ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirThe inverse of a function B A f : exists if f is one-one onto i.e., a bijection and is given by x y f y x f = =) ( ) (1.Consider the sine function with domain R and range [1, 1]. Clearly this function is not a bijection and so it is not invertible.If we restrict the domain of it in such a way that it becomes oneone, then it would become invertible. If we consider sine as afunction with domain((2,2t tand co-domain [1, 1], then it is a bijection and therefore, invertible. The inverse of sinefunction is defined as x x = =u u sin sin1, where(( 2,2t tu and ] 1 , 1 [ x .5.1 Properties of Inverse Trigonometric Functions.(1) Meaning of inverse function(i) x = u sin u =x1sin (ii) x = u cos u =x1cos (iii) x = u tan u =x1tan(iv) x = u cot u =x1cot (v) x = u sec u =x1sec (vi) x = u cosec u =x1cosec(2) Domain and range of inverse functions(i) If , sin x y = then , sin1x y= under certain condition.; 1 sin 1 s s y but x y = sin . 1 1 s s xAgain,21 sint = = y y and21 sint= = y y .Keeping in mind numerically smallest angles or real numbers.2 2t ts s yThese restrictions on the values of x and y provide us with the domain and range for the function x y1sin= .i.e., Domain : ] 1 , 1 [ e xRange:(( e2,2t ty(ii) Let x y = cos , then x y1cos= , under certain conditions 1 cos 1 s s y1 1 s s xt = = y y 1 cos0 1 cos = = y yt s s y 0 {as cos x is a decreasing function in [ t , 0 ];hence 0 cos cos cos s s y tThese restrictions on the values of x and y provide us the domain and range for the function x y1cos= .i.e. Domain: ] 1 , 1 [ e xRange : ] , 0 [ t e y(iii) If x y = tan , then , tan1x y= under certain conditions.Yy = cos1xO(1, t/2)X(1, 0)yOxy = tan1xy = t/2y =t/2Y(1, t/2)y = sin1xO(1, t/2)X1ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirInverse Trigonometrical FunctionsHere, R x R y e e tan ,2 2tant t< < < < y yThus, Domain R x e ;Range |.|\| e2,2t ty(iv) If , cot x y = then x y1cot=under certain conditions, ; cot R x R y e et < < < < y y 0 cotThese conditions on x and y make the function, x y = cotnd onto so that the inverse function exists. i.e.,x y1cot= is meaningful. Domain : R x eRange : ) , 0 ( t e y(v) If , sec x y = then , sec1x y= where 1 | | > x and2, 0tt = s s y yHere, Domain: ) 1 , 1 ( eR xRange:)` e2] , 0 [tt y(vi) If x y = cosec , then x y1cosec=one-one aWhere 1 | |2 2= s s y yt t> x and 0 ,Here, Domain ) 1 , 1 ( eRRange } 0 {2,2(( et tFunctionDomain (D) Range (R)x1sin1 1 s s 2 2tuts s or((x or ] 1 , 1 [2,2t tx1cos1 1x1tans s < < 2 2tutx or ] 1 , 1 [t u s s 0 or ] , 0 [x i.e., R x e or ) , ( t< < or |.|\|2,2t tx1cot < < x i.e., R x e or ) , (xy = sec1xy = t/2(1,t)O (1,0)xy = cosec1x(1, t/2)yO (1, t)y = tOx(0, t/2)y = cot1x t u < < 0 or ) , 0 (tANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirx1sec1 , 1 > s x x ort utu s s = 0 ,2or((\) , 1 [ ] 1 , ( ||.|tt t,2 2, 0x1cosec1 , 1 > s x x or ) , 1 [ ] 1 , (2 2, 0tutu s s = or((\||.|2, 0 0 ,2t t(3) u u =) (sin sin1, Provided that2 2tuts s ,u u =) (cos cos1, Provided that t u s s 0u u =) (tan tan1 , Provided that2 2tut< < ,u u =) (cot cot1, Provided that t u < < 0u u =) (sec sec1, Provided that20tu < s or t uts y xnd 1 < xy+ t n t n1 1If 0 , 0> >y xnd 1 > xyxANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www. nur gty gicl sses.comwww.f cebook.com/ nurgty gisirInverse Trigonometric l Functions(3)||.