + tnº tnº is equl to - x x + x x + x - x (D) - x + x B-. B-. B-. If sin sinb cos cosb + = 0, then the vlue of cot tnb is 0 (D) none of these The vlue of tn 0 + tn + tn 0 tn is 0 (D) If A lies in the third qudrnt nd tn A = 0, then sin A + sina + cosa is equl to 0 (D) 8 B-. If cos q = æ ö ç + è, then cos q in terms of is ø æ ö ç + è ø æ ö æ ö ç + è ç + ø è ø (D) none B-6. B-7. If tn q = tn f +, then the vlue of cos q + sin f is (D) Indeendent of f (cos + sin ) The vlue of is (cos - sin ) tn 0 tn 6 cot (D) cot A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
Section :Hlf Angles Formul nd Rnge of Trigonometry Function C-. The vlue of -tn º tn º + is (D) C-. If cos A = /, then the vlue of 6 cos (A/) sin (A/) sin (A/) is (D) C-. If sinq = sinq then tn q cn hve the vlue equl to C-. If cos q + b sin q = & sin q - b cos q = then + b hs the vlue = 7 (D) none æ 7 ö C-. Given tht cos - sin - = 0ç < <, the vlue of è ø (D) - /Ö Ö (D) Ö/ cot is- C-6. If sin t + cos t = then tn t is equl to: - (D) 6 C-7. If f(q) = sin q + cos q, then rnge of f(q) is é ù ê, ú ë û é ù ê, ú ë û é ù ê, ú ë û (D) None of these C-8. The mximum vlue of + sinq-cosq is 0 (D) Section (D) :Miscellneous D-. cos0 + 8sin70 sin0 sin0 is equl to: sin 80 / (D) none of these D-. If A = tn 6 tn nd B = cot 66 cot 78, then A = B A = B A = B (D) A = B D-. If + b + g =, then tn + tn b + tn g = tn tn b tn g tn tn b + tn b tn g + tn g tn = tn + tn b + tn g = tn tn b tn g (D) tn tn b + tn b tn g + tn g tn = 0 A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
D-. D-*. If x + y = z, then cos x + cos y + cos z cos x cos y cos z is equl to cos z sin z cos (x + y z) (D) In tringle tn A + tn B + tn C = 6 nd tn A tn B =, then the vlues of tn A, tn B nd tn C re,,,,,, 0 (D) none D-6. If tn + cot = then the vlue of tn + cot = + + - + - - (D) none D-7. If < q <, then + + cosq is equl to cosq sinq cosq (D) sinq D-8. If sinx = cos x then cos x ( + cos x) equls to 0 (D) none of these Section (E) :Trigonometric Equtions E-. E-. E-. The generl solution of the eqution, cosx =.cos x - is x = n x = n x = n/ (D) x = n/ Angles A & B re obtuse ngles such tht, tna + tnb + tna tnb =. If A - B = o then t A = º, B = 9º A = º, B = 0º A = 7º, B = º (D) A = 6º, B = º All solutions of the eqution, sinq + tnq = 0 re obtined by tking ll integrl vlues of m nd n in: n + n & m ± n & m ± (D) n & m ± E-. The generl solution of the eqution: tn + tn = is given by: = n = (n + ) = (6n + ) (D) = n E-. sinx - cos x - ssumes the lest vlue for the set of vlues of x given by: x = n + (-) n+ (/6) x = n + (-) n (/6) x = n + (-) n (/) (D) x = n - (-) n (/6) where n Î I E-6. sin q sinq - - cos q cosq - cosq + cot q - tn q cot q = - if: q Î æ 0, ö æ ç q Î ç è ø è, ö æ q Î, ö æ ç (D) q Î ö ç, ø è ø è ø æ ö E-7. The generl solution of the eqution, tn x + tn çx + è ø + tn æ ç è x + ö = is ø n + n Î I n + n Î I n + n Î I (D) none 6 E-8. sin q = sin q. sin q. sin q in 0 q hs: rel solutions rel solutions 6 rel solutions (D) 8 rel solutions. A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
E-9. The generl solution of the eqution, cot q = ( + cot q) is n + (- ) n n + (- ) n n + (- ) n 6 (D) none E-0. The generl solution of sin x + sin x = sin x + sin x is: n ; n Î I n ; n Î I n/ ; n Î I (D) n/ ; n Î I E-. Generl solution of the eqution, cot q - cot q = 0 is q = (n - ) q = (n - ) q = (n - ) (D) none E-. tn x- tn x The set of vlues of x for which + tn x tnx = is f {/} {n + / n =,,...} (D) {n + / n =,,...} E-. The set of ngles between 0 & stisfying the eqution cos q - cos q - = 0 is ì 9 ü í,,, ý î þ ì 9ü í,, ý î þ E-. The eqution sin x + 0 cos x 6 = 0 is stisfied if ( n Î I ) (D) ì 7 7 ü í,,, ý î þ ì 7 9 ü í,,, ý î þ x = n + cos (/) x = n cos (/) x = n ± cos (/) (D) x = n cos (/) E-. Number of solutions of the eqution tnx + secx = cosx lying in the intervl [0, ] is 0 (D) E-6. The number of solutions of the eqution, cot x = cot x + (0 x ) is : sin x 0 (D) E-7. The generl solution of, sin x - sin x + sin x = cos x - cos x + cos x is n + 8 n + 8 æ nö æ (- ) n ç + (D) n + cos è ø - ç ö 8 è ø E-8. The rincil solution set of the eqution cos x = + sinx is ì ü í, ý î 8 8 þ ì ü í, ý î 8 þ ì ü í, ý î 0 þ (D) ì ü í, ý î8 0 þ A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
