CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER ......6 CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRIC...

CENGAGE/GTEWANIMATHSSOLUTIONS

CHAPTERINVERSETRIGONOMETRICFUNCTIONS||TRIGONOMETRY

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1

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_InverseTrigonometircFunctions

Find the principal value of the following (ii)

WatchFreeVideoSolutiononDoubtnut

2


Solve


3


Solve


4


Solvefor if


5


Findthevalueof forwhich isdefined.


cos ec− 1(2) tan− 1( − √3)

cos− 1( )1√2

sin − 1 x ≻ 1

cos − 1 x > cos− 1 x2

x (cot− 1 x)2

− 3(cot− 1 x) + 2 > 0

x cos ec− 1(cosx)

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6


Find the value of (i) (ii) (iii) (iv)


7


Findtherangeof


8


Findthevalueof forwhich


9


If

thenfindthevalueof


sin− 1(2x) cos− 1 √x2 − x + 1tan − 1(x2)

1 + x2

sec− 1(x + )1x

f(x) = ∣∣3 tan − 1 x − cos− 1(0)∣∣ − cos − 1( − 1).

x sec − 1 x + sin− 1 x = .π

2

sin− 1(x2 − 4x + 5) + cos− 1(y2 − 2y + 2)

=π

2xandy.

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10


If thenfindthevalueof


11


If

thenfindthevalueof


12


Findthevaluesof forwhich willhaveatleastonesolution.


13


Find satisfying where represents the greatestintegerfunction.



cos − 1 λ + cos − 1 μ + cos − 1 γ = 3π, λμ + μγ + γλ

sin− 1 x1 + sin− 1 x2 + + sin− 1 xn ≤ − ,

n ∈ N, n = 2m + 1,m ≥ 1,

nπ

2

x11 + x33 + x

55 + ...(m + 1)terms

x22 + x44 + x

66 + .... mterms

a sin− 1 x = |x − a|

x [tan− 1 x] + [cot − 1 x] = 2, []

− 1 − 1

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14 Findthenumberofsolutionoftheequation


15

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_PrincipalValueOfTheFunction

Evaluatethefollowing: (ii)


16


Evaluate the following: 1. 2. 3.


17


Findthevalueof


cos(cos − 1 x) = cos ec(cos ec− 1x).

sin − 1( )sinπ4

cos− 1(cos 2 )π3

tan− 1( )tan π3

sin− 1(sin )2π3

cos − 1(cos )7π6

tan− 1(tan )2π3

sin− 1(sin 5) + cos − 1(cos 10)

+ tan− 1{tan( − 6)} + cot− 1{cot( − 10)}.

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18


Findtheareaboundedby and


19


Findthenumberofsolutionof


20


If thenfindthevaluesof


21


Findthevalueof


22


Findthevalueof


23


Provethat:

y = sin − 1(sin x) x = aξ or x ∈ [0, 100π]

2 tan− 1(tanx) = 6 − x.

cos(2 sin− 1 x) = ,19

x.

sin( cot − 1( − ))12

34

sin( cos − 1( ))14

−19

1 + sin 1 − sin

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24


Solve


25

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_RelatingDifferentInverseTrigonometricFunctions

Find intermsof where


26


Simplify


27


Provethat:

cot− 1( ) = , x∈ (0, )

√1 + sin x + √1 − sin x

√1 + sin x − √1 − sin x

x

2π

4

sin − 1(1 − x) − 2 sin− 1 x =π

2

tan− 1 x

√a2 − x2sin− 1 x ∈ (0, a).

sin cot− 1tan cos − 1x, x > 0

cos ec(tan − 1(cos(cot− 1(sec(sin− 1 a)))))

= √3 − a2,

∈ [0, 1]

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where


28


If thenprovethat


29


Provethat


30


Provethat


31


Provethat:

a ∈ [0, 1]

x < 0, cos− 1 x = π − sin− 1 √1 − x2

cos − 1{ } =1 + x2

cos− 1 x2

tan− 1{ } = , − a< x < a

x

a + √a2 − x212

sin− 1 xa

1 + 1 −

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32


Provethat


33




34


Findtheminimumvalueofthefunction


35


Findtherangeof


sin− 1{ } =+ , 0 < x < 1

√1 + x + √1 − x2

π

4sin− 1 x

2

cos − 1( ) = 2 tan − 1 xn, 0 < x < ∞1 − x2n1 + x2n

x ∈ [ − 1, 0], cos− 1(2x2 − 1) − 2sin − 1 x

f(x) = − cot − 1 xπ2

16 cot − 1( − x)

y = (cot− 1 x)(cot− 1( − x)).