|\|++ = xyy xy x1t n t1 1t ;(4)||.|\|+= xyy xy x1t n t1 1;(5)||.|+n t n1If 0 , 0< xy\|++ = xyy xnd 1 > xyy xy1t n t n t n1 1 1t ;If 0 , 0(6)||.|\|++ xyy xy x1t n1t ;(7)((< >y xnd 1 < xy= t n t n1 1If 0 , 0> + y x(12) }, 1 1 { sin sin sin2 2 1 1 1x y y x y x + = + t If x < 0 , 1 s ynd 12 2> + y x(13) }, 1 1 { sin sin sin2 2 1 1 1x y y x y x + = +1 t If 0 ; 1 < s y xnd 12 2> + y x(14) }, 1 1 { sin sin sin2 2 1 1 1x y y x y x = If 1 ; 1 s s y x nd 12 2s + y x if or 0 > xynd 12 2> + y x .(15) }, 1 1 { sin sin sin2 2 1 1 1x y y x y x = t If 0 1 , 1 0 < s s < y xnd 12 2> + y x .(16) }, 1 1 { sin sin sin2 2 1 1 1x y y x y x = t If 1 0 , 0 1 s < < s y x nd 12 2> + y x .(17) } 1 . 1 { cos cos cos2 2 1 1 1y x xy y x = + , If 1 , 1 s s y xnd 0 > + y x .(18) } 1 1 { cos 2 cos cos2 2 1 1 1y x xy y x = + t , If 1 , 1 s s y x nd 0 s + y xANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www. nur gty gicl sses.comwww.f cebook.com/ nurgty gisir(19) }, 1 1 { cos cos cos2 2 1 1 1y x xy y x + = If , 1 , 1 s s y xnd y x s .(20) }, 1 1 { cos cos cos2 2 1 1 1y x xy y x = If , 0 1 s s y 1 0 s < xnd y x > .Imort nt TisIf ,2t n t n t n1 1 1t= + + z y x then 1 = + +If , t n t n t n1 1 1t = + + z y x then xyz z yIf ,2sin sin sin1 1 1t= + + z y x then 1 22 2 2= + + + xyz z y x .If , sin sin sin1 1 1t = + + z y x then xyz z z y2 2 2= + + .If , 3 cos cos cos1 1 1t = + + z y x then 3 = + +If , cos cos cos1 1 1t = + + z y x then 1 22 2 2= + + + xyz z y x .If ,23sin sin sin1 1 1t= + + z y x then 3 = + +If , sin sin1 1u = + y x then u t = + y x1 1cos cos .If , cos cos1 1u = + y x then u t = + zx yz xy .x= + + .y x x2 1 1 1zx yz xy .zx yz xy .y x1 1sin sin .If ,2t n t n1 1t= + y x then 1 = xy .If ,2cot cot1 1t= + y x then 1 = xy .If , cos cos1 1u = + byxthen u u22222sin cos2= + bybxyx.Ex mle: 15 The v lue of(((||.|\|+ |.|\| 133cos53sin t n1 1is( )176(b)136(c)513(d)617Solution: (d)(((||.|\|+ |.|\| 133cos53sin t n1 1[AMU 2001]= |.|\|+ 32t n43t n t n1 1=||||.|\|+32.4313243t n t n1=((6121217t n t n1=617.Ex mle: 16 = + 31t n21t n1 1[MP PET 1997, 2003; UPSEAT 2003; K rnt k CET 2001]( ) 0 (b)4t(c)2t(d) tANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www. nur gty gicl sses.comwww.f cebook.com/ nurgty gisirInverse Trigonometric l FunctionsSolution: (b)41 t n31.2113121t n31t n21t n1 1 1 1t= =+= + .Ex mle: 17 If , sin32sin31sin1 1 1x = + then x is equ l to( ) 0 (b)92 4 5 (c)92 4 5 +(d)2tSolution: (c)(((+=((( + = + 92 4 5sin9113294131sin32sin31sin[Roorkee 1995]1 1 1 1Therefore,92 4 5 += x .Ex mle: 18 3 cot51sin1 1 + is equ l toK rn t k CET 1995]( )6t(b)4t(c)3t(d)2tSolution: (b) 3 cot51511cot 3 cot51sin1 1 1 1 +|||||.|\|= +4) 12 31 3cot1=( cot2) 3 ( cot ) 2 ( cot1 1 1[MP PET 1993;t= = |.|\|+ = + .Ex mle: 19 If , sin1312cos53sin1 1 1C = |.|\|+ then = C( )5665(b)6524(c)6516(d)6556Solution: (d) Given,1312cos53sin sin1 1 1 + = C135sin53sin sin1 1 1+ = C [Pb. CET 1999])` + =259113516925153sin165566556sin1= |.|\|=C .Ex mle: 20 If , 3 3212cos cos ) (2 1 1)` + + = xxx x f then( )3 32 t= |.