E-9. The number of solutions of the eqution sinx = cosx in [, ] is : 8 (D) 0 é nù E-0. If tn x secx = 0 hs 7 different roots in ê0, ú, n Î N, then gretest vlue of n is ë û 8 0 (D) E-. The solution of cosx = cosx sinx is x = n, n Î I x = n +, n Î I x = n + ( ) n, n Î I (D) x = (n + ) +, n Î I E-. The number of solutions of sinq + sinq + sinq + sinq = 0 in(0, ) is (D) 0 E-. The solution set of the eqution sinq.cosq cosq sinq + = 0 in the intervl (0, ) is ì 7ü í, ý î þ ì ü í, ý î þ ì í,,, î ü ý þ (D) ì ü í,, ý î6 6 6 þ E-. Totl number of solutions of eqution sinx. tnx = cosx belonging to (0, ) re : 7 8 (D) E-. If x Î é 0, ù ê ë ú û, the number of solutions of the eqution sin 7x + sin x + sin x = 0 is: 6 (D) None E-6. If cos ( + x) + sin ( + x) vnishes then the vlues of x lying in the intervl from 0 to re x = /6 or /6 x = / or / x = / or / (D) x = / or / E-7. cos q cos q- = if q = n +, n Î I q = n ±, n Î I q = n ± 6, n Î I (D) q = n + 6, n Î I E-8. If sinx + cosx = y +, xî[0, ], then y x = / x = x = 6 (D) x = / E-9. E-0. If x x + siny = 0, yî[0, ), then x =, y = 0 x =, y = / x =, y = 0 (D) x =, y = / The number of integrl vlues of for which the eqution cos x + sin x = - 7 ossesses solution is (D) A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
Comrehension # PART - II : MISCELLANEOUS OBJECTIVE QUESTIONS If q increses from 0 to then vlue of sinq, tnq nd secq increses while cosq, cotq nd cosecq decreses. The following irs (sinq, cosq), (tnq, secq) nd (secq, cosecq) hve the sme vlue t q =. If A = then cosa sina will be ositive negtive zero (D) Ö. If A = 9 then seca + coseca will be ositive negtive zero (D) not defined. If A = 8 then cota tna will be- ositive negtive zero (D) two Comrehension # sin cos x x Consider the eqution + =, 0 x nd nswer the following questions b + b. From the given eqution sin x cos x = b sin x cosx = b sin x cos x = (D) none of these b. In terms of nd b the vlue of sin x must be b + b 8 8 sin x cos x 6. The vlue of + must be b ( + b) ( + b) -b + b ( + b) (D) none of these (D) none of these Comrehension # Let, b, c, d Î R. Then the cubic eqution of the tye x + bx + cx + d = 0 hs either one root rel or ll three roots re rel. But in cse of trigonometric equtions of the tye sin x + b sin x + c sinx + d = 0 cn ossess severl solutions deending uon the domin of x. To solve n eqution of the tye cosq + b sinq = c. The eqution cn be written s cos (q ) = c/ ( + b ). The solution is q = n + ± b, where tn = b/, cos b = c/ ( + b ). 7. On the domin [, ] the eqution sin x + sin x sinx = 0 ossess only one rel root three rel roots four rel roots (D) six rel roots 8. In the intervl [ /, /], the eqution, cos x + 0tnx + tn x = hs no solution one solution two solutions (D) three solutions 9. tn x = tn x + cosx (0 x ) hs no solution one solution two solutions (D) three solutions A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 6
Comrehension # To solve trigonometric ineqution of the tye sin x ³ where, we tke hill of length in the sine curve nd write the solution within tht hill. For the generl solution, we dd n. For instnce, to solve sinx ³, we tke the hill é ê, ú ù over which solution is ë û 6 7 < x <. The generl solution is n 6 7 < x < n +, n is ny integer. Agin to solve n ineqution of the tye sin x, where, we 6 6 tke hollow of length in the sine curve. (since on hill, sinx is stisfied over two intervls). Similrly cos x ³ or cosx, re solved. 