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36


Findthevalueof


37

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_ComplementaryAngles



38




39


Solve


40


Findtherangeof


sin− 1(sin 5) + cos − 1(cos 10)

+ tan− 1{tan( − 6)} + cot− 1{cot( − 10)}.

sin − 1 x = , f or somex ∈ ( − 1, 1),π

5cos− 1 x.

sin(sin− 1( ) + cos− 1 x) = 1,15

x.

sin − 1 x ≤ cos− 1 x.

f(x) = sin− 1 x + tan − 1 x + cos− 1 x

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41


Findtheminimumvalueof


42


Solve


43


If

thenfind


(sec− 1 x)2

+ (cosec− 1x)2

+ =sin− 1(14)

|x|

sin− 1(2√15)|x|

π

2

α = sin − 1(cos(sin− 1 x))andβ

= cos − 1(sin(cos− 1 x)),

tanα.

tanβ

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44




45


Provethat:


46


Findthevalueof


47


Findthevalueof


sec − 1 x = cos ec− 1y, + .cos− 1 1

x

cos− 1 1y

tan− 1 x + = { , if x > 0− , if x < 0

tan − 1 1x

π

2π

2

sin− 1 x + + cos − 1 x + .sin− 1 1

x

cos − 1 1

x

10

∑r= 1

10

∑s= 1

tan− 1( ).rs

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48


If thenfindthemaximumvalueof


49

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_SumAndDifferenceOfAnglesInTermOfTan^-1`

Provethat


50


Iftwoanglesofatriangleare thenfindthethirdangle.


51


Findthevalueof`tan^(-1)(1/2tan2A)+tan^(-1)(cotA)+tan^(-1)(cot^3A),for0WatchFreeVideoSolutiononDoubtnut

52


Simplify`[(3sin2alpha)/(5+3cos2alpha)]+tan^(-1)[(tanalpha)/4],where-pi/2WatchFreeVideoSolutiononDoubtnut

53


Solvetheequation


sin − 1 xi ∈ [0, 1] ∀i = 1, 2, 3, .28

√sin− 1 x1√cos − 1 x2+ √sin− 1 x2√cos− 1 x3 +

√sin− 1 x3√cos − 1 x4 ++ √sin− 1 x28√cos− 1 x1

+ =cos− 1 4

5

cos− 1(12)

13

cos− 1(33)

65

tan− 1(2)andtan− 1(3),

tan− 1 2x + tan − 1 3x =π

4

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54


Solve


55




56


If isanA.P.withcommondifference thenprovethat



tan − 1 x + sin− 1 x = tan − 1 2x.

x > y > z > 0,

+

+

cot− 1(xy + 1)

x − y

cot− 1(yz + 1)

zy − zcot− 1(zx + 1)

z − x

a1, a2, a3, , an d,

tan[tan − 1( )+ tan− 1( )+ tan− 1( )] =

d

1 + a1a2d

1 + a2a3d

1 + an− 1an

(n − 1)d

1 + a1an

∞

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57Findthevalueof


58

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_MultipleAnlgesInTermsOfTan^-1X

Findthevalueof


59


If ,thenfindthevalueof


60

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_SumAndDifferenceOfAnglesInTermOfSin^-1AndCos^-1

Findthevalueof


∞

∑r= 0

tan− 1( )11 + r + r2

4 − +tan− 1 1

5tan− 1 1

70tan− 1 1

99

(x − 1)(x2 + 1) > 0 sin( − tan − 1 x)12

tan − 1(2x)

1 − x2

+cot− 1 3

4sin− 1 5

13

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61


Solve


62


Solve


63


Solve:


64


If thenshowthat


65


Whichofthefollowinganglesisgreater?


sin − 1 x + sin− 1 2x = .π

3

sin − 1 x ≤ cos− 1 x.

cos− 1( x2 + √1 − x2√1 − )= − cos− 1 x.

12

x2

4

cos− 1 x2

x ∈ (0, ),π2

cos− 1( (1 + cos 2x) + √(sin2 x − 48 cos2 x) sin x)= x − cos− 1(7 cosx)

72

θ1 = sin− 1( ) + sin− 1( ) or θ2= cos − 1( ) + cos − 1( )

45

13

45

13

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66


Findthevalue


67

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_MultipleAngleSInTermsOfSin^-1XAndCos^-1X

If thenprovethat


68


Solve


69


If

thenwhichisgreater.

limn→ ∞

n

∑k= 2

cos − 1( )1 + √(k − 1)k(k + 1)(k + 2)k(k + 1)

f(x) = sin− 1 x limx→

f(3x − 4x3) = π − 3 limx→

sin− 1 x12

12

sin − 1 x − cos− 1 x = sin− 1(3x − 2)

A = 2 tan − 1(2√2 − 1)andB = 3 sin− 1( )+ sin− 1( ),

13

35

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70

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_SomeMoreMultipleAngleFormulas

If ,thenfindthevalueof


71


If isindependentof findthevaluesof


72




73


Findthevalueof



=sin− 1(2x)

1 + x2tan − 1(2x)

1 − x2x.

sin − 1( ) + 2 tan− 1( − )4xx2 + 4

x

2x, x.

