Inverse Trignometric Functions

7/29/2019 Inverse Trignometric Functions

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Chapter 2INVERSE TRIGONOMETRIC

FUNCTIONS

2.1 Overview

2.1.1 Inverse function

Inverse of a function f exists, if the function is one-one and onto, i.e, bijective.

Since trigonometric functions are many-one over their domains, we restrict their

domains and co-domains in order to make them one-one and onto and then find

their inverse. The domains and ranges (principal value branches) of inverse

trigonometric functions are given below:

Functions Domain Range (Principal value

branches)

y = sin1x [1,1]

,2 2

y = cos1x [1,1] [0,]

y = cosec1x R (1,1)

, {0}2 2

y = sec1x R (1,1) [0,]

2

y = tan1x R

,2 2

y = cot1x R (0,)Notes:

(i) The symbol sin1x should not be confused with (sinx)1. Infact sin1x is an

angle, the value of whose sine isx, similarly for other trigonometric functions.

(ii) The smallest numerical value, either positive or negative, of is called theprincipal value of the function.


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INVERSE TRIGONOMETRIC FUNCTIONS 19

(iii) Whenever no branch of an inverse trigonometric function is mentioned, we mean

the principal value branch. The value of the inverse trigonometic function which

lies in the range of principal branch is its principal value.

2.1.2 Graph of an inverse trigonometric functionThe graph of an inverse trigonometric function can be obtained from the graph of

original function by interchangingx-axis andy-axis, i.e, if (a, b) is a point on the graph

of trigonometric function, then (b, a) becomes the corresponding point on the graph of

its inverse trigonometric function.

It can be shown that the graph of an inverse function can be obtained from the

corresponding graph of original function as a mirror image (i.e., reflection) along the

liney =x.

2.1.3 Properties of inverse trigonometric functions

1. sin1 (sinx) =x :

,2 2

x

cos1(cosx) =x : [0, ]x tan1(tanx) =x :

,

2 2x

cot1(cotx) =x : ( )0, x

sec1(secx) =x :

[0, ] 2

x

cosec1(cosec x) =x :

, {0}2 2

x

2. sin (sin1x) =x : x[1,1]cos(cos1x) =x : x[1,1]

tan(tan1x) =x : xRcot(cot1x) =x : xRsec(sec1x) =x : xR (1,1)cosec(cosec1x) =x : xR (1,1)

3.1 11sin cosec x

x

: xR (1,1)

1 11cos sec xx

: xR (1,1)


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20 MATHEMATICS

1 11tan cot xx

: x > 0

= + cot1x : x < 0

4. sin1 (x) = sin1x : x[1,1]cos1 (x) = cos1x : x[1,1]tan1 (x) = tan1x : xRcot1 (x) = cot1x : xRsec1 (x) = sec1x : xR(1,1)cosec1 (x) = cosec1x : xR(1,1)

5. sin1x + cos1x =

2 : x[1,1]

tan1x + cot1x =

2 : xR

sec1x + cosec1x =

2 : x R[1,1]

6. tan1x + tan1y = tan1 1

y

xy

: xy< 1

tan1x tan1y = tan1; 1

1

x yxy

xy

> +

7. 2tan1x = sin1 22

1 : 1 x 1

2tan1x = cos12

21 1

x

: x 0

2tan1x = tan1 22

1

x: 1


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Solution If cos13

2

= , then cos =3

2.

Since we are considering principal branch, [0, ]. Also, since3

2> 0, being in

the first quadrant, hence cos13

2

=

6.

Example 2 Evaluate tan1

sin2

.

Solution tan1

sin2

= tan1

sin

2

= tan1(1) =

4 .

Example 3 Find the value of cos113

cos6

.

Solution cos113

cos6

= cos1 cos(2 )

6

+

=1 cos cos

6

=6

.

Example 4 Find the value of tan19

tan8

.

Solution tan19

tan 8

= tan1 tan 8

+

=1tan tan

8

=

8

Example 5 Evaluate tan (tan1( 4)).

Solution Since tan (tan1x) = x, x R, tan (tan1( 4) = 4.

Example 6 Evaluate: tan1 3 sec1 (2) .


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22 MATHEMATICS

Solution tan1

3 sec1

( 2) = tan1

3 [ sec1

2]

=1 1 2cos

3 2 3 3 3

+ = + =

.

Example 7 Evaluate:1 1 3sin cos sin

2

.

Solution1 1 13 sin cos sin sin cos

2 3

=

1 1 sin

2 6

.