0. Solution to the ineqution sin 6 x + cos 6 x < 6 7 must be n + < x < n + n + < x < n + n n + < x < + (D) none of these 6. Solution to inequlity cos x + cos x + ³ 0 over [, ] is é ù [, ] ê, ú ë 6 6 û é ù [0, ] (D) ê, ú ë û. æ ö Over [, ], the solution of sin çx + + è ø cos x ³ 0 is [, ] é ù ê, ú ë 6 6 û [0, ] é 7ù é ù é ù (D) ê, ú È ë ê, ú È û ë ê, ú û ë û Mtch the Column :. Mtch the following Column-I Given tn = is equl to tn b, then the vlue of sin( + b) sin( - b) Column-II (P) æ ö æ ö If sinç x + + cosç x - + b then the vlue (Q) è 6 ø è ø of + b is equl to cos 90 + sin0 is equl to (R) (D) cos0 + sin Ösin 6 is equl to (S) the vlue of ( + ) (b ) where tnq cotq = nd cosq sinq = b A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 7
Mtch the Column :. Mtch the following Let E = + sina + -sin A, mtch the following vlues of E : Column-I Column-II 0 A 80 () sin 80 A 60 (q) cos 60 A 0 (r) sin (D) 0 A 70 (s) cos. Mtch the following If cosa = m (0 < m < ) where A lies between 0 nd 0 then mtch the following Column-I Column-II sina () - -m + m sin A (q) -m cos A -m (r) (D) tn A +m (s) + 6. If nd b re the roots of the eqution, cos q + b sin q = c then mtch the entries of column- I with the entries of column-ii. Column-I sin + sin b () sin. sin b (q) tn + tn b Column-II (r) b + c c- c+ bc b (D) tn. tn b = (s) + c - + b A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 8
7. Solve the eqution for 'x' given in Column-I nd mtch with the entries of Column-II Column-I Column-II cos x. cos x + sin x. sin x = 0 () n ± sin = sin sin(x + ) sin(x - ) (q) n +, n Î I where is constnt ¹ n. tn x + cot x =. (r) n +, n Î I 8 (D) sin 0 x + cos 0 x = 9 6 cos x. (s) Assersion-Reson Tye n ± Sttement - is true, Sttement - is true ; Sttement - is correct exlntion for Sttement - Sttement - is true, Sttement - is true ; Sttement - is NOT correct exlntion for Sttement - Sttement - is true, Sttement - is flse. (D) Sttement - is flse, Sttement - is true. 8. Sttement- If A, B, C re the ngles of tringle such tht ngle A is obtuse, then tnb tn C > tn B + tn C Sttement-: In ny tringle tna = tn B tn C - 9. Sttement- The number sin8 nd sin re the roots of sme qudrtic eqution with integer co-efficients. Sttement-: If x = 8, then x = 90, if y =, then y = 70 0. Sttement- The number of roots of the eqution sinx= x - x+ is. Sttement-: In [0, ], sinx= hs exctly two solutions.. Sttement-: In (0, ), the number of solutions of the eqution tnq + tn q + tn q = tn q.tnq.tnq is. Sttement-: Ech solution of tn 6q = 0 is solution of tnq + tn q + tn q = tn q.tnq.tnq.. Sttement- If 0 x, 0 y nd cos x. sin y =, then the ossible number of vlues of the ordered ir (x, y) is. Sttement-: sinx, cosx 0.. Sttement- The number of integrl vlues of n so tht sinx(sinx + cosx) = n hs t lest one solution re. Sttement-: + b sinx + b cosx + b A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 9
PART - I : OBJECTIVE QUESTIONS. The vlue of cos cos cos 8 cos 6 cos is : 0 0 0 0 0 cos( /0) 6 6 (D) - 0 + 6. If o A = 0 then A sin is identicl to + sin A + - sin A - + sin A - - sin A + sin A - - sin A (D) - + sin A + - sin A. A mn running on circulr trck t the rte of Km/hr trverses n rc which subtends n ngle of 6 t the centre in 0 seconds. The dimeter of the circle is (Assume is roximtely equl to meters 0 meters 7 meters (D) 00 meters ) 7. - tn A =, - - tn B =, if A nd B re ositive then A + B is (D) none. +b + g =, S = sin ( + b) sin g + sin ( b + g) sin + sin ( g + ) sinb nd, C = cos ( + b) cos g + cos ( b + g) cos + cos ( g + ) cosb then S C equls : - 0 (D) 7 6. cos cos cos cos is equl to - 8 6 - (D) 6 8 7. The vlue of tn0 + tn0 + tn0 tn0 is equl to (D) / 8. 6 cos + cos + cos + cos + cos + cos is equl to- 7 7 7 7 7 7 0 (D) sin( +q) -sin( -q) 9. The exression cos( b-q) -cos( b+q) indeendent of indeendent of q is indeendent of b (D) indeendent of nd b A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 0