|\|f (b)3 32cos 2321t = |.|\|f (c)3 31 t= |.|\|f (d)3 31cos 2311t = |.|\|fSoltuion: ( ,d))` + + = 2 1 11 .2321cos cos ) ( x x x x f= ) cos21(cos cos1 1 1x x , ccording s >21cos1or x1cos which holds for32= x=21cos cos 21 1 x if , cos21cos1 1x < which holds for31= x .ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADwww.f cebook.com/ nurCALL US @ 09818777-622www. nur gty gicl sses.comgty gisirEx mle: 21 = + + 1663t n54cos1312sin1 1 1( ) 0 (b)2t(c) t (d)23tSolution: (c)1663t n43t n512t n1 1 1 + + =1663t n36 2015 48t n1 1 +++ t ( 1 > xy ) = t t 1663t n1663t n1 1.Ex mle: 22 If31sin54sin1 1 + = ond= + 31cos54cos1 1 + = | , then( ) | o < (b) | o =Solution: ( )((( + =251613191154sin1o =||.|\|+=(((+ 153 2 8sin153152 8(c) | o >(d) None of thesesin1 1Since2, 1153 2 8 to < = |.|\| + = | o < .Ex mle: 23 If ,3cos2cos1 1u = + y xthen = + 2 24 cos 12 9 y xy x u( ) u2sin 36 (b) u2cos 36 (c) u2t n 36 (d) None of theseSolution: ( ) u = + 3cos2cos1 1y x u cos91413.22 2=||.|\|||.|\| y x y x) 9 )( 4 ( ) cos 6 (2 2 2y x xy = u u u u2 2 2 2sin 36 ) cos 1 ( 36 4 cos 12 9 = = + Ex mle: 24 The number of solutions of32 sin sin1 1t= + x x is( ) 0 (b) 1 (c) 2 (d) InfinitieSolution: (b) x x1 1 1sin23sin 2 sin =(((y xy x . =431 1 .23sin2 1x x212322xx x =) 1 (432522xx = |.|\|or 3 282= x 7321283= = x , (not7321 ) .Ex mle: 25 If o = |.|\|+ |.|\| byx1 1cos cos , then = + 2222cos2bybxyxo( ) o2sin (b) o2cos (c) o2t n (d) o2cot[UPSEAT 1999; MP PET 1995]ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www. nur gty gicl sses.comwww.f cebook.com/ nurgty gisirInverse Trigonometric l FunctionsSolution: ( ) We h ve o =(((||.|\|||.|\| 222211 1 . cosbyxbyxo cos 1 12222=||.|\|||.|\| byxbxy2 22 2222221 cosby xbyxbxy+ = |.|\| o2 22 2222222 22 21 cos2cosby xbyxbxyby x+ 22222sin2= bybxy = +2o oo o ocos 1 cos= + x.Ex mle: 26 If , b, c be ositive re l numbers nd the v lue ofcc b bbcc b) (t n) (t n1 1+ +++ += ubc b c )t n1+ ++, thenu t n is( ) 0 (b)Solution: (bc b ccc b bbcc b)t n) (t n) (t n1 1 1+ +++ +++ += uLetbcc bs+ +=2(1 (c) c b)[IIT 1981]+ + (d) None of these(2 2 1 2 2 1 2 2 1t n t n t n s c s b s + + = u ) ( t n ) ( t n ) ( t n1 1 1cs bs s + + = u((( + +=2 2 2311t nc s bcs bsbcs cs bs su 0) ( 1) (t n22=(((+ + + +=s c bc bbcs c bs u )] ( [2c bbcs + + =Trick : Since it is n identity so it will be true for ny v lue of1 = = = c bthen, 3 t n 3 t n 3 t n1 1 1t u = + + = 0 t n = u .Ex mle: 27 All ossible v lues of nd q for which431 cos 1 cos cos1 1 1t= + + q holds, is[K rn t k CET 2002]( )21,b,c. Let, 1 = = q (b)21, 1 = > q (c)21, 1 0 = s s q (d) None of theseSolution: (c) q = + 1 cos431 cos cos1 1 1t q ||.|\|= + 1 cos21cos 1 cos cos1 1 1 1(((21121121= = q q ..1q q =1 0q .5.3 Inverse Trigonometric R tios of Multile Angles.(1) ) 1 2 ( sin sin 22 1 1x x x = , If2121s s x (2) ), 1 2 ( sin sin 22 1 1x x x = t If 121s s x(3) ) 1 2 ( sin sin 22 1 1x x x = t , If211s s x (4) ), 4 3 ( sin sin 33 1 1x x x = If2121s sxANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADwww.f cebook.com/ nurCALL US @ 09818777-622www. nur gty gicl sses.comgty gisir(5) ) 4 3 ( sin sin 33 1 1x x x = t , If 121s < x (6) ), 4 3 ( sin sin 33 1 1x x x = t If211 < s x(7) ) 1 2 ( cos cos 22 1 1 = x x , If 1 0 s s x (8) ) 1 2 ( cos 2 cos 22 1 1 = x x t , if 0 1 s s x(9) ) 3 4 ( cos cos 33 1 1x x x = If 121s s x(10) ) 3 4 ( cos 2 cos 33 1 1x x x = t , If2121s s x(11) ) 3 4 ( cos 2 cos 33 1 1x x x + = t , If211 s s x(12) |.