20. INVERSE TRIGONOMETRIC FUNCTIONS 1. INTRODUCTION TO INVERSE TRIGONOMETRY The inverse trigonometric functions are the inverse functions of the trigonometric functions. They are sometimes referred to as cyclometric functions. 2. IMPORTANT DEFINITIONS Given two non-empty sets X and Y, let f:X → Y be a function, such that y = f(x). The set X is called as the domain of f while the set Y is called as the co-domain of f. The set {f(x): x ∈X} is called as range of f.A map f: A → B is said to be one-one or injective, if and only if, distinct elements of A have distinct images in B, i.e. if, and only if, x12≠⇒ x f(x 1 ) ≠ f(x 2 ) , for all x12 ,x∈ A Onto map or Surjective map: A map f: A → B is said to be an onto map or Surjective map if, and only if, each element of B is the image of some element of A, i.e. if, and only if, Range of f = co-domain of f. Objective map: A map f : A → B is an objective map if, and only if, it is both one – one and onto. 3. INVERSE FUNCTIONS If f : X → Y is one-to-one and onto (i.e. f is objective), then, we can define a unique function g : Y → X, such that g(y) = x, where x ∈X is such that y = f (x). Thus, the domain of g = range of f and range of g = domain of f. The function is called the inverse of f and is denoted by f-1. (a) Trigonometric functions are many-one functions but these become one-one, onto, if we restrict the domain of trigonometric functions. Similarly, co-domain is equated to range to make it an onto function. We can say that the inverse of trigonometric functions are defined within restricted domains of corresponding trigonometric functions. (b) Inverse of sin (sine functions) is denoted by sin-1 (arc sine function). We also write it as sin-1 x. Similarly, other inverse trigonometric functions are given by cos-1 x, tan-1 x, sec-1 x, cot-1x and cosec-1 x. 1 (c) Note that sin−1 x ≠ and ( sin−1 x)² ≠ sin−2 x, Also sin−−11 x≠ (sinx) sinx (d) Domain and Range of Inverse Trigonometric Functions: Function Domain Range (Principal value branch) (i) −1 ππ y= sin x −≤≤1x1 − ≤≤y 22 20.2 | Inverse Trigonometric Functions Function Domain Range (Principal value branch) (ii) −1 y= cos x −≤≤1x1 0y≤ ≤π (iii) −∞ <x < ∞ ππ y= tan−1 x − ≤<y 22 (iv) ππ y= cosec−1 x x≥ 1 or x ≤− 1 − ≤≤y ,y ≠ 0 22 (v) π y= sec−1 x x≥ 1 or x ≤− 1 0≤ y ≤π ,y ≠ 2 (vi) −1 y= cot x −∞ <x < ∞ 0y< <π (e) The principal value of an inverse trigonometric function is the value of that inverse trigonometric function which lies in the range of principal branch. MASTERJEE CONCEPTS If no branch of an inverse trigonometric function is mentioned, then it can be implied that the principal value branch of that function. You can remember range as set of angles that have the smallest absolute values satisfying for all the values of domain. Vaibhav Gupta (JEE 2009 AIR 54) 4. TRANSFORMATION OF TRIGONOMETRIC FUNCTIONS TO INVERSE TRIGONOMETRIC FUNCTIONS 4.1 sin x to sin-1 x The graph of an inverse trigonometric function can be obtained from the graph of the original by interchanging x and y axes. Note: It can be shown that the