0. 8{sin 7 sin 60 } + is-. 8sin sin8 sin is equl to 8 (D) 8 (D) 6 æ C ö C A B. If A + B + C = & sin ç A + = k sin, then tn tn isè ø k - k + k + k - k k +. Vlue of exression tn8 + cot8 + tn7 + cot7 is (D) k + k (D) o o o o cos 0 + 8 sin 70 sin0 sin0. o is equl to : sin 80 / (D) None. cotq tnq tnq tnq is equl to- 0 when q 8 when q = 6 8 when q = 6 (D) 0 when q = = 6. cos + cos8 cos cos8 is equl to- 7. If cot ( + b) = 0, then sin ( + b) is equl to- 0 (D) 0 cos cosb sin (D) sinb 8. In right ngled tringle the hyotenuse is times the erendiculr drwn from the oosite vertex. Then the other cute ngles of the tringle re & 6 & 8 8 & (D) & 0 9. If < <, - cos then, + + cos sin - sin + cos - cos is- sin (D) - sin cos6x + 6cosx + cosx + 0 0. The exression is equl to cosx + cosx + 0cosx cos x cos x cos x (D) + cos x cosq. If cos + cos b =, sin + sin b = b nd b = q, then = cosq + b + b b (D) ( + b ) / A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
é ù. If Î ê, ú ë û cos then the vlue of sin +sin - -sin is equl to: (D) none of these æ C ö. If A + B + C = & sin ça + C A B = k sin, then tn tn = è ø k - k + k k + - k k + (D) k + k æ ö æ x ö. If x Î ç, then cos ç - + è ø è sin x + sin x is lwys equl to ø (D) none of these. If three ngles A, B, C re such tht cos A + cos B + cos C = 0 nd if cos A cos B cos C = l ( cos A + cos B + cos C), then vlue of l is : 8 (D) 6 6. In ny tringle ABC, which is not right ngled å cosa.cosecb.cosecc is equl to (D) none of these 7. If x = y cos = z cos, then xy + yz + zx is equl to 0 (D) 8. If cos (A B) = nd tn A tn B =, cos A cos B = sin A sin B = cos (A + B) = (D) sin A cos B = 9. If A + B + C =, then cos A + cosb + cosc is equl to cos A cosb cosc sin A sin B sin C + cos A cos B cos C (D) sin A sin B sin C 0. If A + B + C = & cosa = cosb. cosc then tnb. tnc hs the vlue equl to: / (D) MULTIPLE OPTIONS CORRECT *. If the sides of right ngled tringle re {cos + cosb + cos( + b)} nd {sin + sinb + sin( + b)}, then the length of the hyotenuse is: - b [+cos( - b)] [ - cos( + b)] cos b *. If tn x =, ( ¹ c) - c y = cos x + b sin x cos x + c sin x z = sin x b sin x cos x + c cos x, then + b (D) sin y = z y + z = + c y z = c (D) y z = ( c) + b A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
*. For 0 < q < /, tn q + tn q + tn q = 0 if tn q = 0 tn q = 0 tn q = 0 (D) tn q tn q = *. ( + ) sin + ( ) cos = ( + ) if tn = æ cosa + cosb ö *. The vlue of ç è sin A - sinb ø A -B tn n 0 : n is odd n æ sin A + sinb ö + ç è cosa - cosb ø n is + cot n (D) none A -B (D) : n is even - 6*. The extreme vlues of f(q) = sin q + sin q.cos q + cos q, " q Î R, re 0 6 Ö (D) 8 7*. Which of the following when simlified reduces to unity? - sin æ ö æ ö cotç + cos ç - è ø è ø sin( - ) + cos ( - ) sin -cos tn sin cos + (- tn tn ) + sin (D) (sin + cos ) 8*. If sin =, then the vlue of exression tn sin - cos 9*. If cos(a B) = /, nd tna tnb =, then 8 my be - (D) 8 cos A cosb = cos( A+ B) = - sina sinb = - (D) none of these 8 6 0*. The vlue of cos cos cos cos cos is : 0 0 0 0 0 0 + 6 cos( /0) 6 cos( /0) 6 0 + (D) 6. The solutions set of (cosx ) ( + cosx) = 0 in the intervl 0 x is ì ü í ý î þ ì ü í, ý î þ ì,, cos - æ - öü í ç ý î è øþ (D) none of these A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
. The number of ordered ir (x, y) where x nd y stisfy x + y = / nd cosx + cosy = / is 0 (D) infinity. The number of solution of cos q + sin q + = 0, for (q Î [0, ]) is 0 (D) infinity. The number of vlues of x for which sinx + sinx = is 0 infinite (D) none of these. The number of solutions of cos(x/) = x + x, x Î [0, ] is 0 (D) infinite 6. If cos q + cos q = 0, then æ q = n ± where = cos ö ç 7 - è ø æ q = n ± where = cos ö ç ± 7 - è ø æ q = n ± where = cos ö ç - 7- è ø (D) none of these 7. If sin q + 7 cos q =, then tn (q/) is root of the eqution x - 6x + = 0 6x - x - = 0 6x + x + = 0 (D) x - x + 6 = 0 8. The most generl solution of tnq = nd cosq = is : n + 7 7, n Î I n + ( ) n, n Î I n + 7, n Î I (D) none of these 9.