= − + tan − 1 3x,cos− 1(6x)

1 + 9x2π

2x.

2 + +cos− 1 3

√13

cot− 1(16)

6312

cos− 1 725

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74Solve


75

CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRICFUNCTIONS_Miscellaneous

Findthedomainfor


76


Findthesum


77


Find the number of positive integral solution of the equation



2 cos− 1 x = sin − 1(2x√1 − x2)

f(x) = sin − 1( )1 + x22x

cos ec− 1√10 + cos ec− 1√50 + cos ec− 1√170

+ + cos ec− 1√(n2 + 1)(n2 + 2n + 2)

tan− 1 x + =cos− 1 y

√1 + y2sin− 1 3

√10

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78If`tan^(-1)y=4tan^(-1)x(|x|WatchFreeVideoSolutiononDoubtnut

79


If aretherootsoftheequations

provethat

,where isaninteger.


80


Solveforrealvaluesof


81


Find the set of values of parameter so that the equationhasasolution.


82


If and ,thenfindthevalueof



x1, x2, x3, andx4x4 − x3 sin 2β + x2 cos 2β − x cos β − sin β= 0,

tan− 1 x1 + tan − 1 x2 + tan − 1 x3 + tan − 1 x4

= nπ + ( ) − βπ2

n

x : = 7(sin− 1 x)

3+ (cos − 1 x)3

(tan − 1 x + cot− 1 x)3

a

(sin− 1 x)3

+ (cos− 1 x)3

= aπ3

p > q > 0 pr < − 1 < qr

tan− 1( ) + tan− 1( )+ tan− 1( )

p − q1 + qr

q − r1 + qr

r − p1 + qr

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83

Solvetheequation


84


Solvetheequation


85


Solvetheequation



If thenprovethat

√∣∣sin − 1|cosx|∣∣ + ∣∣cos − 1|sinx|∣∣= sin − 1|cos x| − cos− 1|sinx|, ≤ x

≤ .

−π2

π

2

tan− 1( ) + tan − 1( )= tan − 1( − 7)

x + 1x − 1

x − 1x

sin− 1 6x + sin− 1 6√3x = .−π2

0 < a1 < a2 < .... < an,

− −

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86


87


Let

Thenfindthemaximumandminimumvaluesof


88


is equal to (b) (c) (d)


89


Thevalueof

(a) (b) (c) (d)noneofthese


tan− 1( ) + tan− 1( )+ tan− 1( ) + .......+ tan− 1( ) + tan − 1( )= tan − 1( ).

a1x − yx + a1y

a2 − a11 + a2a1

a3 − a21 + a3a2an − an− 11 + anan− 1

1an

x

y

f(x) = sin x + cos x + tanx + sin− 1 x

+ cos− 1 x + tan − 1 x.f(x).

cos− 1(cos(2cot − 1(√2 − 1))) √2 − 1π

43π4

noneofthese

sin− 1

⎛⎜⎝cot⎛⎜⎝sin

− 1⎛⎜⎝ +

+ sec− 1 √2⎞⎟⎠

⎞⎟⎠⎞⎟⎠is

2 − √34

cos − 1(√12)4

0π

2π

3

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90


Thevalueof isequalto (b) (c) (d)


91


Thevalueof is(a) (b) (c) (d)


92


If thenthemaximumvalueof is6(b) (c)5(d)noneof

these


93


If and areequalfunctions,thenthemaximum

range of value of is (a) (b) (c)

(d)



cos − 1 √ − cos− 1( )23

√6 + 1

2√3

π

3π

4π

2π

6

cos( cos − 1( ))12

18

34

−34

116

14

> , n ∈ N ,cot− 1 n

π

π

6n 7

cos ec− 1(cos ecx) cos ec(cos ec− 1x)

x [ − , − 1] ∪ [1, ]π2

π

2[ − , 0] ∪ [0, ]π

2π

2( − ∞, − 1) ∪ [1, ∞] [ − 1, 0] ∪ [0, 1]

2 01 2 − 1

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94 isequalto5(b)13(c)15(d)6


95


Themaximumvalueof is


96


Fortheequation ,thenumberofrealsolutionis(a)1(b)2(c)0(d)


97


Thenumberofrealsolutionoftheequation

is


98


If

sec2(tan01 2) + cos ec2(cot− 1 3)

f(x) = tan − 1⎛⎜⎝

⎞⎟⎠(√12 − 2)x2x2 + 2x2 + 3

cos− 1 x + cos− 1 2x + π = 0∞

tan− 1 √x2 − 3x + 7 + cos− 1 √4x2 − x + 3= π

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thenthevalueof is(a)1(b) (c) (d)nonofthese