Example 8 Prove that tan(cot1x) = cot (tan1x). State with reason whether the

equality is valid for all values ofx.

Solution Let cot1x = . Then cot =x

or,

tan =2

1 tan 2

x =

So1 1 1 tan(cot ) tan cot cot cot cot(tan )

2 2x x x = = = =

The equality is valid for all values ofx since tan1x and cot1x are true forx R.

Example 9 Find the value of sec1tan

2

y

.

SolutionLet1tan =

2

y, where

,

2 2

. So, tan =2

y,

which gives

24

sec=2

y.

Therefore,

2

1 4sec tan =sec =2 2

yy +

.

Example 10 Find value of tan (cos1x) and hence evaluate tan1 8cos

17

.

Solution Let cos

1

x = , then cos =x, where [0,]


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Therefore, tan(cos1x) =

2 21cos 1 tan = .

cos

x

x=

Hence

2

1

81

178 15tan cos =

817 8

17

=

.

Example 11 Find the value of1 5sin 2cot

12

Solution Let cot15

12

=y . Then coty =5

12

.

Now1 5sin 2cot

12

= sin 2y

= 2siny cosy =12 5

213 13

since cot 0, so , 2

y y <

120

169

Example 12 Evaluate1 11 4cos sin sec

4 3

Solution1 11 4

cos sin sec4 3

=

1 11 3

cos sin cos4 4

+

=1 1 1 11 3 1 3cos sin cos cos sin sin sin cos

4 4 4 4

=

2 23 1 1 3

1 1 4 4 4 4

=

3 15 1 7 3 15 7

4 4 4 4 16

.


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24 MATHEMATICS

Long Answer (L.A.)

Example 13 Prove that 2sin13

5 tan1

17

31=

4

Solution Let sin13

5= , thensin =

3

5, where ,

2 2

Thus tan =3

4, which gives = tan1

3

4.

Therefore, 2sin13

5 tan1

17

31

= 2 tan117

31= 2 tan1

3

4 tan1

17

31

=1 1

32.

174tan tan9 31

1

16

= tan1124 17tan

7 31

=1

24 17

7 31tan24 17

1 .7 31

+

=4

Example 14 Prove that

cot17 + cot18 + cot118 = cot13

Solution We have

cot17 + cot18 + cot118

= tan11

7+ tan1

1

8+ tan1

1

18(since cot1x = tan1

1

x, ifx > 0)

=1 1

1 1

17 8tan tan1 1 18

17 8

+ +

(sincex. y =1 1

.7 8

< 1)


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=1 13 1tan tan

11 18+ =

1

3 1

11 18tan3 1

111 18

+

(sincexy < 1)

=1 65tan

195=

1 1tan3

= cot1 3

Example 15Which is greater, tan 1 or tan1 1?

Solution From Fig. 2.1, we note that tan x is an increasing function in the interval

,2 2

, since 1 >4

tan 1 > tan

4

. This gives

tan 1 > 1

tan 1 > 1 >4

tan 1 > 1 > tan1 (1).

Example 16Find the value of

1 12sin 2 tan cos(tan 3)3

+

.

SolutionLet tan12

3=x and tan1 3 =y so that tanx =

2

3and tany = 3 .

Therefore,

1 12sin 2 tan cos(tan 3)3

+

= sin (2x) + cosy

= 2 2

2 tan 1

1 tan 1 tan

x

x y+

+ + = ( )2

22.

134

1 1 39

++ +

=12 1 37

13 2 26+ = .

/2p/2 p/4/4 p/2/2X

tan xan x

O


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26 MATHEMATICS

Example 17Solve forx

1 11 1tan tan , 01 2

xx x

x

= > +

Solution From given equation, we have 1 11

2tan tan1

x

x

= +

1 1 12 tan 1 tan tan x =

12 3tan

4x

=

1tan6

x

=

1

3x =

Example 18 Find the values ofx which satisfy the equation

sin1x + sin1 (1 x) = cos1x.

Solution From the given equation, we havesin (sin1x + sin1 (1 x)) = sin (cos1x)

sin (sin1x) cos (sin1 (1 x)) + cos (sin1x) sin (sin1 (1 x) ) = sin (cos1x)

2 2 21 (1 ) (1 ) 1 1x x x x+ =

2 22 1 (1 1) 0x x x x x+ =

( )2 22 1 0x x x x =

x = 0 or 2x x2

= 1 x2

x = 0 or x =1

2.