graph of an inverse function can be obtained from the corresponding graph of original function as the mirror image in the line y = x. Y 1 3 -5p -p p 2 2 2 XX’ O 5 -2 p p p 2p p p -1 2 2 Y’ Figure 20.1 (a) y = sin x, x ∈ R and y1≤ ; y= sin−1 x, x ≤ 1, y ∈ [ −π / 2, π / 2] Mathematics | 20.3 4.2 cos x to cos-1 x (b) y=∈≤ cosx,x Rand y 1 y= cos−1 x,x ∈− [ 1,1]andy ∈ [0, π ] Y Y 5p 5p 2 2 2p 3p 3p 2 2 y=x p p p 2 -p -2 2 -1 1 X’ p p X X’ X - O 2 O p p p - p -x 2 p 2 2 2 - p p -3 -3p p 2 2 -2p -5p -5p 2 2 Y’ Y’ y-sin x and y-sin-1(x) y-sin-1(x) Figure 20.2 Y 5p 2 2p 3p 2 p p Y 2 1 X’ -1 1 X -5p p 5p O - 2 X’ 2 p p 2 X -3 - O 3 2 -p -2p p p p p 2 2 2 -1 2 -p Y’ -3p y=cos x 2 -2p -5p 2 Y’ y=cos-1 (x) Figure 20.3 20.4 | Inverse Trigonometric Functions 4.3 tan x to tan-1 x π −1 ππ (c) y= tanx,x ∈ R − x : x = (2 π+ 1) ,n ∈ Z and y ∈ R y= tan x,x ∈ R and y ∈− , 2 22 Y Y 2 p 3p 2 p 2 p p 3p 2 - 1 2 2 2 X’ p p X O O -2 -1 -p p X’ X 2 -1 -2 12 -p 2 -p Y’ Y’ y=tan x Figure 20.4 4.4 cot x to cot-1 x (d) y= cot x,x ∈− R{ x:x =π n , n ∈ Z} and y ∈ R y= cot−1 x,x ∈ R and y ∈π (0, ) Y Y 2 p 3p 2 p 2 p p 3p 2 - 1 2 2 2 X’ p p X O O -2 -1 -p p X’ X 2 -1 -2 12 -p 2 -p Y’ Y’ y=tan x y=tan-1 x Figure 20.5 4.5 sec x to sec-1 x π −1 π (e)(v) y= secx,x ∈− R x : x = (2n + 1) ,n ∈ z andy ∈−− R ( 1,1) y= sec x,x ∈ R −− ( 1,1)andy ∈ [0, π ] 2 2 Mathematics | 20.5 2 Y p 3p 2 2 p p - 1 2 2 X’ p p p X - O 3 p -1 p p X’ X 2 -2 2 2 -2 -1 1 2 -p 2 - Y’ p y = sec x Y’ y = sec-1 x Figure 20.6 4.6 cosec x to cosec-1 x −1 ππ (f) y= cosecx,x ∈ {x : x = n π ,n ∈ Z} andy ∈−− R ( 1,1) y= cosec x,x ∈ R −− ( 1,1)andy ∈− , − {0} 22 Y 2p Y 3p 2 2 p - 3 p p 1 p 2 - 2 2 p p 2p X’ X X O X’ -2 -1 p -1 2 O 12 -2 -p 2 - Y’ p y = cosec x Y’ y = cosec-1 x Figure 20.7 20.6 | Inverse Trigonometric Functions Illustration 1: Find the domain of definition of the function f(x) = 3cos−1 (4 x) −π. (JEE MAIN) Sol: Use the condition that the expression inside the square root is ≥ zero. For domain of f(x) = 3cos−1 (4 x) −π, we must have π 11 4x≥ cos ⇒ 4x ≥ ⇒≥ x …..(i) 3 28 −11 Also −≤1 4x ≤ 1 ⇒ ≤ x ≤ .….(ii) 44 −11 ∴ From (i) and (ii), we get x'∈ 48 MASTERJEE CONCEPTS In case of confusion, try solving problems by replacing inverse functions with angles and applying trigonometric identities. Shrikant Nagori (JEE 2009 AIR 30) Illustration 2: If 0< cos−1 x < 1and 1 + sin(cos − 1 x) + sin 21 (cos −− x) + sin 31 (cos x) + ...... ∞= 2, then find the value of x. (JEE MAIN) Sol: Use summation of infinite GP series. We have 1+sin(cos-1 x) + sin²(cos-1 x) + ……… ∞=2 11 1π 3 ⇒ =⇒=−2 1 sin(cos−−11 x) ⇒ sin(cos x) =⇒cos−1 x = ⇒= x 1− sin(cos−1 x) 2262 2 Illustration 3: Let f(x)= (sin−−−111 [x] ++ tan [x] cot [x]) , where [x] denotes the greatest integer less than or equal to π x. If A and B denote the domain and range of f(x) respectively, find the number of integers in A∪B. (JEE ADVANCED) π Sol: Use tan−−11 [x]+= cot [x] and proceed. 