Page 1 :
MATHEMATICS, , INVERSE TRIGONOMETRIC FUNCTIONS, Inverse function : A function f:AB will have an inverse if its both one-one and onto., Inverse sine function: The inverse sine function is a rule which assigns to each x in [-, ,    , , 1,1] a unique number y in , ,  such that x=siny and we can write y = Sin-1x.,  2 2, Thus y = Sin-1x if and only if x=siny, -1  x  1;, Example:, , (i)Sin-11 =, , Because Sin, , = 0., , y, , (ii), , Sin-1 =, , because Sin =, , (iii), Sin-13 does not exist because 3  [-1,1], Inverse Cosine functions:, Definition: The inverse cosine function is defined by : y = Cos-1x iffdomain [-1,1] and, the range is [0, ], Example: (i) Cos-1 . =, , Because Cos =, , ., , (ii)Cos-1 0 =, , because Cos = 0, , (iii), Cos-13/2 does not exist because 3/2  [-1,1], The inverse tangent function, To obtain the inverse tangent function we consider the restricted tangent function: y =, tanx ;, , < x < . As this function is bijective, its inverse exists and we define the inverse, , tangent function by y = tan-1x iff, , x = tany, x  R and, , <y< ., , The domain of tan-1x is the set of all real numbers and the range is, Example : Tan-1(3) =, , because tan, , ,, , ., , = 3, , Similarly Cosec-1x, Sec-1x and Cot-1x can be defined., Note :, (i), , Sin-1 (-x) = -Sin--1x., , (ii), , Cos-1(-x) =, , (iii), , Tan-1(-x) = - tan-1x, x>0., , (iv), , Cosec-1 (-x) = -Cosec--1x. for x ε R- (-1, 1) (x) Sin-1 x = Cosec-1, , (v), , Sec-1(-x) =, , (vi), , Cot-1(-x) = π- Cot-1x, x εR., , - Cos-1x for x>0., , - Sec-1x for x ε R- (-1, 1)., , (vii) Sin-1, , = Cosec-1x., , (viii) Cos-1, , = Sec-1x, , (ix) Tan-1, , = Cot-1x, ., , (xi) Cos-1 x= Sec-1, (xii) Tan-1 x = Cot-1, , Absolute Learning Academy 1

Page 2 :
MATHEMATICS, , , , Tan(tan-1x )= x, ,, , , , Sec(Sec-1x) = x, , , , Cosec(Cosec-1x) = x, , -< x <, , and, and, , Tan-1(tan x )= x, , <x< ., , Sec-1 (sec x) = x, and, , Cosec-1 (Cosec x) = x, , .,  Cot(Cot-1x) = x, -< x <, and, Cot-1(cot x )= x, To express one inverse trigonometric function in terms of other:, Sin-1x = Cos-1 1  x 2 = Sec-1, , x, 1 x2, 1, =Cosec-1 = tan-1, =Cot-1, x, x, 1 x2, 1 x2, 1, , Standard formulae:, -1  x 1., , (i), , Sin-1x + Cos-1x =, , (ii), , tan-1x + Cot-1x =, , (iii), , Cosec-1x + Sec-1x = , x  -1, x  1., , (iv), , tan-1x + tan-1y = tan-1, , (v), (vi), (vii), , xR, , x y, if x  0, y  0, xy < 1., 1  xy, x y, tan-1x + tan-1y = + tan-1, if x > 0, y > 0, xy > 1., 1  xy, x y, tan-1x - tan-1y = tan-1, if x  0, y  0., 1  xy, 2x, 2 tan-1x = tan-1, if 0  x 1., 1 x2, , Absolute Learning Academy 2