* cosx cos8x - cosx cos9x = 0 if cosx = cos x sin x = 0 sinx = 0 (D) cosx = 0 0.* If sin(x - y) = cos(x + y) = / then the vlues of x & y lying between 0 nd re given by: x = /, y = / x = /, y = / x = /, y = / (D) x = /, y = /.* The eqution sin x. cos x + sin x = sin x. sin x + cos x hs root for which sinx = sinx = cosx = (D) cosx =.* sinx - cos x - ssumes the lest vlue for the set of vlues of x given by: x = n + (-) n+ (/6), n Î I x = n + (-) n (/6), n Î I x = n + (-) n (/), n Î I (D) x = n - (-) n (/6), n Î I.* The generl solution of the eqution cosx. cos6x =, is : x = (n + ), n Î I x = n, n Î I x = (n ), n Î I (D) none of these.* The generl vlue of q, stisfying the eqution cos q + sinq =, is n n + ( ) n n + ( ) n 6 (D) none of these A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
PART - II : SUBJECTIVE QUESTIONS. Prove tht : () sec A ( sin A) tn A = (b) tn A sin A = sin A sec A = tn A sin A (c) cosa coseca - sin A sec A cosa + sin A = cosec A sec A (d) (cosec q sin q) (sec q cos q) (tn q + cot q) = (e) cot q (sec q -) = sec + sin q q. - sin q + sec q (f) ( + cot A + tn A) (sin A cos A) = sec A cosec A - cosec A sec A sin q tnq ( - tnq) + sin q sec (g) (+ tnq) q = sin q (+ tnq). If the rcs of the sme length in two circles subtend ngles 7 nd 0 t the centre, find the rtio of their rdii.. Find the rdin mesures corresonding to the following degree mesures (i) (ii) 0 (iii) 0. Find the degree mesures corresonding to the following rdin mesures (i) (ii) (iii) 7 (iv) 6. Prove tht : () sin 6 + cos tn = 9 (c) cos + sec + tn = 7 (b) sin + cosec cos = 0 6 6 (d) cot + cosec + tn = 6 6 6 6 (e) sin + cos + sec = 0 6. Find the vlue of : (i) cos 0 (ii) sin (iii) tn 0 (iv) cot ( ) 7. cos( + q)cos( -q) æ ö sin( - q)cosç + q è ø = cot q. 8. cosq + sin (70 + q) sin (70 q) + cos (80 + q) = 0. æ ö 9. cos ç + q è ø cos ( + q) é æ ö ù êcot ç - q + cot( + q) ú ë è =. ø û A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
æ A ö 0. sin ç + è 8 ø æ A ö sin ç - è 8 ø æ = ö ç è ø s i n A q 9 q q. cos q cos cos q cos = sin q sin.. cos² + cos² ( + b) - cos cos b cos ( + b) = sin²b.. For ll vlues of, b, g rove tht, cos + cos b + cos g + cos ( + b + g) = cos + b b + g g +. cos. cos. x x. If tn x =, < x <, find the vlue of sin nd cos.. If cos ( + b) = ; sin ( - b) = &, b lie between 0 &, then find the vlue of tn. 6. If sin (q + ) = & sin (q + b) = b (0 <, b, q < /) then find the vlue of cos ( - b) - b cos( - b) 7. Prove tht, sin x. sin x + cos x. cos x = cos x. 8. rove tht ì æ - ö ü ï- cot ç ï è ø í + cos cot ý ï æ - ö + cot ç ï ïî è ø ïþ 9 sec = cosec. 9. Prove tht tn - tn - cot-cot = cot. 0. Find the gretest nd lest vlue of y (i) y = 0 cos²x - 6 sin x cos x + sin²x (ii) y = + sin x + cos x (iii) æ ö y = cos çq + è + cos q + ø. If f is the exterior ngle of regulr olygon of n sides nd q is ny constnt, then rove tht sin q + sin (q + f) + sin (q +f) +... u to n terms = 0. If x + y + z = show tht, sin x + sin y + sin z = cosx cosy cosz.. If x + y = + z, then rove tht sin x + sin y sin z = sin x sin y cos z.. Prove tht : (i) cosa coseca - sin A sec A cosa + sin A = cosec A sec A (ii) sec - tn cos = cos sec + tn (iii) cos A + sin A cosa + sin A + cos A - sin A cos A - sin A = A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 6
. Show tht: (i) cot 7 or tn 8 = ( ) ( + ) + or + + + 6 (ii) tn = + - - 6. 