99


If then isequalto(b) (d)


100


Range of is (a) (b)

(c) (d)noneofthese


101


Thevalueof isequalto (b) (c) (d)


sin− 1(x − 1) + cos − 1(x − 3)

+ tan− 1( ) = cos− 1 k + π,x2 − x2

k −1

√2

1

√2

sin − 1 x = θ + βand sin − 1 y = θ − β, 1 + xy sin2 θ + sin2 βsin2 θ + cos2 β cos2 θ + cos2 θ cos2 θ + sin2 β

f(x) = sin− 1 x + tan − 1 x + sec− 1 x ( , )π4

3π4

[ , ]π4

3π4

{ , }π4

3π4

( lim )n−→∞

(tan − 1 x))) −1 π2

−1

√2

1

√2

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102


Rangeof is(a) (b) (c) (d)


103


If , where denotes the greatest integer functions,thenthecompletesetofvaluesof is (b) (d)noneofthese


104


Complete solution set of where denotes thegreatest integer function, is equal to (a) (b) (d)


105


The number of integral values of for which the equationhasasolutionis1(b)2(c)3(d)4


106


Therangeofvalueof forwhichtheequation hasa

solutionis(a) (b)(0,1) (d)


tan − 1( )2x1 + x2

[ − , ]π4

π

4( − , )π

2π

2( − , )π

2π

4

[ , ]π4

π

2

[cot − 1 x] + [cos− 1 x] = 0 []x (cos 1, 1) cos 1, cos 1) (cot 1, 1)

[cot− 1 x] + 2[tan− 1 x] = 0, [](0, cot 1) (0, tan 1) (tan 1, ∞)

(cot 1, tan 1)

ksin− 1 x + tan− 1 x = 2k + 1

p sin cos− 1(cos(tan − 1 x)) = p

( − , )1√2

1

√2(c)( , 1)1

√2( − 1, 1)

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107


Thesumofthesolutionoftheequation

is0(b) (c) (d)2


108


Complete solution set of is (a)

(b) (c) (d)noneofthese


109


The value of is equal to (a) (b) (d)noneofthese


110


Thevalueoftheexpression

2 sin− 1 √x2 + x + 1 + cos− 1 √x2 + x

=3π2

−1 1

tan2(sin− 1 x) > 1 ( − 1, − ) ∪ ( , 1)1√2

1

√2

( − , )~{0}1√2

1

√2( − 1, 1)~{0}

sin − 1(sin 12) + cos − 1(cos 12) zero 24 − 2π4π − 24

sin− 1( ) + cos− 1( )+ tan− 1( ) + sin − 1(cos 2)

sin(22π)7

cos(5π)3

tan(5π)7

17π −

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is(a) (b) (c) (d)


111


Thevalueof where ,is

equalto (b) (c) (d)


112


If ,thenthevalueof

isequalto(a) (b) (c)0(d)


113


is equal to


− 217π42

−2 − 2−π21

noneofthese

sin − 1(cos(cos− 1(cos x) + sin− 1(sin x))), x ∈ ( , π)π2

π

2−π π −

π

2

α ∈ ( − , − π)3π2

tan− 1(cot α) − cot− 1(tan α) + sin − 1(sin α)

+ cos− 1(c0sα)2π + α π + α π − α

tan− 1[ ]cosx1 + sinx

− , f or x ∈ ( − , )π4

x

2π

23π2

− , f or x ∈ ( − , )π4

x

2π

2π

2− , f or x ∈ ( − , )π

4x

2π

2π

2

− , f or x ∈ ( − , )π4

x

23π2

π

2

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114


If and isconstant, thenisequalto (b) (c) (d)


115


Thetrigonometricequation hasasolutionforallrealvalues(b)

(d)


116


holdif(a) (b) (c)(d)


117


Thevalueof

isequalto0(b)1(c) (d)noneofthese



There exists a positive real number of satisfying Then the

f(x) = x11 + x9 − x7 + x3 + 1 f(sin − 1(sin 8) = α, αf(tan − 1(tan8) α α − 2 α + 2 2 − α

sin− 1 x = 2sin − 1 a

|a| <1a

|a| ≤1

√2< |a| <

12

1

√2

sin− 1(sin 5) > x2 − 4x x = 2 − √9 − 2π x = 2 + √9 − 2πx > 2 + √9 − 2π x ∈ (2 − √9 − 2π, 2 + √9 − 2π)

tan(sin− 1(cos(sin− 1 x)))tan

(cos− 1(sin(cos − 1 x))), wherex ∈ (0, 1),−1

x cos(tan − 1 x) = x.2 2 4

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118valueof (b) (c) (d)


119


Thevalueof

(b) (d)noneofthese


120


If is equal to 1 (b) 2 (c) (d)

noneofthese


121


The value of is equal to (a) (b)