Example 19 Solve the equation sin16x + sin1 6 3x =2

Solution From the given equation, we havesin1 6x =1sin 6 3

2


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sin (sin1 6x) = sin1sin 6 3

2

6x = cos (sin1 6 3x)

6x = 21 108x . Squaring, we get

36x2 = 1 108x2

144x2 = 1 x= 1

12

Note that x = 1

12is the only root of the equation asx =

1

12does not satisfy it.

Example 20 Show that

2 tan11 sin cos

tan .tan tan2 4 2 cos sin

= +

SolutionL.H.S. =1

2 2

2 tan .tan

2 4 2tan

1 tan tan2 4 2

1 1

22since 2 tan tan

1xx =

=1

2

2

1 tan22tan

21 tan

2tan

1 tan 21 tan2

1 tan2

+

+

=

2

1

2 22

2 tan . 1 tan2 2

tan

1 tan tan 1 tan2 2 2

+


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28 MATHEMATICS

=

2

1

2 2 2

2 tan 1 tan2 2

tan

1 tan 1 tan 2 tan 1 tan2 2 2 2

+ + +

=

2

2 2

1

2

2 2

2 tan 1 tan2 2

1 tan 1 tan

2 2tan1 tan 2 tan

2 2

1 tan 1 tan2 2

+ +

+

+ +

=1 sin costan

cos sin

+

= R.H.S.

Objective type questions

Choose the correct answer from the given four options in each of the Examples 21 to 41.

Example 21 Which of the following corresponds to the principal value branch of tan1?

(A) ,2 2

(B) ,2 2

(C) ,2 2

{0} (D) (0, )

Solution (A) is the correct answer.

Example 22The principal value branch of sec1 is

(A) { }, 02 2

(B) [ ]0,

2

(C) (0, ) (D) ,2 2


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Solution (B) is the correct answer.

Example 23 One branch of cos1 other than the principal value branch corresponds to

(A)3

,2 2

(B) [ ]3

, 22

(C) (0, ) (D) [2, 3]

Solution (D) is the correct answer.

Example 24The value of1 43

sin cos 5

is

(A)3

5

(B)

7

5

(C)

10

(D)

10

Solution (D) is the correct answer.1 140 3 3sin cos sin cos 8

5 5

+ = +

=1 13 3

sin cos sin sin5 2 5

=

=1sin sin

10 10

=

.

Example 25The principal value of the expression cos1 [cos ( 680)] is

(A)2

9

(B)

2

9

(C)

34

9

(D)

9

Solution (A) is the correct answer. cos1 (cos (680)) = cos1 [cos (720 40)]

= cos1 [cos ( 40)] = cos1 [cos (40)] = 40 =2

9

.

Example 26The value of cot (sin1x) is

(A)21+

(B) 21

x

x+


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30 MATHEMATICS

(C)1

x(D)

21 x

x

.

Solution (D) is the correct answer. Let sin1x = , then sin =x

cosec =1

cosec2 = 21

x

1 + cot2 = 21

cot =2

1 x.

Example 27If tan1x =10

for some xR, then the value of cot1x is

(A)5

(B)

2

5

(C)

3

5

(D)

4

5

Solution (B) is the correct answer. We know tan1x + cot1x =2

. Therefore

cot1x =2

10

cot1x =2

10

=

2

5

.

Example 28The domain of sin1 2x is

(A) [0, 1] (B) [ 1, 1]

(C)1 1

,2 2

(D) [2, 2]

Solution(C) is the correct answer. Let sin12x = so that 2x = sin .

Now 1 sin 1, i.e., 1 2x 1 which gives1 1

2 2x .

Example 29The principal value of sin13

2

is


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(A) 23 (B)

3 (C) 4

3 (D) 5

3 .


1 1 13sin sin sin sin sin

2 3 3 3

= = = .

Example 30The greatest and least values of (sin1x)2 + (cos1x)2 are respectively

(A)2 25

and4 8

(B) and

2 2

(C)2 2

and4 4

(D)

2

and04

.

Solution (A) is the correct answer. We have

(sin1x)2 + (cos1x)2 = (sin1x + cos1x)2 2 sin1x cos1x

=2 1 12sin sin

4 2x

= ( )2

21 1

sin 2 sin4

x x

+

= ( )2

21 12 sin sin

2 8x x

+

=

2 2

12 sin4 16

x +

.

Thus, the least value is

2 2

2 i.e.16 8

and the Greatest value is

2 2

22 4 16

+

,

i.e.25

4

.

Example 31Let = sin1(sin ( 600), then value of is


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32 MATHEMATICS

(A)3 (B)

2 (C) 2

3 (D) 2

3 .