6. Prove tht, tn + tn + tn + 8 cot 8 = cot. 7. Clculte the following without using trigonometric tbles: (i) tn 9 - tn 7 - tn 6 + tn 8 (ii) cosec 0 sec 0 (iii) ésec cos 0 ù sin0 ê + sin ú ë sin û (iv) cot 70º + cos 70º (v) tn 0 - tn 0 + tn 70 8. Let A, A,..., A n be the vertices of n n-sided regulr olygon such tht; Find the vlue of n. A = +. A AA AA 9. Prove tht % cos = sin b + cos( + b) sin sinb + cos( + b) 0. If cos (b - g) + cos (g - ) + cos ( - b) = -, rove tht cos + cos b + cos g = 0, sin + sin b + sin g = 0.. If x cosq + by sinq xsin q = b, cos q Show tht (x) / + (by) / = ( b ) / bycosq = 0. sin q m+ n. If m tn (q - 0 ) = n tn (q + 0 ), show tht cos q = (m-n). tn + tn g sin + sin g. If tn b = + tn.tn g, rove tht sin b = + sin. sin g.. If sin x + sin y = & cos x + cos y = b, show tht, b sin (x + y) = + b nd tn x-y - -b = ± + b. If sin (q + ) = & sin (q + b) = b (0 <, b, q < /) then find the vlue of cos ( - b) - b cos( - b) 6. If P n = cos n q + sin n q nd Q n = cos n q sin n q, then show tht P n P n = sin q cos q P n Q n Q n = sin q cos q Q n nd hence show tht P = sin q cos q Q = cos q sin q. A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 7
Trignometric Eqution 7. Wht re the most generl vlues of q which stisfy the equtions: () sinq= (b) tn (x - ) = (c) tn q = - (d) cosec q = (e) cot q = cosec q 8. Solve sin 9q = sin q 9. Solve cot q + tn q = cosec q 0. Solve sin q = cos q. Solve cot q = tn 8q. Solve cot q - tn q =. Solve cosec q = cot q +. Solve tn q tn q =. Solve tn q + tn q + tn q tn q = 6. Solve sinq + sin q + sin q = 0. 7. Solve cos q + sin q = cos q + sin q 8. find ll vlues of q between 0º nd 80º stisfying the eqution cos 6q + cos q + cos q + = 0. 9. Solve cos x + cos x + cos x =. 0. Solve sin nq - sin (n -) q = sin q, where n is constnt nd n¹ 0,. find the most generl solution of the following : (i) sin6x = sinx sinx (ii) secx secx =. If sina = sinb nd cosa = cosb, find ll the vlue of A in terms of B.. Solve tnx + tnx = 0. Solve ( tnq) ( + sinq) = + tnq.. Solve sinx + cosx =. 6. Solve sinq =, tnq = 7. Solve the equlity: sin x + cos x + sin x = 0 8. Find ll vlue of q, between 0 &, which stisfy the eqution; cos q. cos q. cos q = /. 9. Find the generl solution of the eqution, + tn x cot x + cot x tn x = 0 60. Solve for x, the eqution - 8tnx = 6 tn x, where < x <. 6. Determine the smllest ositive vlue of x which stisfy the eqution, + sinx - cosx = 0. æ ö 6. sin çx + = + 8sin x. cos x è ø 6. Find the number of rincil solution of the eqution, sin x sin x + sin x = cos x cos x + cos x. æ ö ç + log(cos x+ sin x) è ø log (cos x-sin x) 6. Find the generl solution of the trigonometric eqution - =. 6. Find ll vlues of q between 0 & 80 stisfying the eqution; cos 6 q + cos q + cos q + = 0. A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 8
PART-I IIT-JEE (PREVIOUS YEARS PROBLEMS). Let f (q) = sin q (sin q + sin q). Then [IIT-JEE-000, Scr., /] f (q) ³ 0 only when q ³ 0 f (q) 0 for ll rel q f (q) ³ 0 for ll rel q (D) f (q) 0 only when q 0.. The mximum vlue of (cos ). (cos )... (cos n ) under the restrictions, 0,,..., n nd (cot ). (cot )... (cot n ) = is: [IIT-JEE - 00, Scr - (- M), ] / n/ / n / n (D). If + b = nd b + g =, then tn equls [IIT-JEE - 00,Scr - (- M), ] (tn b + tn g) tn b + tn g tn b + tn g (D) tn b + tn g. The number of integrl vlues of ' k ' for which the eqution 7 cos x + sin x = k + hs solution is: [IIT-JEE-00, Scr., (, )/90] 8 0 (D). If sin = / nd cos q = /, then the vlues of + q (if q, re both cute) will lie in the intervl [IIT-JEE-00, Scr., (, )/8] é ù ê, ú ë û é ù ê, ú ë û é ù ê, ú ë 6 û é (D) ê ë, 6 ù ú û 6. Find the rnge of vlues of ' t ' for which sin t = - x + x é ù, t Î x - x - ê-, ú. ë û [IIT-JEE - 00, Min, (-M), 60] 7. In n equilterl tringle, coins of rdii unit ech re ket so tht they touch ech other nd lso the sides of the tringle. Are of the tringle is [IIT-JEE - 00, Scr- (, ), 8] + 6 + + 7 (D) + 7 8. cos ( b) = nd cos ( + b) =, where, b Î [, ]. Pirs, b which stisfy both the equtions is/ e re [IIT-JEE-00, Scr., (, )/8] 0 (D) æ ö 9. Let q Î ç0, nd t è ø = (tn q) tn q, t = (tn q) cot q, t = (cot q) tn q nd t = (cot q) cot q, then [IIT-JEE - 006, Min - (, ), 8] t > t > t > t t < t < t < t t > t > t > t (D) t > t > t > t A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 9
