INVERSE TRIGONOMETRIC FUNCTIONS

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1 INVERSE TRIGNMETRIC FUNCTINS INTRDUCTIN In chapter we learnt that only one-one and onto functions are invertible. If a function f is one-one and onto then its inverse exists and is denoted by f. We shall notice that trigonometric functions are not one-one over their whole domains and hence their inverse do not exist. It often happens that a given function f may not be one-one in the whole of its domain but when we restrict it to a part of the domain it may be so. If a function is one-one on a part of its domain it is said to be inversible on that part only. If a function is inversible in serveral parts of its domain it is said to have an inverse in each of these parts. In this chapter we shall notice that trigonometric functions are not one-one over their whole domains and hence we shall restrict their domains to ensure the existence of their inverses and observe their behaviour through graphical representation. The inverse trigonometric functions play a very important role in calculus and are used extensively in science and engineering. REMARK Let a real function f : D f R f where D f = domain of f and R f = range of f be invertible then f : R f D f is given by f (y) = x iff y = f (x) for all x D f and y R f. Thus the roles of x and y are just interchanged during the transition from f to f In fact graph of f = {(x y) : y = f (x) for all x D f } and graph of f = {(y x) : x = f (y) for all y R f } = {(y x) : y = f (x) for all x D f }. Thus (x y) graph of f iff (y x) graph of f. The point (y x) is the reflection of the point (x y) in the line y = x therefore the graph of f can be obtained from the graph of f by reflecting it through the line y = x.. INVERSE TRIGNMETRIC FUNCTINS. Inverse sine function Consider the sine function f defined by f (x) = sin x D f (domain of f) = R and R f (range of f ) = [ ]. Table for the graph of f : x 0 y 0 0 0

2 MATHEMATICS II A portion of the graph is shown in fig... y = y = Fig... Note that the horizontal line y = meets its graph in many points so f is not one-one. But if we restrict the domain from to both inclusive we observe that in this part of the domain f is one-one. Therefore the function y = f (x) = sin x with D f = and R f = [ ] has an inverse function called the inverse sine function or the arc sine function denoted by sin. Thus y = sin x iff x = sin y and y. Domain of sin x is [ ] and its range is. REMARKS The graph of sin x is shown in fig... Note that it is one-one function. Fig... (a) Besides there exist other intervals where the sine function is one-one and hence has an inverse function but here by sin x we shall always mean the function sin : [ ] defined above (unless stated otherwise). The portion of the curve for which y is known as the principal value branch of the function y = sin x and these values of y are known as the principal values of the function y = sin x. (b) The graph of sin function can be obtained from the graph of the original function by interchanging the roles of x and y i.e. if (a b) is a point on the graph of sine function then (b a) becomes the corresponding point on the graph of inverse sine function. The graph of sin function is the mirror image along the line y = x of the corresponding original function. This can be visualised by looking the graphs of y = sin x and y = sin x in the same axes as shown in fig... Fig... y = sin x y = sin x

3 INVERSE TRIGNMETRIC FUNCTINS (c) It may be noted that besides there exist other intervals such as etc. where the sine function is one-one and hence has an inverse function. Thus the range of other branches are etc.. Inverse cosine function Consider the cosine function f defined by f (x) = cos x D f = R and R f = [ ]. A portion of the graph of cos x is shown in fig... Fig... Clearly f is not one-one but if we restrict the domain to [0 ] f is one-one and so it has an inverse function called the inverse cosine function or the arc cosine function denoted by cos. Thus y = cos x iff x = cos y and y [0 ]. Domain of cos x is [ ] and its range is [0 ]. The graph of cos x is shown in fig... Note that it is one-one function. The portion of the curve for which 0 y is known as the principal value branch of the function y = cos x and these values of y are known as the principal values of the function y = cos x. Note that the range of other branches of cos x are [ ] [ ] [ 0] etc.. Inverse tangent function Consider the tangent function f defined by Fig... f (x) = tan x D f = R except odd multiples of and R f = R. A portion of the graph of tan x is shown in fig... Clearly f is not one-one but if we restrict the domain to f is one-one and so it has an inverse function called the inverse tangent function or the arc tangent function denoted by tan. Thus y = tan x iff x = tan y and y. Fig...

4 MATHEMATICS II Domain of tan x is R and its range is. A portion of the graph of tan x is shown in fig..7. Note that it is one-one function. The portion of the curve for which < y < is known as the principal value branch of the function y = tan x and these values of y are known as the principal values. Fig..7.. Inverse cotangent function Consider the cotangent function f defined by f (x) = cot x D f = R except even multiples of and R f = R. A portion of the graph of cot x is shown in fig..8. Clearly f is not one-one but if we restrict the domain to (0 ) f is one-one and so it has an inverse function called inverse cotangent function or arc cotangent function denoted by cot. Thus y = cot x iff x = cot y and y (0 ). Fig..8. Domain of cot x is R and its range is (0 ). A portion of the graph of cot x is shown in fig..9. Note that it is one-one function. The portion of the curve for which 0 < y < is known as the principal value branch of the function y = cot x and these values of y are known as the principal values.. Inverse secant function Consider the function f defined by f (x) = sec x Fig..9. D f = R except odd multiples of R f = ( ] [ ). and A portion of the graph of sec x is shown in fig..0. Clearly f is not one-one but if we restrict the domain to 0 f is one-one and so it has an inverse function called inverse secant function or arc secant function denoted by sec. Thus y = sec x iff x = sec y and y 0. Fig..0.