|\|= 21 112t n t n 2xxx , if 1 1(13) |.|s < \|+ = 21 112t n t n 2xxx t , If 1 > x.x(14)||\|+ = 21 112t n t n 2xxx t , If 1 < x(15) |.|\|+= 21 112sin t n 2xxx , If 1 1.|s s \|+ = 21 112sin t n 2xxx t , If 1 > x(17) |.|\|+ = 21 11x(16)|2sin t n 2xxx t , If 1 < x||.|(18)\|+= 221 111cos t n 2xxx , If < s x 0(19)||.|\|+ = 221 111cos tan 2xxx , If 0 s < ||.|\|= x(20)231 13 13tan tan 3xx xx , If3131< x(22)||.|\|+ = 231 13 13tan tan 3xx xx t , If31 < x(23)axx ax12 21sin tan =((((24)axx a ax x a12 23 21tan 3) 3 (3tan =(((25)2 12 22 21cos2141 11 1tan xx xx x + =((( + + + t(26) xxx1 1cos2111tan =+Example: 28 ), cosec ( tan ) (cos tan 22 1 1x x = then x =[UPSEAT 2002](a)2t(b) t (c)6t(d)3tSolution: (d) ) (cos tan 21x) ec (cos tan2 1x=x xxx xx2 2121sin1sincossin1tancoscostan.|2212= |\|= |.|\| 31 cos 2t= = x x .Example: 29 The solution set of the equation x x1 1tan 2 sin = is[AMU 2002](a) {1, 2} (b) } 2 , 1 { (c) } 0 , 1 , 1 {21, 1 {(d) } 0 ,ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirInverse Trigonometrical FunctionsSoltuion: (c) x x1 1tan 2 sin = 21 112sin sinxxx+= xxx=+212 03= x x } 0 , 1 , 1 { 0 ) 1 )( 1 (Exmaple: 30)`||.|\|++||.|\| 2212111cos21tan sin = = +x x x x .xxxxis equal to(a) 0 (b) 1 (c) 221Solution: (b))`+ |.|\| x x1 1tan 2 tan 22sint=2sint= 1.Example: 31 If ,12tan11cos12sin2122121xxbbaa=[Kurukshetra CEE 2001](d)++ then = x[EAMCET 1989](a) a (b) b (c)abb a+1(d)abb a+1Solution: (d) Put | u tan , tan = = b a and , tan = x then reduced form is) 2 (tan tan ) 2 (cos cos ) 2 (sin sin1 1 1 | u = | u | u = = 2 2 2Taking tan on both sides, we get ) tan( | u = tan | u| utantan . tan 1tan tan=+Substituting these values, we get xabb a=+1Example: 32 =(( |.|\|4 51tan 2 tan1t[IIT 1984](a)717(b)717 (c)177(d)177Solution: (d)(((((=(( |.|\| ) 1 ( tan251152tan tan4 51tan 2 tan1 1 1t=17712511125tan . tan ) 1 ( tan125tan tan1 1 1=||||.|\|+=(( .Example: 33 = + 991tan701tan51tan 41 1 1[Roorkee 1981](a)2t(b)3t(c)4t(d) None of theseSolution: (c)991tan701tan51tan 41 1 1 + =991tan701tan251152tan 21 1 1 + (((((=991tan701tan125tan 21 1 1+ |.|\|=99 1tan701tan14425165tan1 1 1+ ((((( ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisir=991tan701tan119120tan1 1 1 + |.|\|=(((((++ 701.9911701991tan119120tan1 1= |.|\| + 693129tan119120tan1 1=2391tan119120tan693129tan119120tan1 1 1 1 = =4) 1 ( tan23911191201239 1119120tan1 1t= =((((( + .Example: 34 The value of = +||.|\||.|\|) 231tan12 (tan cos2 sin1[AMU 1999](a)1516(b)1514(c)1512(d)1511Solution: (b) )] 2 2 ( [tan cos31tan 2 sin1 1 +((|.|\|= )] 2 2 ( [tan cos91132tan sin1 1 +(((((] 2 2 cos[tan43tan sin1 1 +((= =1514315331cos53sin1= +((cossin1=+(( .Example: 35(( +((+ baba1 1cos214tan cos214tant tequal to(a)ba 2(b)ab 2(c)[MP PET 1999]ba(d)abSolution: (b) Letbaba= =u u cos cos1(( +((+ baba1 1cos214tan cos214tant t=tttt+++1111, where2tanu= tabtt 2cos21) 1 (222= =+=u.Example: 362391tan51tan 41 1 is equal to(a) t (b)2t(c)3t(d)4tSolution: (d) Since,21 112tan tan 2xxx= 25115[UPSEAT 1995]2tan51tan51tan1=((22 241 1= 57610012420tan2410tan 21 1= = 119120tan1 =So,2391tan119120tan2391tan51tan 41 1 1 1 = 2391.11912012391 119120tan1+=120 ) 239 119 (119 ) 239 120 (tan1+ =41 tan1t= =.Example: 37 The formula xxx1221tan 211cos =+holds only for(a) R x e (b) 1 | | s x (c) ] 1 , 1 ( e xSolution: (d) If , 1 = x LHS = ,2tRHS = |.