Page 3 :
MATHEMATICS, ,  x  y  z  xyz ,  . But these are valid for restricted, (viii) tan-1x + tan-1y + tan-1z = tan-1 ,  1  xy  yz  zx , values of x,y,z only., , , , , , (ix), , Sin-1x + Sin-1y = Cos-1 1  x 2 . 1  y 2  xy , x  0, y  0., , (x), , Cos-1x + Cos-1y = Cos-1 xy  1  x 2 . 1  y 2 , x  0, y  0., , (xi), Sin-1x + Cos-1y = Cos-1, Important results, , , , , y, , , ,, , 1  x 2  x. 1  y 2, , x  0, y  0., , , , (1), , Sin-1x - Sin-1y = Sin-1 x 1  y 2  y. 1  x 2 , x  0, y  0., , (2), , Cos-1x - Cos-1y = Sin-1 y 1  x 2  x. 1  y 2 , x  0, y  0., , (3), , Sin-1x - Cos-1y = Cos-1, , (4), , If tan-1x + tan-1y + tan-1z =, , (5), , If Cos-1x +Cos-1y + Cos-1z =, , (6), , 2 Sin-1x = Cos-1(1-2x2), x  0, , (7), , 2 Cos-1x = Cos-1(2x2-1), x  0 of course x  1., , (8), ,  2x , 2 tan-1x = tan-1 ,  0  x < 1., 2, 1  x , , tan-1x, , =, , , xy , , 1 x2 . 1 y2, , , ,, , x  0, y  0., , then x+y+z = xyz, then x2+y2+z2 +2xyz=1, , 1  x2, 2x ,  2x , -1, -1, ,  =Sin ,  = Cos , 2, 2, 2, 1  x , 1  x , 1  x, , , tan-1, , (9), , 2, , (10), , Sin-1x + Cos-1y =, , , , , , then x2+y2 =1., CLASS TEST:, , 1, 119, is equal to,  tan 1, 5, 120, b., a. , , 1. 4 tan 1, , 2. The value of sin(2 sin 1 (0.8)) is equal to, 0, b. sin 1.6 0, a. sin 1.2, , c., , d., , c. 0.48, , d. 0.96, , 3. The value of sin(cot 1 (cot(tan 1 x))) is, a., , x2  2, x2 1, , b., , x2 1, x2  2, , 4.The solution of tan-1x + 2cot-1 x =, a., , b., , c., , c., , x, x2 1, , d., , 1, x2  2, , is, , d., , 1, 1  1  3, 5. If 2 tan x  5 tan  x   2 then x=, , Absolute Learning Academy 3

Page 4 :
MATHEMATICS, a., , b., , 6.If, a. 0, , d. 2  1, , c. 1, , +, , +, , = 3 ,then, c. 3/4, , b. 3, , 7.If, , +, , =, , a. 1 b. 0, 8.The value of, a. b., c., 9.If, a. 1, , ,then x + y + xy is equal to, c. 2, +, d., , +, 0, , +, , is, d. 1, , =, , d. 3, is, , and x = y, then, c. 3, , b. 2, , is equal to, d. 4, , 3, 9, , the value of x100  y 100  z 100  101, is, 2, x  y 101  z 101, b. 1, c. 2, d. 3, , 10.If sin 1 x  sin 1 y  sin 1 z , a. 0, , 11.The value of tan2( sec, a. 15, b.11, 12.If, , - 2., , 2) + cot2(cosec –1 3) is, c. 13, , b. 1, and, , a., , d. 0, , + 1 =0, then the value of x is, , a. 0, 13.If, , –1, , c., , d., , are two angles of a triangle then the remaining angle is, b., , c. d., , 14.The value of, , is, , a .1, , b.0, , c., , 15., , d., , =, a. b.1, , 16.If sec, , -1x, , c., = cosec -1y ,then, , a. b. c. 17., , +, a. 10, , +, , =, , d., =, , ,then x =, b.12, , 18.If, a.3, , d., , b.0, , c. 13, = 3 then xy + yz +zx =, c. -3, , d. 14, , d. 1, Absolute Learning Academy 4

Page 5 :
MATHEMATICS, 19.If 3, ,  2x  , + 2 tan 1 , then x =, , 2 , 1 x  3, d. 1, , -4, , a. b. c.2, 20.If, , = x and, = y ,-1, a.x + y =2, b.x-y =2, , 21.If, , +, , =, , a.1, 6, , ,then x =, , b. 0,1, , , , , 22. The value of tan 2 tan, a., , k 1 then the correct relationship is, c.x+y =, d.x- y =, , 7, 17, , 1 1, , 5, , c. ,, , , , d. , 1, , , , 4 , 1, b., 17, , c.7, , 23. The number of solutions of the equation tan, a. 1, , 1, , d.-7, , x x  1  sin 1 x 2  x  1 , , b. 2, , c. 3, , , 1 1, 1 1 , 2, cos, , sin, =, , 5, 5 , 2 6, 1, a., b., 5, 5, , d. 4, , , 2, , 24. cos, , c., , 1, 5, , d., , 25. If cos ec 1 x  2 cot 1 7  cos 1 (3 5) then the value of x is, b. 125 117, c. 24 7, a. 44 117, , c) 2, , 2.If 4sin-1x + cos-1x =  , then x =, a) 2, b) 1, 2, , 2 sin, , 1 5, , 13, , , , =, , a) 51, , d. 5 3, , HOME TEST, , 1.The value of cos (2sin-1 1 3 ) is, a) 7, b)  7, 9, 9, , 3.tan, , 6, 5, , c) 1, b), , 78, , 4.If cos-1 p + cos-1 1  p + cos-1 1  q = 3  ,, 4, 1, a) 1, b), 2, , , , , , d) none of these, , 3, , d) 2, , 3, , 60, 69, , c), , 5, , 100, 119, , d), , 120, 119, , then q =, c), , 1, , 3, , d), , 1, , 2, ,  1 , 5.If cos-1x =  , (0<x<1) and sin-1 2x 1 - x 2  sec 1  2,  = 2  3 then tan-1(2x) =, 2, x, , 1, , , , Absolute Learning Academy 5