5 INVERSE TRIGNMETRIC FUNCTINS 7 Domain of sec x is ( ] [ ) and its range is 0 A portion of the graph of sec x is shown in fig... Note that it is one-one function. The portion of the curve for which 0 y. y is known as the principal value branch of the function y = sec x and these values of y are called the principal values.. Inverse cosecant function Consider the function f defined by Fig... f (x) = cosec x D f = R except even multiples of and R f = ( ] [ ). A portion of the graph of cosec x is shown in fig... Clearly f is not one-one but if we restrict the domain to 0 0 f is one-one and so it has an inverse function called inverse cosecant function or arc cosecant function denoted by cosec. Thus y = cosec x iff x = cosec y y 0 0. Domain of cosec x is ( ] [ ) and its range is 0 0. A portion of the graph of cosec x shown in fig... Note that it is one-one function. The portion of the curve for which Fig... y y 0 is known as the principal value branch of the function y = cosec x and these values of y are known as principal values. Fig... Example. Find the principal values of : (i) cos ILLUSTRATIVE EAMPLES cosec () sin. (NCERT) (NCERT) (C.B.S.E. 00) Solution. (i) Let cos = x 0 x = cos x cos x = x = cos =. cos x = cos

6 8 MATHEMATICS II Let cosec () = x x x 0 cosec x = cosec x = cosec x = cosec () =. Let sin = x x sin x = sin x = sin sin x = sin x = sin =. Example. Find the principal values of : (i) cos tan ( ) cot ( ) (iv) sec ( ) (NCERT) (NCERT) (C.B.S.E. 00) Solution. (i) Let cos cos x = = x 0 x cos x = cos cos x = cos = cos x = cos =. Let tan ( ) = x < x < tan x = tan x = tan tan x = tan x = Let cot ( ) = x 0 < x < tan ( ) =. cot x = cot x = cot cot x = cot = cot x = cot ( ) =. (iv) Let sec ( ) = x 0 x x sec x = sec x = sec sec x = sec sec x = sec x = sec ( ) =.

7 INVERSE TRIGNMETRIC FUNCTINS 9 Example. Find the principal values of : (i) sin sin cos cos cos cos 7 (iv) tan tan 7. (NCERT) (NCERT) (C.B.S.E. 0 09) (C.B.S.E. 0) Solution. (i) Let sin sin = x x sin x = sin = sin x = sin sin =. Let cos cos = x 0 x = sin cos x = cos = cos + = cos x = cos cos =. Let cos cos 7 = x 0 x cos x = cos 7 = cos = cos x = cos cos 7 =. (iv) Let tan tan 7 = x < x < tan x = tan 7 = tan + = tan x = tan tan 7 =. Example. Using principal values find the values of : (i) tan sec ( ) tan () + cos (C.B.S.E. 0) + sin. (NCERT) Solution. (i) Let tan = x < x < tan x = = tan x =. Let sec ( ) = y 0 y y sec y = = sec = sec = sec y =. tan sec ( ) = x y = =. Let tan () = x < x < tan x = = tan Let cos = y 0 y x =. cos y = = cos = cos = cos y =.

8 70 MATHEMATICS II Let sin = z z sin z = = sin = sin z =. tan () + cos + sin = x + y + z = + ( + 8 ) 9 = = =. Example. Evaluate : sin sin. (C.B.S.E. 0 08) Solution. Let sin = x x sin x = = sin = sin x = sin =. sin Example. (i) If tan If cot sin = sin = sin + = sin =. = x find the value of cos x. = x find the values of sin x and cos x. Solution. (i) Given tan = x tan x = < x <. For this value of x sec x is +ve. sec x = + = + tan x = + = cos x = 9. Given cot = x cot x = 0 < x <. For this value of x cosec x is +ve. cosec x = + = + cot x = + = sin x =. Also cos x = cos x sin x. sin x = cot x. sin x =. =. Example 7. Evaluate the following : (i) sin cos sin cot (NCERT Examplar Problems) tan cos. Solution. (i) Let cos = x 0 x cos x = 0 x. For this value of x sin x 0 sin x = cos x = 9 = = =. sin cos = sin x = sin x cos x =.. =.