|(d) ) , 1 [ + e x\| 22t. So, the fomula does not hold.ANURAG TYAGI CLASSES (ATC)10C-82,VASUNDHRA, GHAZIABAD.BRANCH: SAHIBABADCALL US @ 09818777-622gtyagisirwww.anuragtyagiclasses.comwww.facebook.com/anuraInverse Trigonometrical FunctionsIf , 1 < x the angle on the LHS is in the second quadrant while the angle on the RHS is 2 (angle in the fourth quadrant), whichcannot be equal.If , 1 > x the angle on the LHS is in the second quadrant while the angle on the RHS is 2 (angle in the first quadrant) and these twomay be equal.If 0 1 < < x , the angle on the LHS is positive and that on the RHS is negative and the two cannot be equal.Example: 3821 112sin tan 2xxx++ is independent of x , then(a) ) , 1 [ + e x (b) ] 1 , 1 [ e x (c) 1 , ( e x ] (d) None of theseSolution: (a) Let u tan = x . Then ) 2 (sin sintan 1tan 2sin12sin12121uuu =+=+ xx)12sin121 1u u++2 (sin sin 2tan 2 =+xxxIf21 112sin tan 2 ,222 xxx++ s s tut= = = +x1tan 4 2 2 u u independent of x.If21 112sin tan 2 ,222 xxx++ s s tu tt= u t u u t u 2 2 )] 2 [sin( sin 21 + = += t = independent of x.(( e4,4t tu but((e43,4t tu and from the principal value of x1tan.|.|\| e2,2t tu . Hence, |.|\|e2,4t tu|.|\|e2,4t tu t =++ 21 112sin tanxxx .Also at tt t t tu = + = + =++ = 2 21 sin4. 212sin tan 2 ,4121 1xxx .The given function = t = constant if |.|e2,4t tu . i.e., ) , 1 [ + e x .Example: 39 The number of positive integral solutions of the equation103sin1cos tan121 1 =++yyx or 3 tan cot tan1 1 1 = + y xis[Roorkee 1993](a) One (b) Two (c) Zero (d) None of theseSolution: (b) 3 tan1tan tan1 1 1 = +yx ory1tan1 = x1 1tan 3 tan orxxy 3 13tan1tan1 1+= xxy+=33 1As , x y are positive integers, 1 = x , 2 and corresponding , 2 = y 7Solutions are ) 7 , 2 ( ), 2 , 1 ( ) , ( = y x .ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisir1. The domain of x1sinis[Roorkee Screening 1993](a) ) , ( t t (b) [ 1, 1] (c) ) 2 , 0 (2. The range of xt(d) ) , ( 1tanis(a) |.|\|2,tt.|[DCE 2002](b) |\|2,2t t(c) ) , ( t t (d) ) , 0 ( t3. x x1 1cos sin + is equal to[Pb. CET 1997; DCE 2002](a)4t(b)2t(c) 1 (d) 14. =)`+ 21cos21sin sin1 1[EAMCET 1985](a) 0 (b) 1 (c) 2 (d) 15. The value of |.|\|+ |.|\| 35cos sin35cos cos1 1t tis(a)2t(b)35t(c)310t(d) 06. =(([UPSEAT 2003]|.|\| + |.|\| 71sin71cos cos1 1[EAMCET 2003](a)31 (b) 0 (c)31(d)947. The value of x x1 1cot 2 tan + is[Kurukshetra CEE 1998](a)3t(b)6t(c)32t(d) t 28. If ,32cot 2 tan1 1t= + x x then = x[Karnataka CET 1999](a) 2 (b) 3 (c) 3 (d)1 31 3+9. If , cos sin 41 1t = + x x then x is equal to(a) 0 (b)21(c)23 (d)2110. =((+ 5[UPSEAT 2001]1sin51cos 2 cos1 1[IIT 1981](a)56 2(b)56 2 (c)51(d)51B Ba as si ic c L Le ev ve el lProperties of Inverse Trigonometrical Function141ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirInverse Trigonometrical Functions11. The value of |.|\|533cos sin1tis(a)53t(b)57t(c)10t(d)10t12. If , ) 1 ( sin 21sec1 1t = + |.