Solution(A) is the correct answer.

1 1 10sin sin 600 sin sin

180 3

=

=1 2sin sin 4

3

=1 2sin sin

3

=1 1sin sin sin sin

3 3 3

= =

.

Example 32The domain of the functiony = sin1(x2) is

(A) [0, 1] (B) (0, 1)

(C) [1, 1] (D)

Solution(C) is the correct answer. y = sin1 (x2) siny = x2

i.e. 1 x2 1 (since 1 siny 1) 1 x2 1

0 x2 1

1 . . 1 1x i e x

Example 33The domain of y = cos1(x2 4) is

(A) [3, 5] (B) [0, ]

(C) 5, 3 5, 3 (D) 5, 3 3, 5

Solution(D) is the correct answer. y = cos1 (x2 4 ) cosy =x2 4i.e. 1 x2 4 1 (since 1 cosy 1)

3 x2 5

3 5x

5, 3 3, 5x

Example 34The domain of the function defined byf(x) = sin1x + cosxis


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(A) [1, 1] (B) [1, + 1]

(C) ( ) , (D)

Solution (A) is the correct answer. The domain of cos is Rand the domain of sin1 is

[1, 1].Therefore, the domain of cosx + sin1x is R [ ]1,1 , i.e., [1, 1].

Example 35The value of sin (2 sin1 (.6)) is

(A) .48 (B) .96 (C) 1.2 (D) sin 1.2

Solution (B) is the correct answer. Let sin1 (.6) = , i.e., sin = .6.

Now sin (2) = 2 sin cos = 2 (.6) (.8) = .96.

Example 36If sin1x + sin1y =2

, then value of cos1x + cos1y is

(A)2

(B) (C) 0 (D)

2

3

Solution (A) is the correct answer. Given that sin1x + sin1y =2 .

Therefore,1 1 cos cos

2 2 2x y

+ =

cos1x + cos1y =2

.

Example 37The value of tan1 13 1cos tan

5 4

+

is

(A)19

8(B)

8

19(C)

19

12(D)

3

4

Solution (A) is the correct answer. tan1 13 1cos tan

5 4

+

= tan1 14 1tan tan

3 4

+


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34 MATHEMATICS

= tan tan 11

4 119 193 4 tan tan

4 1 8 81

3 4

+ = =

.

Example 38The value of the expression sin [cot1 (cos (tan1 1))] is

(A) 0 (B) 1 (C)1

3(D)

2

3.

Solution(D) is the correct answer.

sin [cot1 (cos4

)] = sin [cot1

1

2]=

1 2 2sin sin

3 3

=

Example 39The equation tan1x cot1x = tan11

3

has

(A) no solution (B) unique solution

(C) infinite number of solutions (D) two solutions

Solution (B) is the correct answer.We have

tan1x cot1x =6

and tan1x + cot1x =

2

Adding them, we get 2tan1x =2

3

tan1x =3

i.e., 3x = .

Example 40If 2 sin1x + cos1x , then

(A) ,2 2

= = (B) 0, = =

(C)3

,2 2

= = (D) 0, 2 = =


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Solution (B) is the correct answer. We have2

sin1x 2

2

+

2

sin1x +

2

2

+

2

0 sin1x + (sin1x + cos1x)

0 2sin1x + cos1x

Example 41The value of tan2 (sec12) + cot2 (cosec13) is

(A) 5 (B) 11 (C) 13 (D) 15


tan2 (sec12) + cot2 (cosec13) = sec2 (sec12) 1 + cosec2 (cosec13) 1

= 22 1 + 32 2 = 11.

2.3 EXERCISE

Short Answer (S.A.)

1. Find the value of1 15 13

tan tan cos cos6 6

+

.

2. Evaluate1 3

cos cos2 6

.

3. Prove that1cot 2cot 3 7

4

.

4. Find the value of1 1 11 1

tan cot tan sin23 3

.

5. Find the value of tan12

tan3

.

6. Show that 2tan1

(3) =

2

+

1 4tan3

.


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36 MATHEMATICS

7. Find the real solutions of the equation

( )1 1 2

tan 1 sin 12

x x x x+ + + + = .

8. Find the value of the expression sin ( )1 11

2tan cos tan 2 23

+

.

9. If 2 tan1 (cos ) = tan1 (2 cosec ), then show that =

4,

where n is any integer.

10. Show that1 11 1cos 2tan sin 4tan

7 3

=

.

11. Solve the following equation ( )1 13

cos tan sin cot4

x =

.