9 INVERSE TRIGNMETRIC FUNCTINS 7 Let cot For this value of x cosec x > 0. = x 0 < x < cot x = 0 < x <. cosec x = + = + cot x = sin x =. Now cos x = cos x sin x. sin x = cot x. sin x =. =. sin cot = sin x = sin x cos x =.. = 0 9. Let cos = x cos = x 0 x i.e. 0 x cos x = 0 x tan + tan x x = + tan x = tan x + ( ) tan x = tan x = + tan x = + ( Q for 0 x tan x 0) tan x = + = tan cos =. Very short answer type questions ( to ) : EERCISE. Find the principal values of the following ( to ) :. (i) sin (NCERT) cos tan ( ).. (i) sec (NCERT) cot. (i) cos. (i) cot (NCERT) sec ( ) (NCERT) cosec ( ) (NCERT) tan ( ) cosec ( ) (C.B.S.E C) sin ( ). (NCERT). (i) cos cos sin sin tan tan (NCERT) (C.B.S.E. 0). (i) cos cos tan tan 9 8 cosec cosec (C.B.S.E. 0).

10 7 MATHEMATICS II 7. Show that : (i) tan tan what is its value? cos cos what is its value? sin sin. What is its value? 8. Using principal values evalaute the following : cos cos + sin sin. (C.B.S.E. 0 08) 9. Find the value of sec (tan ( )). 0. Find the domain of the following functions : (i) sin x + sin x sin ( x).. Find the domain of the function cos (x ).. Find the domain of the function sec (x ).. Write the range of one branch of sin x other than the principal branch. (Sample Paper). Write the range of one branch of cos x other than the principal branch.. If cot = x find the value of cos x.. (i) If tan = x find the values of cos x and sin x. If cot = x find the values of sin x and cos x. 7 If tan x = sin find the value of x. 7. Evaluate the following : (i) cot (tan ) sin sin cos cos + 8. Prove the following : (i) sin + = cos tan ( ) + cos 9. Using principal values find the values of : (i) cos sin tan () + cos tan ( ) cot ( ) (NCERT Examplar Problems) = (C.B.S.E. 0) (C.B.S.E. 0) (C.B.S.E. 0) 0. Find the values of the following : (iv) cosec ( ) + cot (i) tan cos sin tan sin cos. (C.B.S.E. Sample Paper) (C.B.S.E. 0)

11 INVERSE TRIGNMETRIC FUNCTINS 7. Find the value of : (i) tan tan + cos cos tan + cot. Evaluate the following : (i) cos sin + tan sin cosec cos (NCERT Examplar Problems) (NCERT Examplar Problems) sin sin. (C.B.S.E. 0). PRPERTIES F INVERSE TRIGNMETRIC FUNCTINS In this section we shall prove some important properties of inverse trigonometric functions. These results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined. Some results may not be valid for all values of the domains of inverse trigonometric functions. In fact they will be valid only for some values of x for which inverse trigonometric functions are defined. However we may not go into the details of these values of x in the domain.. (i) sin (sin x) = x x cos (cos x) = x x tan (tan x) = x x R (iv) cot (cot x) = x x R (v) sec (sec x) = x x (vi) cosec (cosec x) = x x. Proof. (i) Let sin x = y x = sin y x = sin (sin x) ( Q y = sin x) Thus sin (sin x) = x for all x in [ ] i.e. x. We leave the proofs of other parts for the reader.. (i) sin (sin x) = x x cos (cos x) = x x [0 ] tan (tan x) = x x (iv) cot (cot x) = x x (0 ) (v) sec (sec x) = x x 0 (vi) cosec (cosec x) = x x 0 0. Proof. (i) Let sin x = y x = sin y x = sin (sin x) ( Q y = sin x) Thus sin (sin x) = x for all x. (It may be noted that when x sin x is one-one and hence its inverse function exists.) We leave the proofs of other parts for the reader.. (i) sin ( x) = sin x x cos ( x) = cos x x tan ( x) = tan x x R (iv) cot ( x) = cot x x R (v) cosec ( x) = cosec x x (vi) sec ( x) = sec x x. Proof. (i) Let sin x = y x = sin y y x = sin y y x = sin ( y) y sin ( x) = y sin ( x) = sin x x. Let cos x = y x = cos y 0 y

12 INVERSE TRIGNMETRIC FUNCTINS 07. (i). (i). (i) 7. (i) 0. (i) [ ] [0 ].. [ ]. 7. (i) 0. (i). (i) 7. (v) x ANSWERS EERCISE.. (i). (i). (i) 9. (i). (i) ( ] [ ).. (i) 7 ; ; EERCISE.. (i) (iv) x tan x (iv) tan x cos x (vi) cos x (vii) sin x (viii) cos x (ix) sin x (x) + tan x (xi) sin x a x + y xy 0. (i) (iv) 7 (xii) tan x a. (i) (iv) ± 9 (vi) 7 (vii) a + b (viii) ab. (i) 0 7. (i) 0 8. (i) 0. (i) ±. (i) 8. (i) 9. (i) (iv) x (v) ± sin x (v) EERCISE.... (i). n n + where n is any integer. x = y = 0 7. x = y = ; x = y = 7 CHAPTER TEST (iv) ab ±

This is called the horizontal displacement of also known as the phase shift.

This is called the horizontal displacement of also known as the phase shift. sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for