|\| xthen x equals(a)21(b) 1 (c)2t(d) None of these13. The value of |.|[AMU 1988]\|||.|\| 21sin23sin1 1is[MP PET 2003](a)o45 (b)o90 (c)o15 (d)o3014. The value of )) 2 (tan cos(tan1 is[AMU 2002](a)51(b)51 (c) 2 cos (d) 2 cos 15. The value of x which satisfies the equation||.|\|= 103sin tan1 1x is[Pb. CET 1999](a) 3 (b) 3 (c)31(d)3116. If , cosec sec1 1y x =then = + y x1cos1cos1 1[Orissa JEE 2002](a) t (b)4t(c)2t (d)2t17. If ,1cos1u = |.|\|xthen = u tan[MNR 1978; MP PET 1989](a)112 x(b) 12+ x (c)21 x (d) 12 x18. If5sin1t=x for some ), 1 , 1 ( e x then the value of x1cosis[DCE 1997; Karnataka CET 1996; IIT 1992](a)103t(b)105t(c)107t(d)109t19. ) cosec ( sec1xis equal to(a) ) (sec cosec1x(b) x cot (c) t20. ) (cos tan1xis equal to(a)xx21 (b)21 x[Kurukshetra CEE 2001](d) None of these[IIT 1993]x+(c)xx21 +(d)21 x ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisir21. =) sin(cot1x[MNR 1987; MP PET 2001; DCE 2002](a)21 x + (b) x (c)2 / 3 2) 1 (+ x (d)212) 1 (+ x22. =) cos(tan1x[MP PET 1988; MNR 1981](a)21 x + (b)211x +(c)21 x + (d) None of these23. =((|.|\|257cos cot1[Karnataka CET 1994](a)2425(b)725(c)2524(d) None of these24. The value of x1 1cos tan cot sin is equal to[Bihar CEE 1974](a) x (b)2t(c) 1 (d) None of these25. =((|.|\|2143tan sin(a)53(b)35(c)259(d)92526. = +[EAMCET 1983]) 31c(a)(cot osec ) 2 (tan sec2 1 2[EAMCET 2001]5 (b) 13 (c) 15 (d) 627. If , sin cos1 1x x > then(a) 0 < x (b) 0 1 < < x (c)210 < s x (d)211 < s x28. If , 0 ) (sin ) (cos2 1 2 1> x x then(a)21< x (b) 2 1 < < x (c)210 < s x (d)211 < s x29. The greatest and the least values of3 1 3 1) (cos ) (sin x x + are(a)2,2t t (b)8,83 3t t (c)87,323 3t t(d) None of these30. If x satisfies the equation , 0 22> t t then there exists a value for(a) x1sin(b) x1cos(c) x1sec(d) None of these31. If , tan sec ) (1 1x x x f + = then ) (x f is real forA Ad dv va an nc ce e L Le ev ve el lANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirInverse Trigonometrical Functions(a) ] 1 , 1 [ e x (b) R x e (c) ) , 1 [ ] 1 , (se32. If==nrrn x211, sin t then==nrrx21(a) n (b) 2n (c)2) 1 ( + n n(d) None of these + e x(d) None of the33.52t is the principal value of(a) )57(cos cos1t(b) )57(sin sin1t(c) )57(sec sec1t(d) None of these34. The number of real solutions of ) , ( y x ; where t t, sin | |1s s = =x x y x y is(a) 2 (b) 1 (c) 3 (d) 435. The set of values of k for which 0 ) 4 (sin sin1 2> + kx x for all real x is(a) | (b) ) 2 , 2 ( (c) R (d) None of these36. xx xx x1 122 2 1cos2cos41 . 121cos =)`2 2 ), (cos cos + holds for(a) 1 | | s x37.1t =z y(a)38..|(b) R x eIf , 3 cos cos cos1+then = + + zx yz xy0 (b) 1 (c) 3 (d) 3The value of |1+x\| 31tan21tan tan1 1is(a)65(b)67(c)61(d)7139. = |.|\|+ |.|\| 12(c) 1 0[AMU 2001]s s x(d) 0 1s s [Karnataka CET 2003]x2tan111tan1 1[DCE 1999](a) |.|\|13233tan1(b) |.|\|21tan1(c) |.