(c) (d)noneofthese



cos − 1( )isx22

π

10π

52π5

4π5

cos ec2( )+ sec2( tan − 1( ))isequa < o

α3

212

tan− 1 αβ

β3

212

β

α(α + β)(α2 + β2) (α + β)(α2 − β2) (α + β)(α2 + β2)

sin − 1 x + sin− 1 y = , thenπ

21 + x4 + y4

x2 − x2y2 + y212

2 tan− 1(cos ec tan − 1 x − tan cot− 1x) cot − 1 xcot− 1 1

xtan − 1 x

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122If then (b) (d)


123


If then the value of

will be (b) (c) (d)


124


If then is equal to (a) (b)

(c) (d)


125


The solution of the inequality is (b)

(d)noneofthese


= 40tan − 1(√1 + x2 − 1)

xx = tan 20 x = tan 40 x =

tan 1

40x = tan 80

sin − 1 a + sin− 1 b + sin− 1 c = π,

a√(1 − a2) + b√(1 − b2) + √(1 − c2) 2abc abc abc12

abc13

a sin − 1 x − b cos − 1 x = c, a sin − 1 x + b cos − 1 x 0πab + c(b − a)

a + bπ

2πab + c(a − b)

a + b

(log) sin−1 x> ( log ) 1 / 2 cos −1 x12x ∈ [ ]0, 1

√2

x ∈ [ , 1]1√2

x ∈ ( )0, 1√2

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126


For , is true when belongs to (a)

(b) (c) (d)


127


If (a) (b) (c) (d)


128


Meanandstandarddeviationof100observationsofarefoundtobe40and10.Ifatthetimeofcalculationtwoobservationsarewronglytakenas30and70insteadof3and27respectively.Findthecorrectstandarddeviation.


129


The number of integer satisfying is

(a) (b) (c) (d)



0 < θ < 2π sin− 1(sin θ) > cos − 1(sin θ) θ

( , π)π4

(π, )3π2

( , )π4

3π4

( , 2π)3π4

∣∣sin− 1 x∣∣ + ∣∣cos− 1 x∣∣ = , thenπ

2x ∈ R [ − 1, 1] [0, 1] φ

x sin− 1|x − 2| + cos − 1(1 − |3 − x|) =π

21 2 3 4

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130 If then isequal to (a) (b)3 (c) (d)


131


The number of solutions of the equation

is0(b)1(c)2(d)3


132


areidenticalfunctionsif(a) (b) (c) (d)


133


Thevalueof forwhich

hasarealsolutionis (b) (c) (d)


tan − 1 x + 2 cot− 1 x = ,2π3

x,√3 − 1

√3 + 1√3

√2

cos− 1( ) − cos− 1 x = + sin− 1 x1 + x22x

π

2

f(x) = tan − 1 x + tan− 1( ); g(x)= sin − 1 x + cos− 1 x

1x

x ∈ R x > 0 x ∈ [ − 1, 1] x ∈ [0, 1]

aax2 + sin− 1(x2 − 2x + 2)

+ cos− 1(x2 − 2x + 2) = 0π

2−π

22π

−2π

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134


If then isequalto (b) (c) (d)


135


If

thenthevalueof is1(b) (c) (d)


136


Theleastandthegreatestvaluesof are(a) (b)

(c) (d)noneofthese


137


If

thenvalueof isequalto0(b) (c) (d)noneofthese


sin − 1( ) + sin− 1( ) = ,5x

12x

π

2x

713

43

13137

cos− 1 √p + cos− 1 √1 − p + cos− 1 √1 − q

= ,3π4

q1

√2

13

13

(sin − 1 x)3

+ (cos− 1 x)3

,−π2

π

2

,−π3

8π3

8,

π3

327π3

8

tan− 1(sin2 θ − 2 sin θ + 3)

+ cot− 1(5sec ^ (2y) + 1) = ,π

2cos2 θ − sin θ −1 1

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138


Theproductofallvaluesof satisfyingtheequation

9(b) (c) (d)


139


is (b) (c) (d)


140


If

(b) (c) (d)noneofthese


141


If`y=tan^(-1)1/2+tan^(-1)b,(0

x

sin− 1 cos( )= cot(cot− 1( )) + is

2x2 + 10|x| + 4

x2 + 5|x| + 3

2 − 18|x|

9|x|π

2−9 −3 −1

tan− 1( ) − tan− 1( )xy

x − yx + y

π

2π

3π

4or

π

43π4

+ = ,

thenx2 =

tan− 1(a + x)

a

tan− 1(a − x)

a

π

6

2√3a √3a 2√3a2

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142


If are natural numbers such that then thenumberoforderedtriplets thatsatisfytheequationis0(b)1(c)2(d)Infinitesolutions