0. The number of solutions of the ir of equtions sin q cos q = 0, cos q sin q = 0 in the intervl [0, ] is [IIT-JEE-007, Per-, (, )/8] zero one two (D) four. For 0 < q < /, the solution(s) of 6 6 æ (m-) ö æ m ö å cosecç q + cosec + = m= ç q is/re : è ø è ø (D) [IIT-JEE 009, (, )/8].* If sin x cos x + =, then [IIT-JEE - 009,Per-, (, ), 80] tn x = tn x = (D) sin 8 x cos 8 x + = 8 7 sin 8 x cos 8 x + = 8 7. The mximum vlue of the exression sin q + sin qcosq + cos q is æ ö n. The number of vlues of q in the intervl ç, such tht q ¹ è for n = 0, ±, ± nd ø tnq = cot q s well s sin q = cos q is [IIT-JEE-00, Per-, (, 0)/8]. Let P = { q : sinq - cosq = cosq} nd Q { q : sinq cosq sinq} P ÌQ nd Q - P ¹Æ Q Ë P = + = be two sets. Then [IIT-JEE-0, Per-/80] P Ë Q (D) P = Q PART-II AIEEE (PREVIOUS YEARS PROBLEMS). If is root of cos q + cos q = 0, < <, then sin is equl to [AIEEE - 00] () () () 8 () 8. The eqution sin x + b cos x = c, where c > + b hs [AIEEE - 00] () unique solution () infinite number of solutions () no solution () None of the bove. If y = sin q + cosec q, q ¹ 0, then [AIEEE - 00] () y = 0 () y () y ³ () y ³. If sin ( + b) =, sin ( b) = then tn( + b) tn( + b) is equl to [AIEEE - 00] () () () 0 () None of these A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 0
. If tnq = then sin q is [AIEEE - 00] () - but not () - or () but not - () None of these tn º 6. The vlue of is [AIEEE - 00] + tn º () () () () xy 7. sin q = is true if nd only if [AIEEE - 00] (x + y) () x y ¹ 0 () x = y, x ¹ 0 () x = y () x ¹ 0, y ¹ 0 8. In DABC, medins AD nd BE re drwn. If AD =, ÐDAB = 6 & ÐABE =, then the re of DABC is [AIEEE - 00] 8 6 6 () squre unit () squre unit () squre unit () squre unit 7 9. Let, b be such tht < b <. If sin + sin b = nd cos + cos b =, then the vlue of 6 6 æ cos ç è b ö ø is [AIEEE - 00] () 0 () 0 () 6 6 () 6 6 æ P ö æ Q ö 0. In tringle PQR, Ð R =. If tn ç nd tn ç re the roots of x è ø è + bx + c = 0; ¹0 then ø [AIEEE - 00] () b = + c () b = c () c = + b () = b + c. If 0 < x < nd cos x + sin x =, then tn x is [AIEEE-006] () 7 æ () ö ç + 7 è () ø + 7 7 (). A tringulr rt is enclosed on two sides by fence nd the third side by stright river bnk. The two sides hving fence re of sme length x. The mximum re enclosed by the rk is [AIEEE-006] () x x () () 8 x () x. Let A nd B denote the sttements [AIEEE-009] A : cos + cos b + cos g = 0 B : sin + sin b + sin g = 0 If cos (b g) + cos (g ) + cos ( b) =, then : A is flse nd B is true both A nd B re flse both A nd B re true (D) A is true nd B is flse A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
. Let cos( + b) = nd let sin( b) =, where 0, b. Then tn = () 6 9 () () 0 7 () 6. If tn q =, then sin q is [AIEEE 00] but not or but not (D) None of these 6. If is root of cos q + cos q = 0, < <, then sin is equl to [AIEEE 00] 8 (D) 8 7. If sin ( + b) =, sin( b) =, then tn( + b) tn ( + b) is equl to [AIEEE 00] zero (D) none of these 8. If y = sin q + cosec q, q ¹ 0, then [AIEEE 00] y = 0 y y ³ (D) y ³ 9. The eqution sin x + b cos x = c where c > + b hs [AIEEE 00] unique solution infinite number of solutions no solution (D) none of the bove 7 0. Let, b be such tht < b <. If sin + sin b = nd cos + cos b =, then the vlue of cos 6 6 æ - ç è b ö ø is [AIEEE 00] 0 0 6 6 (D) 6 6. The number of vlues of x in the intervl [0, ] stisfying the eqution sin x + sin x = 0 is 6 (D) [AIEEE 006]. If 0 < x < nd cos x + sin x =, then tn x is [AIEEE 006] ( - 7 ) ( + 7 ) ( + 7 ) ( - 7 ) (D) tna cota. The exression + -cot A -tn A cn be written s [JEE MAINS 0] () sin A cos A + () sec A cosec A + () tn A + cot A () sec A + cosec A A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