|\|33132tan1(d) None of these40. If , sin sin sin1 1 1t = + + c b a then the value of ) 1 ( ) 1 ( ) 1 (2 2 2c c b b a a + + will be[UPSEAT 1999](a) abc 2 (b) abc (c) abc21(d) abc3141.1A y= If , tan tan tan1 1x then = A[MP PET 1988]B Ba as si ic c L Le ev ve el lSum and Difference of Inverse Trigonometrical FunctionANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisir(a) y x (b) y x + (c)xyy x+1(d)xyy x+142. If43 tan 2 tan1 1t= + x x then x =[Roorkee 1978, 1980; MNR 1986; Karnataka CET 2002](a) 1 (b)61(c)61, 1 (d) None of these43. If , cos54sin53cos1 1 1x = then = x[AMU 1978](a) 0 (b) 1 (c) 1 (d) 244. If , cot cot cot1 1 1x = + | o then = x[MP PET 1992](a) | o + (b) | o (c)| oo|++ 1(d)| oo|+145. = + 198tan53tan43tan1 1 1[AMU 1976, 1977](a)4t(b)3t(c)6t(d) None of these46. =((+ 21tan31tan cos1 1[MP PET 1991; MNR 1990](a)21(b)23(c)21(d)4t47. =+ y xy xyx1 1tan tan(where 0 > > y x )(a)4t (b)4t(c)43t(d) None of these48.(([EAMCET 1992]+ 32tan54cos tan1 1=(a)176(b)617(c)167(d)71649. |.|\|+ |.|\|[IIT 1983; EAMCET 1988; MP PET 1990; MNR 1992] 92tan41tan1 1=(a) |.|[EAMCET 1994]\|53cos211(b) |.|\|53sin211(c) |.|\|21tan1(d) Both (a) and (c)50. 3 cot51sin1 1 + is equal to; MP PET 1993](a)6t(b)4t[Karnataka CET 1995(c)3t(d)2t51. If 1 cos51sin sin1 1= |.|\|+ x , then x =[UPSEAT 1994]ANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirInverse Trigonometrical Functions(a) 1 (b) 0 (c)54(d)5152.2) 11t= x x(a)53.3623tan1 21 2tan11tan1=+A solution of the equation( tan ) 1 ( tan1+ +is[Karnataka CET 1993]x = 1 (b) x = 1 (c) x = 0 (d) t = xIf ,1 1 ++xxxxthen x =[ISM Dhanbad 1973](a)83,43 (b)83,43(c)83,34(d) None of these54. If ,2 45sec5sin1 1t= |.|\|+ coxthen = x[EAMCET 1983](a) 4 (b) 5 (c) 1 (d 355. = |.|\|+ |.|\| 71tan53sin1 1[Karnataka CET 1994](a)4t(b)2t(c) |.|\|54cos1(d) t56. If 3 tan , 2 tan1 1 are two angles of a triangle, then the third angle is(a)4t(b)43t(c)4t(d) None of these57. If = + + z y x1 1 1cos cos cos t then[Roorkee 1994](a) 02 2 2= + + + xyz z y x (b) xyz z y x 22 2 2+ + + =0 (c) 12 2 2= + + + xyz z y x (d) 1 22 2 2= + + + xyz z y x58. If ,2tan tan tan1 1 1t= + + z y x then(a) 0 = + +xy (d) 0 159. If , tan1 1 1t = + + z y x thenzx yz xy1 1 1+ + =(a) 0 (b) 1xyz1(d) xyz[Karnataka CET 1996]xyz z y x (b) 0 = + + += + + zx yz xytan tanxyz z y x(c) 0 1 = + + +zx yz[MP PET 1991](c)60. If we consider only the principal values of the inverse trigonometric functions, then the value of||.|\| ) 17 (4sin2 51cos tan1 1is[IIT 1994](a)329(b)329(c)293(d)293A Ad dv va an nc ce e L Le ev ve el lANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisir61. The sum of first 10 terms of the series ....... 21 cot 13 cot 7 cot 3 cot1 1 1 1+ + + + is[Karnataka CET 1996](a) |.|\|65tan1(b) ) 100 ( tan1 (c) |.|\|56tan1(d) |.|\|1001tan162. Sum of infinite terms of the series +((+ +((+ +((+ 433 cot432 cot431 cot2 1 2 1 2 1.is(a)4t(b) 2 tan1 (c) 3 tan1 (d) None of these63. .......11tan .........181tan71tan31tan21 1 1 1+ |.|\|+ ++ + + + n nto is equal[Karnataka CET 2000](a)2t(b)4t(c)32t(d) 064. If , sin sin sin1 1 1t = + +z y2x z(a)65.||.|x2zthen ),2 2 2 2 2y y x K z1 (b) 2The sum of( 42 2y x(c)the\|+ + +||.|\|+||.|\|+ ) 1 (1sin .... ..........122 3sin61 2sin21sin1 1 1 1n nn nis(a)4t(b)3t4 4 4z y x + + = + + + where K =4 (d) None of theseinfinite series[UPSEAT 1997](c)2t(d) t66. If sum of the infinite series ..... ) 2 . 2 cot( ) 2 . 2 ( cot ) 2 . 2 ( cot ) 1 . 