143


Thevalueof suchthat aretheanglesofatriangleis

(b) (c) (d)


144


Thenumberofsolutionsoftheequation is2

(b)3(c)1(d)0


145


If

x, y, z cot− 1 x + cot− 1 y = cot− 1 z(x, y, z)

α , , sin− 1 αsin− 1 2

√5

sin − 1 3

√10−1

√2

12

1

√3

1

√2

tan − 1(1 + x) + tan− 1(1 − x) =π

2

cot− 1 x + cot− 1 y + cot− 1 z = , x, y, z

> 0andxy < 1,

π

2

1 1 1

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then is also equal to (b) (d) none of

these


146


If


147


If


148


If

isequalto (b) (c)0(d)noneofthese


149


(d)


x + y + z + +1x

1y

1z

xyz xy + yz + zx

cos − 1 x + cos− 1 y + cos − 1 z = π, then x2 + y2 + z2 + xyz = 0x2 + y2 + z2 + 2xyz = 0 x2 + y2 + z2 + xyz = 1 x2 + y2 + z2 + 2xyz = 1

tan − 1 x + tan− 1 y + tan − 1 z = , thenπ

2x + y + z − xyz = 0

x + y + z + xyz = 0 xy + yz + zx + 1 = 0 xy + yz + zx − 1 = 0

x2 + y2 + z2 = r2, then tan− 1( )+ tan− 1( ) + tan − 1( )

xy

zr

yz

xr

xz

yr

ππ

2

n

∑r= 1

sin − 1( )isequa < o√r − √r − 1√r(r + 1)

tan− 1(√n) −π

4

tan− 1(√n + 1) −π

4tan− 1(√n) tan− 1(√n + 1)

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150


Thevalueof


151


Thesumofseries

is (b) (d)


152


isequalto (b) (c) (d)noneofthese

+ + +

+ ∞equals

tan − 1 47

tan − 1 419

tan − 1 439

tan− 1 467

tan− 1 1 + +tan− 1 1

2tan− 1 1

3tan − 1 + cot− 1 3 cot− 1 1 +

cot − 1 12

cot− 1 13

cot− 1 1 + tan − 1 3

sec− 1 √2 + +

+ + sec− 1 √

sec− 1(√10)3

sec − 1(√50)7

(n2 + 1)(n2 − 2n + 2)

(n2 − n + 1)2

tan− 1 1 n tan− 1(n + 1) tan− 1(n − 1)

tan( + cos− 1 x)+ tan( − cos − 1 x), x ≠ 0,

π

412

π

412

x 2x2x

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153


The value of (b) (c) (d)

independentof


154


If then is (b) (c)

(d)


155


If (a) (b) (c)

(d)



tan − 1( ) − cot− 1( )isx cosθ1 − x sin θ

cosθx − sin θ

2θ θθ

2θ

cot − 1(√cosα) − tan− 1(√cos α) = x, sin xtan2 α

2cot2 α

2tan2 α

cotα2

sin− 1[ ] = tan− 1 x, thenx =12

3 sin 2θ5 + 4 cos 2θ

tan 3θ 3 tan θ

( )tanθ13

3 cot θ

−

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156The value is equal to (b)

(d)


157


If then isequalto1(b)2(c)3(d)


158


If

then isequalto (a) (b) (c) (d)


159


If

then isequalto (b) (c) (d)


160


If

,where then isequalto0(b) (c) (d)noneofthese

2 tan − 1[√ ]a − ba + b

tan θ2

cos− 1( )a cos θ + ba + b cos θ

cos− 1( )a + b cos θa cosθ + b

cos− 1( )a cos θa + b cos θ

cos − 1( )b cos θa cosθ + b

3 tan− 1( ) − = ,12 + √3

tan − 1 1

x

tan− 1 13

x

√2

sin− 1( ) + sin− 1( )= 2 tan− 1 x,

2a1 + a2

2b1 + b2

x [a, b, ∈ (0, 1)]a − b1 + ab

b

1 + abb

1 + aba + b1 − ab

3 sin− 1( ) − 4 cos− 1( )+ 2tan − 2( ) = ,where|x| < 1,

2x1 + x2

1 − x2

1 + x2

2x1 − x2

π

3

x1

√3−

1

√3√3 −

√34

x1 = 2tan − 1( ), x2= sin − 1( )