NCERT BOARD QUESTIONS. Prove tht tna + sec A - + sina = tn A - sec A + cosa. If sin -cos+ sin = y, then rove tht + cos+ sin + sin is lso equl to y.. If m sin q = n sin (q + ), then rove tht tn (q + ) cot = m + n m- n. If cos ( + b) = nd sin ( - b) =, where lie between 0 nd, find the vlue of tn. If tn x = b, then find the vlue of + b -b + - b + b q 9q 6. Prove tht cos q cos - cos q cos =sin 7q sin 8q. 7. If cos q + b sin q = m nd sin q - b cos q = n, then show tht + b = m + n. 8. Find the vlue of tn º0'. 9. Prove tht sin A = sin A cos A - cosa sin A. 0. If tn q + sin q = m nd tn q - sin q = n, then rove tht m - n = sin q tnq.. If tn (A + B) =, tn (A B) = q, then show tht tn A = + q - q. If cos + cosb = 0 = sin + sin b, then rove tht cos + cos b = cos ( + b).. If sin(x+ y) + b tnx = sin(x-y) - b, then show tht = tny b. If tn q = sin-cos sin+ cos, then show tht sin + cos = cosq.. If sin q + cos q =, then find the generl vlue of q. A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
6. Find the most generl vlue of q stisfying the eqution tn q = nd cosq = 7. If cot q + tn q = cosecq, then find the generl vlue of q. 8. If sin q = cos q, where 0 q, then find the vlue of q. 9. If sec x cos x + = 0, where 0 < x, then find the vlue of x. 0. If sin (q + ) = nd sin (q + b) = b, then rove tht cos ( b) -b cos ( - b) = - - b. If cos (q + f) = m cos (q - f), then rove tht tn q = - m cot f. + m. Find the vlue of the exression 6 6 [sin ( -) + sin ( +)] - {sin ( + ) + sin ( -)]. If cos q + b sin q = c hs nd b t its roots, then rove tht tn + tnb = b + c. If x = sec f - tn f nd y = cosec f + cot f then show tht xy + x - y + = 0. If q lies in the first qudrnt nd cosq = 8,then find the vlue of cos (0º + q) + cos (º - q) + cos (0º - q). 7 6. Find the vlue of the exression cos 7 + cos + cos + cos 8 8 8 8 7. Find the generl solution of the eqution cos q + 7 sin q 6 = 0 8. Find the generl solution of the eqution sin x - sin x + sin x = cos x - cos x + cos x 9. Find the generl solution of the eqution ( -) cosq+ ( + ) sin q = A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
EXERCISE # PART # I A-. A-. A-. A-. (D) A-. A-6. A-7. (D) A-8. (D) A-9. A-0. A-. B-. B-. B-. B-. B-. B-6. B-7. B-8. B-9. (D) B-0. B-. B-. B-. B-. B-. B-6. (D) B-7. (A, B, C, D) C-. C-. C-. C-. C-. C-6. (B, C) C-7. C-8. D-. D-. D-. D-. D-*. (A, B) D-6. D-7. (D) D-8. E-. E-. E-. E-. E-. E-6. E-7. E-8. (D) E-9. E-0. E-. E-. E-. E-. E-. E-6. E-7. E-8. E-9. E-0. (D) E-. (D) E-. (D) E-. (D) E-. (D) E-. E-6. E-7. E-8. E-9. (D) E-0. (D) PART # II..... 6. 7. (D) 8. 9. 0.. (D). (D). A-q B-rs C-rs D-. A-q, B- C-s D-r. (A-q) (B-r) (C-s) (D-) 6. (A-r) (B-s) (C -) (D-q) 7. (A-s) (B-) (C-q) (D-r) 8. (D) 9. 0... (D). A-79 Indr Vihr, Kot Rjsthn 00 Pge No. #
EXERCISE # PART # I. (D). (D).. (D). 6. 7. 8. 9. 0. (D)..... 6. 7. 8. 9. 0...... 6. 7. 8. 9. (D) 0. *. (A,C) *. (B, C) *. (C, D) *. (B, D) *. (B,C) 6*. (A, B) 7*. (A, B, D) 8*. (B,D) 9*. (A,C) 0*...... 6. 7. 8. 9.* (A, B, C) 0.* (B, D).* (A, B, C, D).* (A, D).* (A, C).* (A, C) PART # II 0. (i) y mx = ; y min = (ii) y mx = ; ymin = (iii) y mx = 0; y min = 7. (i) (ii) (iii) (iv) (v) 8. n = 7. - - b n n 7. () n + (-),nî I (b) n+ +,nî I (c) n -,nî I (d) n + (-),nî I (e) n ±,nîi m 8.,m Î I or (m + ) æ ö,m ÎI 9. n ±,nîi 0. ç n +,nîi orn -,nîi 0 è ø æ ö. ç n +,nîi è ø9 æ ö. ç n +,n ÎI è ø. n +, n Î I æ ö. (n + ),nîi. ç n +,nîi 6 è ø 6. n æ ö,n ÎI orç n ±,n ÎI è ø n 7. n, n Î I or +,nî I 8. 0º, º, 90º, º, 0º 6 9. x = (n + ),nî I or x = (n +) nîiorx = n ±,nîi 6 A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 6
0. m, m Î I or m,mîi n- æ ö or ç m +,mîi è øn. (i) (n + ) 6, (m + ), k (ii) (n + ) 0, (m + ). A = n + B. x = n, n ± tn. q = n, n. x = n +, n 6. q = n + 6 7. x = n n or x = + 7, n Î I 7 8 8 8.,,,,, 7 8 8 8 8 60. ;,, + where tn = 6. x = /6 6. x = n + or n 7 + ; n Î I 6. 0 solutions 6. x = n + 6. 0º, º, 90º, º, 0º EXERCISE # PART # I é ù..... 6. ê-, - ú ë 0 û È é ù ê, ú ë0 û 7. 8. (D) 9. 0..* (A, B)... (D) PART # II. (). (). (). (). () 6. () 7. () 8. () 9. () 0. (). (). ().. (). 6. 7. 8. (D) 9. 0.. (D).. () EXERCISE #. 6. 6. cosx cosx 7 q= n+ 7. q= n± 8. 8. +. q = + - - n n ( ) q=, 9. x =,,. 6. æ - ö + 7ç è ø 6. 7. n± 8. n + 8 9. q = n± + A-79 Indr Vihr, Kot Rjsthn 00 Pge No. # 7