2 ( cot4 3 1 2 1 2 1+ + + + is equal to(a)5t(b)4t(c)3t(d)2t67.23sin1t= +z y1011009z yz y+ + +(a)68.4 3,2= If ,sin sin1 1+ x then the value of101 101100 100xx+ is equal to0 (b) 3 (c) 3 (d) 9If2 1, , x x x x are roots of the equation , 0 sin cos 2 cos 2 sin3 4 + | | | | x x x xthen = + + + 41312111tan tan tan tan(a) | (b) |t2x x x x(c) | t (d) | 69. Ifna a a a ........, , , ,3 2 1is an A.P. with common difference d, then =(((||.|\|++ +||.|\|++||.|\|+ n na ada ada ad113 212 111tan ..........1tan1tan tan(a)na ad n+1) 1 ((b)na ad n11) 1 (+(c)na and11 +(d)11a aa ann+B Ba as si ic c L Le ev ve el lInverse Trigonometric Ratios of Multiple AnglesANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirInverse Trigonometrical Functions70. a1tan 3is equal to[MP PET 1993](a)2313 13tanaa a++(b)2313 13tanaa a+(c)2313 13tanaa a+(d)2313 13tanaa a71. If , tan1x A= then = A 2 sin(a)212xx(b)[MNR 1988; UPSEAT 2000]212xx(c)212xx+(d) None of these72. =) 8 . 0 sin 2 sin(1[MNR 1980](a) 0.96 (b) 0.48 (c) 0.64 (d) None of these73. If ,91) sin 2 cos(1=x then = x[Roorkee 1975] [MNR 1980](a) Only32(b) Only32 (c)32,32 (d) Neither32nor3274. = |.|\|+ |.|\| 51tan 21715cos1 1[EAMCET 1981](a)2t(b) |.|\|221171cos1(c)4t(d) None of these75. = |.|\|+ |.|\| 71tan31tan 21 1[EAMCET 1983](a) |.|\|2949tan1(b)2t(c) 0 (d)4t76.31tan54sin1= +21[Dhanbad Engg. 1971](a)2t(b)3t(c)4t(d) None of these77. If xbbaa12121tan 212sin12sin =+++, then x =[MNR 1984](a)abb a+1(b)abb+ 1(c)abb 1(d)abb a+178. If ,31tan1tan3 21tan 31 1 1 = ||.|\|+xthen x equals[Pb. CET 2001; AMU 1992](a) 1 (b) 2 (c) 3 (d) 379.(((+ + + x xx xsin 1 sin 1sin 1 sin 1cot1=[UPSEAT 1986](a) x t (b) x t 2 (c)2t(d)2x tANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisir80. = |.|\|54cos21sin1[Karnataka CET 2003](a)101(b)101 (c)101(d)10181. =(((+2tan tan 21ub ab a[ISM Dhanbad 1976](a) |.|\|++uucoscoscos1b ab a(b) |.|\|++b ab auucoscoscos1(c) |.|\|+uucoscoscos1b aa(d) |.|\|+b abuucoscoscos182.] )2o o(a).|If x = [(cos tan ] ) [(cos cot/ 1 1 2 / 1 1. Then = x sin|\|2tan2o(b) |.|\|2cot2o(c) o tan.|(d) |\|2coto83. The value of)` +)` +|[AIEEE 2002].|\| kx k xxkx k xx2 212 216cos cos3sin sint t, |.|\|> < < 0 , 22where k k xkis(a)||.|\|+ +2 22 2122tank xk xk xk x(b)||.|\|+ +2 22 2122tank xk xk xk x(c)||.|\|+ +2 22 212 2 22 2tank xk xk xk x(d) None of these84. Solution of equation 0 )}] tan 2 {cot( cos 2 sin[1 1= x is[UPSEAT 1998; Roorkee 1992](a) 1 = x only (b) 2 1 = x only (c) ) 2 1 (se85. The greater of the two angles ) 1 2 2 ( tan 21 =A and |.|\|+ |.|\|= 53sin3 = x only (d) All of the1sin 31 1B is[IIT 1992](a) B (b) A (c) C (d) None of these86. =(((||.|\|35cos21tan1[Roorkee 1986](a)25 3 (b)25 3 +(c)5 32(d)5 32+***A Ad dv va an nc ce e L Le ev ve el lANURAG TYAGI CLASSES (ATC) 10C-82,VASUNDHRA, GHAZIABAD. BRANCH: SAHIBABADCALL US @ 09818777-622www.anuragtyagiclasses.comwww.facebook.com/anuragtyagisirInverse Trigonometrical FunctionsANURAG TYAGI CLASSES (ATC)1216317b b b212235d b d414255c b b616275a b b8182a a c418d a2336a c4356d a6376b c83d b519b c2437c d4457a b6477b a84d620c b2538d c4558b d6578b b857b d2639c c4659b d6679d c868b d2740b b4760c d6780a c9101112131415c a d d a a a28293031323334c a c c d d a48495051525354d a b d d b d68697071727374d d a d b d a150Assignment (Basic & Advance Level) Inverse Trigonometrical Functions