1 + x1 − x

1 − x2

1 + x2x ∈ (0, 1), x1 + x2 2π π

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161


If

(a) (b) (c) (d)cot


162


Iftheequation`x^3+bx^2+cx+1=0,(bWatchFreeVideoSolutiononDoubtnut

163


Thevalueof isequalto


164


If

where isequalto1(b)2(c)4(d)noneofthese

u = cot− 1 √tanα − tan − 1 √tan α,

then tan( , )isequa < oπ4

u

2√tan α √cot α tan α α

sin − 1[x√1 − x − √x√1 − x2]

sin− 1 x + sin− 1 y + sin− 1 z = π, thenx4

+ y2 + z4 + 4x2y2z2 = K(x2y2 + y2z2

+ z2x2),K

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165


If

isequalto


166


Let =0,thenthenumberofvaluesof satisfying

theequationis1(b)2(c)3(d)4


167


Which of the following is the solution set of the equation

(a)(0.1)(b) (c) (d)


CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRIC

f(x) = sin − 1( x − √1 − x2), −≤ x ≤ 1, thenf(x)

√32

12

12

∣∣∣∣∣

tan − 1 x tan − 1 2x tan − 1 3x

tan − 1 3x tan − 1 x tan − 1 2x

tan − 1 2x tan − 1 3x tan − 1 x

∣∣∣∣∣

x

2 cos− 1 x = cot− 1( )?2x − 12x√1 − x2

( − 1, 1) − {0} ( − 1, 0)

( − 1, 1)

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168

FUNCTIONS_SumAndDifferenceOfAnglesInTermOfSin^-1AndCos^-1

Thenumberofsolutionofequation

is3(b)1(c)2(d)Noneofthese


169


Thesolutionsetoftheequation

is(a) (b) (c) (d)


170


Theequation hasonenegativesolutiononepositivesolution

nosolutionmorethanonesolution


171


If`|cos^(-1)((1-x^2)/(1+x^2))|WatchFreeVideoSolutiononDoubtnut

172


If

for at least one real then (b) (d)



sin− 1 x + n sin− 1(1 − x) = , wheren

> 0,m ≤ 0,

mπ

2

sin− 1 √1 − x2 + cos − 1 x

= − sin− 1 xcot− 1(√1 − x2)

x[ − 1, 1] − {0} (0, 1) ∪ { − 1} ( − 1, 0) ∪ {1} [ − 1, 1]

3− 1x − πx − = 0π

2

22π/sin ( − 1)x − 2(a + 2)π /sin ( − 1 )x + 8a< 0

x, ≤ a < 218

a < 2 a ∈ R − {2}

a ∈ [0, ] ∪ (2, ∞)18

2

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173 If are the roots of equation , then whichfollowingreal?(a) (b) (c) (d)both and


174


is equal to (a) (b) (c)

(d)


175


If isequalto


176


If ,then isequalto(a) (b)

α, β(α < β) 6x2 + 11 = x + 3 = 0cos− 1 α sin − 1 β cos ec− 1α cot− 1 α cot− 1 β

2 tan− 1( − 2) −cos t− 1( )−35

−π +cos − 1 3

5

− + tan− 1( − )π2

34

−π cot− 1( − )34

x < 0, then tan − 1 x

−1 < x < 0 cos − 1 x sec− 1( )1x

π − sin− 1 √1 + x2

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(c) (d) .


177


If

then hasamaximumvalueof2hasamaximumvalueof016differentDarepossiblehasaminimumvalueof


178


Indicatetherelationwhichcanholdintheirrespectivedomainforinfinitevaluesof (b) (d)


179


If where thenthepossiblevaluesof

is(are)3(b)2(c)4(d)8



π + tan− 1( )x√1 − x2

cot− 1( )x√1 − x2

(sin− 1 x + sin − 1w)(sin − 1 y + sin − 1 z)

= π2,D = ∣∣xN1yN3zN3wN4 ∣∣(N1,N2,N3, N4 ∈ N)

−2

x.tan∣∣tan− 1 x∣∣ = |x| cot∣∣cot− 1 x∣∣ = |x| tan− 1|tan x| = |x|sin∣∣sin− 1 x∣∣ = |x|

cot − 1( ) > ,n2 − 10n + 21. 6π

π

6xy < 0 z

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180If where then the possible

valuesof is(are) (b) (c) (d)


181


The value of such that the length of the longest interval in which the

function isconstantis is/are(a)8(b)4

(c)12(d)16


182


Which of the following pairs of function/functions has same graph?


183


If and then(a)

(b) (c) (d)


184


If



z = sec − 1(x + ) + sec − 1(y + ),1x

1y

xy < 0,

z8π10

7π10

9π10

21π20

k(k > 0)f(x) = sin− 1|sinkx| + cos− 1(cos kx)

π

4

y = tan(cos− 1 x); y =√1 − x2

xy = tan(cot− 1 x); y =

1x

y = sin(tan− 1 x); y =x

√1 − x2y = cos(tan − 1 x); y = s ∈ (cot− 1 x)

sin − 1 x + sin− 1 y =π

2sin 2x = cos 2y, x = + √ −π

812

π2

64

y = √ − −12

π2

64π

12x = + √ −π

1212

π2

64y = √ − −1

2π2

64π

8

cos − 1 x + cos− 1 y + cos − 1 z = π, then x2 + y2 + z2 + 2xyz = 12(sin − 1 x + sin− 1 y + sin− 1 z) = cos− 1 x