2 For domain of f(x), we must have −1 ≤ [x] ≤ 1 ⇒− 1 ≤ x < 2, so set A = [-1,2) 2 −1 π −−11π f(x)= sin [x] + As tan [x]+ cot [x] = , ∀∈ x A π 2 2 So, set B = {0, 1, 2} = Range of f(x). Now, A∪=− B [ 1,2) ∪ {0,1,2} =− [ 1,2] Hence, number of integers in ( AB∪ )=4 5.PROPERTIES/IDENTITIES OF INVERSE TRIGONOMETRIC FUNCTIONS 5.1 Complementary Angles π (a) sin−−11 x+ cos x = , ∀ x ∈− [ 1,1] 2 Mathematics | 20.7 π (b) tan−−11 x+ cot x = , ∀∈ x R 2 π (c) sec−−11 x+ cosec x = , ∀ x ∈ ( −∞ , − 1] ∪ [1, ∞ ) 2 5.2 Negative Arguments (a) sin−−11 (− x) =− sin x, ∀ x ∈− [ 1,1] (b) cos−−11 (− x) =π− cos x, ∀ x ∈− [ 1,1] (c) tan−−11 (− x) =− tan x, ∀∈ x R (d) cot−−11 (− x) =π− cot x, ∀ x ∈ R (e) sec−−11 (− x) = π − sec x, ∀ x ∈ ( −∞ , − 1] ∪ [1, ∞ ) (f) cosec−−11 (− x) = − cosec x ∀ x ∈ ( −∞ , − 1) ∪ [1, ∞ ) 5.3 Reciprocal Arguments 1 (a) cosec−−11 x= sin ; x ≥ 1 (Both the functions are identical) x 1 and sin−−11 x= cosec ; x ≤≠ 1,x 0 (Both the functions are not identical) x 1 (b) sec−−11 x= cos ; x ≥ 1 (Both the functions are identical) x 1 and cos−−11 x= sec ; x ≤ 1 (Both the functions are not identical) x −−111 −1 1 (c) tan x= cot , x∈∞ (0, ) = − π+cot , x∈ ( −∞ ,0) , x x −−111 −1 1 and cot x= tan , x ∈∞ (0, ) = π+ tan , x∈ ( −∞ ,0) x x 5.4 Forward Inverse Identities − − (a) y= sin(sin1 x) = x,x ∈− [ 1,1],y ∈− [ 1,1], y is (b) y= cos(cos1 x) = x,x ∈− [ 1,1],y ∈− [ 1,1], y is aperiodic aperiodic y y 1 1 y=x y=x o o -1 45 x -1 45 x O 1 O 1 -1 -1 Figure 20.8 20.8 | Inverse Trigonometric Functions (c) y = tan ( tan−1 x )=x, x ∈ R, y ∈ R, y is aperiodic (d) y = cot (cot−1 x) = x, x ∈ R, y ∈ R, y is aperiodic y y y=x y=x 45o 45o x x O O Figure 20.9 (e) y = cosec ( cosec−1 x )=x, x≥≥ 1, y 1 ,y is aperiodic (f) y = sec (sec−1 x) = x, x≥≥ 1, y 1 , y is aperiodic y y y=x y=x o 1 1 45 -1 -1 x x O 1 O 1 -1 -1 y=x y=x Figure 20.10 Also, cos(sin−12 x)= 1 − x sin(cos−12 x)= 1 − x −1 1 2 cos tan x = −1 1x− ( ) 2 tan(cos x) = 1x+ x x x sin( tan−1 x) = tan( sin−1 x) = 1x+ 2 1x− 2 5.5 Inverse Forward Identities −1 ππ (a) y= sin (sinx) = x, x ∈ R ,y ∈− , , Periodic with period 2 π 22 y p/2 y=( y= +x p p+x) y=x p-x y=2 -/2 45o 3/2 y=x-2 p p p x - /2 2 -2p -3p/2 p O p p p -/p 2 Figure 20.11 Mathematics | 20.9 (b) y= cos−1 (cosx) = x,x ∈ R, y ∈π [0, ], Periodic with period 2 π y y = -x p y=2 -x /2 y=x p p y=x+2p x -2 - 2 p p -/p 2 O p/2 p p Figure 20.12 −1 π ππ (c) y= tan (tanx) = x,x ∈− R (2n − 1) n ∈ I , y ∈− , , Periodic with period π 2 22 y p/2 - y=x 3 p y=x- p -2 y=x+p 2 p 2 p x -3 O 2 p -p p p p 2 2 p/2 Figure 20.13 (d) y= cot−1 (cotx) = x,x ∈− R {n π }, y ∈ (0, π ), periodic with π y p y=x y=x+2p y=x+ y=x- 2 pO p p p p 2 x -2 O 2 p -p p p Figure 20.14 −1 ππ (e) y= cosec (cosec x)=∈− x R {n π ,n ∈ I}, y ∈− ,0 ∪ 0, y is periodic with period 2π 22 y p/2 y=( y= p+x) y=x p-x o -/2 45 3/2 y=x-2p p p x -3 /2 - O /2 p p p p 2p -/p 2 Figure 20.15 20.10 | Inverse Trigonometric Functions (f) y= sec−1 (sec x)= x, y is periodic, y p y = y= - x 2 - y=x p x y=x+2p - x -2 -3 - p O p p 3 2 p p p 2 p p 2 2 2 Figure 20.16 π ππ x∈− R (2n − 1) n ∈ I ,y ∈ 0, ∪ , π with period 2π 2 22 1 (i) tan−1 (cotx)= π− x for x ∈ [0, π ] 2 1 (ii) sin−1 (cosecx)= π− x for x ∈ [0, π ] 2 −1 11 (iii) sec (cosx)= π− x for x ∈ 0, π .