Long Answer (L.A.)

12. Prove that

2 2

1 1 2

2 2

1 1 1tan cos

4 21 1

x xx

x x

13. Find the simplified form of1 3 4cos cos sin

5 5x

, wherex

3,

4 4

.

14. Prove that1 1 18 3 77sin sin sin

17 5 85

.

15. Show that1 1 15 3 63sin cos tan

13 5 16 .

16. Prove that1 1 11 2 1

tan tan sin4 9 5

+ = .

17. Find the value of1 11 14 tan tan

5 239.


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18. Show that11 3 4 7

tan sin2 4 3

and justify why the other value

4 7

3+

is ignored?

19. Ifa1, a

2, a

3,...,a

nis an arithmetic progression with common difference d, then

evaluate the following expression.

1 1 1 1

1 2 2 3 3 4 1

tan tan tan tan ... tan1 1 1 1 n n

d d d d

a a a a a a a a

+ + + + + + + +

.

Objective Type Questions

Choose the correct answers from the given four options in each of the Exercises from

20 to 37 (M.C.Q.).

20. Which of the following is the principal value branch of cos1x?

(A)

,2 2

(B) (0, )

(C) [0, ] (D) (0, )

2

21. Which of the following is the principal value branch of cosec1x?

(A)

,2 2

(B) [0, ]

2

(C)

,2 2

(D)

,2 2

{0}

22. If 3tan1

x + cot1

x = , thenx equals

(A) 0 (B) 1 (C) 1 (D)1

2.

23. The value of sin133

cos5

is

(A)3

5(B)

7

5

(C)

10

(D)

10


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38 MATHEMATICS

24. The domain of the function cos1 (2x 1) is

(A) [0, 1] (B) [1, 1]

(C) ( 1, 1) (D) [0, ]

25. The domain of the function defined byf(x) = sin1 1x is

(A) [1, 2] (B) [1, 1]

(C) [0, 1] (D) none of these

26. If cos1 12

sin cos 05

x + =

, thenx is equal to

(A)1

5(B)

2

5(C) 0 (D) 1

27. The value of sin (2 tan1 (.75)) is equal to

(A) .75 (B) 1.5 (C) .96 (D) sin 1.5

28. The value of1 3cos cos

2

is equal to

(A)2

(B)

3

2

(C)

5

2

(D)

7

2

29. The value of the expression 2 sec1 2 + sin11

2

is

(A)

6(B)

5

6(C)

7

6(D) 1

30. If tan1x + tan1y =4

5, then cot1x + cot1y equals

(A)

5(B)

2

5(C)

3

5

(D)

31. If sin1

21 1

2 2 2

2 1 2cos tan

1 1 1

a a x

a a x

, where a,x ]0, 1, then

the value ofx is

(A) 0 (B)

2

a(C) a (D) 2

2

1

a

a


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32. The value of cot1 7cos

25

is

(A)25

24(B)

25

7(C)

24

25(D)

7

24

33. The value of the expression tan11 2

cos2 5

is

(A) 2 5 (B) 5 2

(C)5 2

2

(D) 5 2

1cosHint :tan

2 1 cos

=

+

34. If |x | 1, then 2 tan1x + sin1 22

1

x

x

is equal to

(A) 4 tan1x (B) 0 (C)2

(D)

35. If cos1 + cos1 + cos1 = 3, then ( + ) + (+ ) + ( + )equals

(A) 0 (B) 1 (C) 6 (D) 12

36. The number of real solutions of the equation

11 cos2 2 cos (cos )in ,2x x

+ = is

(A) 0 (B) 1 (C) 2 (D) Infinite

37. If cos1x > sin1x, then

(A)1

12

< (B)1

02

x 0


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40 MATHEMATICS

Fill in the blanks in each of the Exercises 38 to 48.

38. The principal value of cos11

2

is__________.

39. The value of sin13

sin5

is__________.

40. If cos (tan1x + cot1 3 ) = 0, then value ofx is__________.

41. The set of values of sec1

1

2

is__________.

42. The principal value of tan1 3 is__________.

43. The value of cos114

cos3

is__________.

44. The value of cos (sin1x + cos1x), |x| 1 is______ .

45. The value of expression tan

1 1sin cos

2

x x +

,whenx =3

2is_________.

46. Ify = 2 tan1x + sin1 22

1 x

for allx, then____


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54. The minimum value ofn for which tan1 ,4

n n>

N , is valid is 5.

55. The principal value of sin11 1

cos sin2

is3

.

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Inverse Trignometric Functions