+ cos− 1 y + cos− 1 z

xy + yz + zx = x + y + z − 1 (x + ) + (y + ) + (z + ) ≥ 61x

1y

1z

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185

FUNCTIONS_SumAndDifferenceOfAnglesInTermOfTan^-1`

If

thevalueof is(a) (b) (c) (d)


186


If isindependentof then (b) (c)`0


187


If

and then(a) for (b) for

(c) for (d) for


188


tan− 1(x2 + 3|x| − 4)

+ cot− 1(4π + sin − 1 sin 14) = , thenπ

2sin− 1 sin 2x 6 − 2π 2π − 6 π − 3 3 − π

2 tan− 1 x +sin− 1(2x)

1 + x2x, x > 1 x < − 1

α = tan − 1( ), β= 2 sin− 1( )

4x − 4x3

1 − 6x2 + x2

2x1 + x2

= k,tanπ

8α + β = π x ∈ [ ]1, 1

kα + β x ∈ ( − k, k)

α + β = π x ∈ [ ]1, 1k

α + β = 0 x ∈ [ − k, k]

2tan(tan− 1(x) + tan− 1(x3)), wherex ∈ R

− { − 1, 1},2x

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isequalto


189


Let

then


190


If

terms, then (c) (d)


191


Equation is satisfied by (a) exactly one value of(b)exactlytwovalueof (c)exactlyonevalueof (d)exactlytwovalueof


2x1 − x2

t(2 tan− 1 x) tan(cot − 1( − x) − cot − 1(x)) tan(2 cot− 1 x)

α = som− 1( ), β = cos− 1( )andγ= tan − 1( )

3685

45

815

cotα + cotβ + cotγ = cotα cotβ cot γtan α tanβ + tan β tanγ + tan α tan γ = 1tan α + tanβ + tanγ = tanα tanβ tan γcotα cotβ + cotβ cot γ + cotα cot γ = 1

Sn = cot− 1(3) + cot − 1(7) + cot − 1(13)

+ cot− 1(21) +..n

S10 = tan − 1( )56 S∞ =π

4S6 = sin− 1( )45

S20 = cot− 1 1. 1

1 + x2 + 2xsin(cos − 1 y) = 0 xx y y

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192


To theequation hasonlyone real

root,then(a) (b) (c) (d)


193


Thesolutionsetofinequality

is thenthevalueof is____


194


If

thenthevalueof is_______


195


Thenumberofvaluesof forwhich

22π/cos− 1 x − (a + )2π /cos −1 x − a2 = 012

1 ≤ a ≤ 3 a ≥ 1 a ≤ − 3 a ≥ 3

(cot− 1 x)(tan− 1 x) + (2 − )cot− 1 x− 3tan − 1 x − 3(2 − ) > 0

π

2π

2(a, b), cot− 1 a + cot− 1 b

x = sin− 1(a6 + 1) + cos− 1(a4 + 1)

− tan− 1(a2 + 1), a ∈ R,sec2 x

x

4 6

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where`0lt=|x|


196


If`0WatchFreeVideoSolutiononDoubtnut

197


If

thenthevalueof is_____.


198


If range of function is thenthevalueof is_______>



sin− 1(x2 − + )+ cos− 1(x4 − + ..) = ,

x4

3x6

9

x8

3x12

9π

2

tan− 1(x + ) − tan − 1(x − )= ,

3x

3x

tan− 1 6x

x4

f(x) = sin− 1 x + 2tan − 1 x + x2 + 4x + 1 [p, q],(p + q)

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199

FUNCTIONS_SumAndDifferenceOfAnglesInTermOfSin^-1AndCos^-1

If

areleastnon-negativeanglessuchthat`uWatchFreeVideoSolutiononDoubtnut

200


Absolutevalueofsumofallintegersinthedomainof

is_______


201


Theleastvalueof is________


202


Let If satisfies the equationthenthevalueof is_________


203


Numberofsolutionsofequation


cos− 1(x) + cos − 1(y) + cos − 1(z)

= π(sec2(u) + sec4(v) + sec6(w)), whereu,v, w

f(x) = cot − 1 √(x + 3)x+ cos− 1 √x2 + 3x + 1

(1 + sec− 1 x)(1 + cas− 1x)

cos− 1(x) + cos− 1(2x) + cos− 1(3x)beπ. xax3 + bx2 + cx − c1 = 0, (b − a − c)

sin(cos− 1(tan(sec− 1 x))) = √1 + xis/are__

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CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER ......6 CENGAGE_MATHS_TRIGONOMETRY_INVERSE TRIGONOMETRIC...

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