Page 6 :
MATHEMATICS, a) , 6.2, , tan-1, , b) , , 4, (-3) =, , a) - , , +tan-1 3, , 7.sin-1(cos (-105o)) =, these, , , , 1, 8.tan 1 2 tan, , 12, , 5, , , , c) , , 6, 4, , b), , tan-1 3, , 4, , d) , , 2, , 3, d)  +tan-1  4 3 , ,  3, , c) , , +tan-1 4, , a) 75o, , b) 15o, b)  3, , = a) 5 3, , c) 105o, , c) 2 3, , d) none of, , d) none of these, , 9.The sum of tan-1 1 2 + tan-1 18 + tan-1 118 +……………is, a), , , , b) , , 2, , c), , , , c), , 1, , 3, , d), , , , d), , 3, , 4, , 10.If cos-1x = cot-1 4 3 + tan-1 1 7 , then x =, a), , 1, , b), , 2, , 11.The value of tan-1, , 3/2, , 2, , 5, , 1, 1, 1, + tan-1, + tan-1, + ………upto  is, 2, 2, 111, 1 2  2, 1  3  32, , c) 3 2, , b)  4, , a)  2, , d) 0, , 12.tan-1 (-1) + tan-1 (-2) + tan-1 (-3) =, a) , , b), , , , c) - , , 2, , d) 0, ,  x  2,  x  2, 1 4, =tan-1 (0) ,, then x =,  - tan-1 ,  - tan, x, x, x, , , , , a)  2, b)  1, c) 2, d)  2, 14.cos-1 2, + tan-1 13 =, a)  2, b)  4, c) , 5, 13.If tan-1 , , 15 The value of tan[cos, a), , 5, 12, , 1, , 12, 5, , c) , ,  1  sin x  1  sin x , ,  1  sin x  1  sin x , , 16.cot-1 , , a), , 5, 4 11, ,  1, sin, , b), , , , , , 4, , 5, (0)  sin 1 (1)  sin 1 ( )] is, 13, , b) , , 17.The value of sin, , d), , 5, , 4, , = a), ,  tan 1, , 4, 35, , 1, 18.The value of sin 1 2 sin 817, ,  is, , 5, 12, , , x, , b) 2  -x, , d), , 12, 5, , c), , x, , x, , d)  - 2, , 2, , 5,  is, 11 , c), , 55, 8, , a), , 17, , d) none of these, b), , 1, 17, , c), , 15, , 17, , d), , 18, , 17, , 19.Solution of the equation tan (cos-1x) = sin (cot-1 1 2 ) is, a) x= , , 7, , 3, , b) x= , , 5, , c) x=  2, , 3 5, , 3, , d) none of these, Absolute Learning Academy 6

Page 7 :
MATHEMATICS, 20.Solution of the equation 2 tan 1 cos x   tan 1 2 cos ecx  is, , , , a) 2n , b) n , c) n , d) none of these, 4, 4, 2, 21. If sin(sin 1 1 5  cos 1 x)  1 then x=a.1, b. 0, c. 4 5, 22. The value of, , 23. If, , is, , +, , =, , (a), , then x =, , d. 1 5, , (b), , (a) 1, , (c), , (b), , 4, , (c), , (d), , 3, , (d), , 5, 24. The value of, (a)60o, 25. If, , +, (c) -260o, , (b) 80o, , then, , is, (d) -240o, , (a), , (b), , (c), , (d), , 26. The greatest and least values of (sin 1 x) 3  (cos 1 x) 3 are, ,  , ,  3 7 3, , c. , ,, ,, 8 8, 32 8, , 3, 7 ,   cos 1  cos, 27. The principal value of sin 1  ,  is, , 2, 6, , , , , a) 5 6, b.  2, c. 3 2, a) , , b. , , 3 3, , ,, 2 2, , 28., , 2 tan 1 (cos ec tan -1 3  tan cot 1 3) is equal to, , 29., , tan[tan 1 (1 7)  tan 1 (1 13)] is equal to, , a)  16, , 1, a, 11, b, 21, d, , 2, b, 12, c, 22, a, , 3, d, 13, a, 23, c, , d. none of these, , c.  3, , b.  6, , b. 9 2, a) 2 9, 1, 30. The value of sin(tan x) is equal to, x, 1, a., b., 1 x2, 1 x2, Questions, Answers, Questions, Answers, Questions, Answers, , d. none of these, , d.  2, , c. 7 9, , c., , ANSWERS, 4, 5, 6, d, d, a, 14, 15, 16, b, a, c, 24, 25, 26, b, b, c, , d. 9 7, , x, , d., , 1 x2, 7, d, 17, c, 27, b, , 8, c, 18, b, 28, c, , 9, d, 19, b, 29, a, , 1, 1 x2, 10, c, 20, b, 30, c, , Absolute Learning Academy 7