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RD Sharma solutions for Class 12 Mathematics chapter 4 - Inverse Trigonometric Functions

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) - Shaalaa.com

Chapter 4: Inverse Trigonometric Functions

Ex. 4.1Ex. 4.2Ex. 4.3Ex. 4.4Ex. 4.5Ex. 4.50Ex. 4.6Ex. 4.7Ex. 4.8Ex. 4.9Ex. 4.10Ex. 4.11Ex. 4.12Ex. 4.13Ex. 4.14Ex. 4.15Ex. 4.16

Chapter 4: Inverse Trigonometric Functions Exercise 4.1 solutions [Pages 6 - 7]

Ex. 4.1 | Q 1.1 | Page 6

Find the principal value of the following:

`sin^-1(-sqrt3/2)`

Ex. 4.1 | Q 1.2 | Page 6

Find the principal value of the following:

`sin^-1(cos  (2pi)/3)`

Ex. 4.1 | Q 1.3 | Page 6

Find the principal value of the following:

`sin^-1((sqrt3-1)/(2sqrt2))`

Ex. 4.1 | Q 1.4 | Page 6

Find the principal value of the following:

`sin^-1((sqrt3+1)/(2sqrt2))`

Ex. 4.1 | Q 1.5 | Page 6

Find the principal value of the following:

`sin^-1(cos  (3pi)/4)`

Ex. 4.1 | Q 1.6 | Page 6

Find the principal value of the following:

`sin^-1(tan  (5pi)/4)`

Ex. 4.1 | Q 2.1 | Page 7

`sin^-1  1/2-2sin^-1  1/sqrt2`

Ex. 4.1 | Q 2.2 | Page 7

`sin^-1{cos(sin^-1  sqrt3/2)}`

Ex. 4.1 | Q 3.1 | Page 7

Find the domain of the following function:

`f(x)=sin^-1x^2`

 

Ex. 4.1 | Q 3.2 | Page 7

Find the domain of the following function:

`f(x) = sin^-1x + sinx`

Ex. 4.1 | Q 3.3 | Page 7

Find the domain of the following function:

`f(x)sin^-1sqrt(x^2-1)`

Ex. 4.1 | Q 3.4 | Page 7

Find the domain of the following function:

`f(x)=sin^-1x+sin^-1 2x`

Ex. 4.1 | Q 4 | Page 7

If `sin^-1 x + sin^-1 y+sin^-1 z+sin^-1 t=2pi` , then find the value of x2 + y2 + z2 + t2 

Ex. 4.1 | Q 5 | Page 7

If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,`  find the value of x2 + y2 + z2 

Chapter 4: Inverse Trigonometric Functions Exercise 4.1, 4.2 solutions [Page 10]

Ex. 4.1 | Q 1 | Page 10

Find the domain of definition of `f(x)=cos^-1(x^2-4)`

Ex. 4.2 | Q 2 | Page 10

Find the domain of  `f(x) =2cos^-1 2x+sin^-1x.`

Ex. 4.2 | Q 3 | Page 10

Find the domain of `f(x)=cos^-1x+cosx.`

Ex. 4.2 | Q 4.1 | Page 10

​Find the principal values of the following:
`cos^-1(-sqrt3/2)`

Ex. 4.2 | Q 4.2 | Page 10

​Find the principal values of the following:

`cos^-1(-1/sqrt2)`

Ex. 4.2 | Q 4.3 | Page 10

​Find the principal values of the following:

`cos^-1(sin   (4pi)/3)`

Ex. 4.2 | Q 4.4 | Page 10

​Find the principal values of the following:

`cos^-1(tan  (3pi)/4)`

Ex. 4.2 | Q 5.1 | Page 10

For the principal value, evaluate of the following:

`cos^-1  1/2+2sin^-1  (1/2)`

Ex. 4.2 | Q 5.3 | Page 10

For the principal value, evaluate of the following:

`sin^-1(-1/2)+2cos^-1(-sqrt3/2)`

Ex. 4.2 | Q 5.4 | Page 10

For the principal value, evaluate of the following:

`sin^-1(-sqrt3/2)+cos^-1(sqrt3/2)`

Chapter 4: Inverse Trigonometric Functions Exercise 4.3 solutions [Page 14]

Ex. 4.3 | Q 1.1 | Page 14

Find the principal value of the following:

`tan^-1(1/sqrt3)`

Ex. 4.3 | Q 1.2 | Page 14

Find the principal value of the following:

`tan^-1(-1/sqrt3)`

Ex. 4.3 | Q 1.3 | Page 14

Find the principal value of the following:

`tan^-1(cos  pi/2)`

Ex. 4.3 | Q 1.4 | Page 14

Find the principal value of the following:

`tan^-1(2cos  (2pi)/3)`

Ex. 4.3 | Q 2.1 | Page 14

For the principal value, evaluate of the following:

`tan^-1(-1)+cos^-1(-1/sqrt2)`

Ex. 4.3 | Q 2.2 | Page 14

For the principal value, evaluate of the following:

`tan^-1{2sin(4cos^-1  sqrt3/2)}`

Ex. 4.3 | Q 3.1 | Page 14

Evaluate the following:

`tan^-1 1+cos^-1 (-1/2)+sin^-1(-1/2)`

Ex. 4.3 | Q 3.2 | Page 14

Evaluate the following:

`tan^-1(-1/sqrt3)+tan^-1(-sqrt3)+tan^-1(sin(-pi/2))`

Ex. 4.3 | Q 3.3 | Page 14

Evaluate the following:

`tan^-1(tan  (5pi)/6)+cos^-1{cos((13pi)/6)}`

Chapter 4: Inverse Trigonometric Functions Exercise 4.4 solutions [Page 18]

Ex. 4.4 | Q 1.1 | Page 18

Find the principal value of the following:

`sec^-1(-sqrt2)`

Ex. 4.4 | Q 1.2 | Page 18

Find the principal value of the following:

`sec^-1(2)`

Ex. 4.4 | Q 1.3 | Page 18

Find the principal value of the following:

`sec^-1(2sin  (3pi)/4)`

Ex. 4.4 | Q 1.4 | Page 18

Find the principal value of the following:

`sec^-1(2tan  (3pi)/4)`

Ex. 4.4 | Q 2.1 | Page 18

For the principal value, evaluate the following:

`tan^-1sqrt3-sec^-1(-2)`

Ex. 4.4 | Q 2.2 | Page 18

For the principal value, evaluate the following:

`sin^-1(-sqrt3/2)-2sec^-1(2tan  pi/6)`

Ex. 4.4 | Q 3.1 | Page 18

Find the domain of `sec^(-1)(3x-1)`.

Ex. 4.4 | Q 3.2 | Page 18

Find the domain of `sec^(-1) x-tan^(-1)x`

Chapter 4: Inverse Trigonometric Functions Exercise 4.5, 4.50 solutions [Page 21]

Ex. 4.5 | Q 1.1 | Page 21

​Find the principal value of the following:

`cosec^-1(-sqrt2)`

Ex. 4.5 | Q 1.2 | Page 21

​Find the principal value of the following:

cosec-1(-2)

Ex. 4.5 | Q 1.3 | Page 21

​Find the principal value of the following:

`\text(cosec)^-1(2/sqrt3)`

Ex. 4.5 | Q 1.4 | Page 21

​Find the principal value of the following:

`cosec^-1(2cos  (2pi)/3)`

Ex. 4.5 | Q 2 | Page 21

Find the set of values of `cosec^-1(sqrt3/2)`

Ex. 4.5 | Q 3.1 | Page 21

For the principal value, evaluate the following:

`sin^-1(-sqrt3/2)+\text{cosec}^-1(-2/sqrt3)`

Ex. 4.50 | Q 3.2 | Page 21

For the principal value, evaluate the following:

`sec^-1(sqrt2)+2\text{cosec}^-1(-sqrt2)`

Ex. 4.5 | Q 3.3 | Page 21

For the principal value, evaluate the following:

`sin^-1[cos{2\text(cosec)^-1(-2)}]`

Ex. 4.5 | Q 3.4 | Page 21

For the principal value, evaluate the following:

`cosec^-1(2tan  (11pi)/6)`

Chapter 4: Inverse Trigonometric Functions Exercise 4.6 solutions [Page 24]

Ex. 4.6 | Q 1.1 | Page 24

Find the principal value of the following:

`cot^-1(-sqrt3)`

Ex. 4.6 | Q 1.2 | Page 24

Find the principal value of the following:

`cot^-1(sqrt3)`

Ex. 4.6 | Q 1.3 | Page 24

Find the principal value of the following:

`cot^-1(-1/sqrt3)`

Ex. 4.6 | Q 1.4 | Page 24

Find the principal value of the following:

`cot^-1(tan  (3pi)/4)`

Ex. 4.6 | Q 2 | Page 24

Find the domain of `f(x)=cotx+cot^-1x`

Ex. 4.6 | Q 3.1 | Page 24

Evaluate the following:

`cot^-1  1/sqrt3-\text(cosec)^-1(-2)+sec^-1(2/sqrt3)`

Ex. 4.6 | Q 3.2 | Page 24

Evaluate the following:

`cot^-1{2cos(sin^-1  sqrt3/2)}`

Ex. 4.6 | Q 3.3 | Page 24

Evaluate the following:

`\text(cosec)^-1(-2/sqrt3)+2cot^-1(-1)`

Ex. 4.6 | Q 3.4 | Page 24

Evaluate the following:

`tan^-1(-1/sqrt3)+cot^-1(1/sqrt3)+tan^-1(sin(-pi/2))`

Chapter 4: Inverse Trigonometric Functions Exercise 4.7 solutions [Pages 42 - 43]

Ex. 4.7 | Q 1.01 | Page 42

`sin^-1(sin  pi/6)`

Ex. 4.7 | Q 1.02 | Page 42

`sin^-1(sin  (7pi)/6)`

Ex. 4.7 | Q 1.03 | Page 42

`sin^-1(sin  (5pi)/6)`

Ex. 4.7 | Q 1.04 | Page 42

`sin^-1(sin  (13pi)/7)`

Ex. 4.7 | Q 1.05 | Page 42

`sin^-1(sin  (17pi)/8)`

Ex. 4.7 | Q 1.06 | Page 42

`sin^-1{(sin - (17pi)/8)}`

Ex. 4.7 | Q 1.07 | Page 42

`sin^-1(sin3)`

Ex. 4.7 | Q 1.08 | Page 42

`sin^-1(sin4)`

Ex. 4.7 | Q 1.09 | Page 42

`sin^-1(sin12)`

Ex. 4.7 | Q 1.1 | Page 42

`sin^-1(sin2)`

Ex. 4.7 | Q 2.1 | Page 42

Evaluate the following:

`cos^-1{cos(-pi/4)}`

Ex. 4.7 | Q 2.2 | Page 42

Evaluate the following:

`cos^-1{cos  (5pi)/4}`

Ex. 4.7 | Q 2.3 | Page 42

Evaluate the following:

`cos^-1{cos  ((4pi)/3)}`

Ex. 4.7 | Q 2.4 | Page 42

Evaluate the following:

`cos^-1{cos  (13pi)/6}`

Ex. 4.7 | Q 2.5 | Page 42

Evaluate the following:

`cos^-1(cos3)`

Ex. 4.7 | Q 2.6 | Page 42

Evaluate the following:

`cos^-1(cos4)`

Ex. 4.7 | Q 2.7 | Page 42

Evaluate the following:

`cos^-1(cos5)`

Ex. 4.7 | Q 2.8 | Page 42

Evaluate the following:

`cos^-1(cos12)`

Ex. 4.7 | Q 3.1 | Page 42

Evaluate the following:

`tan^-1(tan  pi/3)`

Ex. 4.7 | Q 3.2 | Page 42

Evaluate the following:

`tan^-1(tan  (6pi)/7)`

Ex. 4.7 | Q 3.3 | Page 42

Evaluate the following:

`tan^-1(tan  (7pi)/6)`

Ex. 4.7 | Q 3.4 | Page 42

Evaluate the following:

`tan^-1(tan  (9pi)/4)`

Ex. 4.7 | Q 3.5 | Page 42

Evaluate the following:

`tan^-1(tan1)`

Ex. 4.7 | Q 3.6 | Page 42

Evaluate the following:

`tan^-1(tan2)`

Ex. 4.7 | Q 3.7 | Page 42

Evaluate the following:

`tan^-1(tan4)`

Ex. 4.7 | Q 3.8 | Page 42

Evaluate the following:

`tan^-1(tan12)`

Ex. 4.7 | Q 4.1 | Page 42

Evaluate the following:

`sec^-1(sec  pi/3)`

Ex. 4.7 | Q 4.2 | Page 42

Evaluate the following:

`sec^-1(sec  (2pi)/3)`

Ex. 4.7 | Q 4.3 | Page 42

Evaluate the following:

`sec^-1(sec  (5pi)/4)`

Ex. 4.7 | Q 4.4 | Page 42

Evaluate the following:

`sec^-1(sec  (7pi)/3)`

Ex. 4.7 | Q 4.5 | Page 42

Evaluate the following:

`sec^-1(sec  (9pi)/5)`

Ex. 4.7 | Q 4.6 | Page 42

Evaluate the following:

`sec^-1{sec  (-(7pi)/3)}`

Ex. 4.7 | Q 4.7 | Page 42

Evaluate the following:

`sec^-1(sec  (13pi)/4)`

Ex. 4.7 | Q 4.8 | Page 42

Evaluate the following:

`sec^-1(sec  (25pi)/6)`

Ex. 4.7 | Q 5.1 | Page 42

Evaluate the following:

`\text(cosec)^-1(\text{cosec}  pi/4)`

Ex. 4.7 | Q 5.2 | Page 42

Evaluate the following:

`cosec^-1(cosec  (3pi)/4)`

Ex. 4.7 | Q 5.3 | Page 42

Evaluate the following:

`cosec^-1(cosec  (6pi)/5)`

Ex. 4.7 | Q 5.4 | Page 42

Evaluate the following:

`cosec^-1(cosec  (11pi)/6)`

Ex. 4.7 | Q 5.5 | Page 42

Evaluate the following:

`cosec^-1(cosec  (13pi)/6)`

Ex. 4.7 | Q 5.6 | Page 42

Evaluate the following:

`cosec^-1{cosec  (-(9pi)/4)}`

Ex. 4.7 | Q 6.1 | Page 43

Evaluate the following:

`cot^-1(cot  pi/3)`

Ex. 4.7 | Q 6.2 | Page 43

Evaluate the following:

`cot^-1(cot  (4pi)/3)`

Ex. 4.7 | Q 6.3 | Page 43

Evaluate the following:

`cot^-1(cot  (9pi)/4)`

Ex. 4.7 | Q 6.4 | Page 43

Evaluate the following:

`cot^-1(cot  (19pi)/6)`

Ex. 4.7 | Q 6.5 | Page 43

Evaluate the following:

`cot^-1{cot (-(8pi)/3)}`

Ex. 4.7 | Q 6.6 | Page 43

Evaluate the following:

`cot^-1{cot  ((21pi)/4)}`

Ex. 4.7 | Q 7.01 | Page 43

Write the following in the simplest form:

`cot^-1  a/sqrt(x^2-a^2),|  x  | > a`

Ex. 4.7 | Q 7.02 | Page 43

Write the following in the simplest form:

`tan^-1{x+sqrt(1+x^2)},x in R `

Ex. 4.7 | Q 7.03 | Page 43

Write the following in the simplest form:

`tan^-1{sqrt(1+x^2)-x},x in R `

Ex. 4.7 | Q 7.04 | Page 43

Write the following in the simplest form:

`tan^-1{(sqrt(1+x^2)-1)/x},x !=0`

Ex. 4.7 | Q 7.05 | Page 43

Write the following in the simplest form:

`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`

Ex. 4.7 | Q 7.06 | Page 43

Write the following in the simplest form:

`tan^-1sqrt((a-x)/(a+x)),-a<x<a`

Ex. 4.7 | Q 7.07 | Page 43

Write the following in the simplest form:

`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`

Ex. 4.7 | Q 7.08 | Page 43

Write the following in the simplest form:

`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`

Ex. 4.7 | Q 7.09 | Page 43

Write the following in the simplest form:

`sin^-1{(sqrt(1+x)+sqrt(1-x))/2},0<x<1`

Ex. 4.7 | Q 7.1 | Page 43

Write the following in the simplest form:

`sin{2tan^-1sqrt((1-x)/(1+x))}`

Chapter 4: Inverse Trigonometric Functions Exercise 4.8 solutions [Page 54]

Ex. 4.8 | Q 1.1 | Page 54

Evaluate the following:

`sin(sin^-1  7/25)`

 

Ex. 4.8 | Q 1.2 | Page 54

Evaluate the following:

`sin(cos^-1  5/13)`

Ex. 4.8 | Q 1.3 | Page 54

Evaluate the following:

`sin(tan^-1  24/7)`

Ex. 4.8 | Q 1.4 | Page 54

Evaluate the following:

`sin(sec^-1  17/8)`

Ex. 4.8 | Q 1.5 | Page 54

Evaluate the following:

`cosec(cos^-1  3/5)`

Ex. 4.8 | Q 1.6 | Page 54

Evaluate the following:

`sec(sin^-1  12/13)`

Ex. 4.8 | Q 1.7 | Page 54

Evaluate the following:

`tan(cos^-1  8/17)`

Ex. 4.8 | Q 1.8 | Page 54

Evaluate the following:

`cot(cos^-1  3/5)`

Ex. 4.8 | Q 1.9 | Page 54

Evaluate the following:

`cos(tan^-1  24/7)`

Ex. 4.8 | Q 2.1 | Page 54

Prove the following result

`tan(cos^-1  4/5+tan^-1  2/3)=17/6`

Ex. 4.8 | Q 2.2 | Page 54

Prove the following result

`cos(sin^-1  3/5+cot^-1  3/2)=6/(5sqrt13)`

Ex. 4.8 | Q 2.3 | Page 54

Prove the following result

`=tan(sin^-1  5/13+cos^-1  3/5) = 63/16`

Ex. 4.8 | Q 2.4 | Page 54

Prove the following result

`sin(cos^-1  3/5+sin^-1  5/13)=63/65`

Ex. 4.8 | Q 3 | Page 54

Solve: `cos(sin^-1x)=1/6`

Chapter 4: Inverse Trigonometric Functions Exercise 4.9 solutions [Pages 58 - 59]

Ex. 4.9 | Q 1.1 | Page 58

Evaluate:

`cos{sin^-1(-7/25)}`

Ex. 4.9 | Q 1.2 | Page 58

Evaluate:

`sec{cot^-1(-5/12)}`

Ex. 4.9 | Q 1.3 | Page 58

Evaluate:

`cot{sec^-1(-13/5)}`

Ex. 4.9 | Q 2.1 | Page 58

Evaluate:

`tan{cos^-1(-7/25)}`

Ex. 4.9 | Q 2.2 | Page 58

Evaluate:

`cosec{cot^-1(-12/5)}`

Ex. 4.9 | Q 2.3 | Page 58

Evaluate:

`cos(tan^-1  3/4)`

Ex. 4.9 | Q 3 | Page 59

Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`

Chapter 4: Inverse Trigonometric Functions Exercise 4.10 solutions [Page 66]

Ex. 4.10 | Q 1.1 | Page 66

Evaluate: 

`cot(sin^-1  3/4+sec^-1  4/3)`

Ex. 4.10 | Q 1.2 | Page 66

Evaluate:

`sin(tan^-1x+tan^-1  1/x)` for x < 0

Ex. 4.10 | Q 1.3 | Page 66

Evaluate:

`sin(tan^-1x+tan^-1  1/x)` for x > 0

Ex. 4.10 | Q 1.4 | Page 66

Evaluate:

`cot(tan^-1a+cot^-1a)`

Ex. 4.10 | Q 1.5 | Page 66

Evaluate:

`cos(sec^-1x+\text(cosec)^-1x)`,|x|≥1

Ex. 4.10 | Q 2 | Page 66

If `cos^-1x + cos^-1y =pi/4,`  find the value of `sin^-1x+sin^-1y`

Ex. 4.10 | Q 3 | Page 66

If `sin^-1x+sin^-1y=pi/3`  and  `cos^-1x-cos^-1y=pi/6`,  find the values of x and y.

Ex. 4.10 | Q 4 | Page 66

If `cot(cos^-1  3/5+sin^-1x)=0`, find the values of x.

Ex. 4.10 | Q 5 | Page 66

If `(sin^-1x)^2+(cos^-1x)^2=(17pi^2)/36,`  Find x

Ex. 4.10 | Q 6 | Page 66

`sin(sin^-1  1/5+cos^-1x)=1`

Ex. 4.10 | Q 7 | Page 66

`sin^-1x=pi/6+cos^-1x`

Ex. 4.10 | Q 8 | Page 66

`4sin^-1x=pi-cos^-1x`

Ex. 4.10 | Q 9 | Page 66

`tan^-1x+2cot^-1x=(2x)/3`

Ex. 4.10 | Q 10 | Page 66

`5tan^-1x+3cot^-1x=2x`

Chapter 4: Inverse Trigonometric Functions Exercise 4.11 solutions [Page 82]

Ex. 4.11 | Q 1.1 | Page 82

Prove the following result:

`tan^-1  1/7+tan^-1  1/13=tan^-1  2/9`

Ex. 4.11 | Q 1.2 | Page 82

Prove the following result:

`sin^-1  12/13+cos^-1  4/5+tan^-1  63/16=pi`

Ex. 4.11 | Q 1.3 | Page 82

Prove the following result:

`tan^-1  1/4+tan^-1  2/9=sin^-1  1/sqrt5`

Ex. 4.11 | Q 2 | Page 82

Find the value of `tan^-1  (x/y)-tan^-1((x-y)/(x+y))`

Ex. 4.11 | Q 3.01 | Page 82

Solve the following equation for x:

`tan^-1  2x+tan^-1  3x = npi+(3pi)/4`

Ex. 4.11 | Q 3.02 | Page 82

Solve the following equation for x:

tan−1(x + 1) + tan−1(x − 1) = tan−1`8/31`

Ex. 4.11 | Q 3.03 | Page 82

Solve the following equation for x:

tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x

Ex. 4.11 | Q 3.04 | Page 82

Solve the following equation for x:

 tan−1`((1-x)/(1+x))-1/2` tan−1x = 0, where x > 0

Ex. 4.11 | Q 3.05 | Page 82

Solve the following equation for x:

 cot−1x − cot−1(x + 2) =`pi/12`, > 0

Ex. 4.11 | Q 3.06 | Page 82

Solve the following equation for x:

tan−1(x + 2) + tan−1(x − 2) = tan−1 `(8/79)`, x > 0

Ex. 4.11 | Q 3.07 | Page 82

Solve the following equation for x:

`tan^-1  x/2+tan^-1  x/3=pi/4, 0<x<sqrt6`

Ex. 4.11 | Q 3.08 | Page 82

Solve the following equation for x:

`tan^-1((x-2)/(x-4))+tan^-1((x+2)/(x+4))=pi/4`

Ex. 4.11 | Q 3.09 | Page 82

Solve the following equation for x:

`tan^-1(2+x)+tan^-1(2-x)=tan^-1  2/3, where  x< -sqrt3 or, x>sqrt3`

Ex. 4.11 | Q 3.1 | Page 82

Solve the following equation for x:

`tan^-1  (x-2)/(x-1)+tan^-1  (x+2)/(x+1)=pi/4`

Ex. 4.11 | Q 4 | Page 82

Sum the following series:

`tan^-1  1/3+tan^-1  2/9+tan^-1  4/33+...+tan^-1  (2^(n-1))/(1+2^(2n-1))`

Chapter 4: Inverse Trigonometric Functions Exercise 4.12 solutions [Page 89]

Ex. 4.12 | Q 1 | Page 89

Evaluate: `cos(sin^-1  3/5+sin^-1  5/13)`

Ex. 4.12 | Q 2.1 | Page 89

`sin^-1  63/65=sin^-1  5/13+cos^-1  3/5`

Ex. 4.12 | Q 2.2 | Page 89

`sin^-1  5/13+cos^-1  3/5=tan^-1  63/16`

Ex. 4.12 | Q 2.3 | Page 89

`(9pi)/8-9/4sin^-1  1/3=9/4sin^-1  (2sqrt2)/3`

Ex. 4.12 | Q 3.1 | Page 89

Solve the following:

`sin^-1x+sin^-1  2x=pi/3`

Ex. 4.12 | Q 3.2 | Page 89

Solve the following:

`cos^-1x+sin^-1  x/2=pi/6`

Chapter 4: Inverse Trigonometric Functions Exercise 4.13 solutions [Page 92]

Ex. 4.13 | Q 1 | Page 92

If `cos^-1  x/2+cos^-1  y/3=alpha,` then prove that  `9x^2-12xy cosa+4y^2=36sin^2a.`

Ex. 4.13 | Q 2 | Page 92

Solve the equation `cos^-1  a/x-cos^-1  b/x=cos^-1  1/b-cos^-1  1/a`

Ex. 4.13 | Q 3 | Page 92

Solve `cos^-1sqrt3x+cos^-1x=pi/2`

Ex. 4.13 | Q 4 | Page 92

Prove that: `cos^-1  4/5+cos^-1  12/13=cos^-1  33/65`

Chapter 4: Inverse Trigonometric Functions Exercise 4.14 solutions [Pages 115 - 116]

Ex. 4.14 | Q 1.1 | Page 115

Evaluate the following:

`tan{2tan^-1  1/5-pi/4}`

Ex. 4.14 | Q 1.2 | Page 115

Evaluate the following:

`tan  1/2(cos^-1  sqrt5/3)`

Ex. 4.14 | Q 1.3 | Page 115

Evaluate the following:

`sin(1/2cos^-1  4/5)`

Ex. 4.14 | Q 1.4 | Page 115

Evaluate the following:

`sin(2tan^-1  2/3)+cos(tan^-1sqrt3)`

Ex. 4.14 | Q 2.01 | Page 115

`2sin^-1  3/5=tan^-1  24/7`

Ex. 4.14 | Q 2.02 | Page 115

`tan^-1  1/4+tan^-1  2/9=1/2cos^-1  3/2=1/2sin^-1(4/5)`

Ex. 4.14 | Q 2.03 | Page 115

`tan^-1  2/3=1/2tan^-1  12/5`

Ex. 4.14 | Q 2.04 | Page 115

`tan^-1  1/7+2tan^-1  1/3=pi/4`

Ex. 4.14 | Q 2.05 | Page 115

`sin^-1  4/5+2tan^-1  1/3=pi/2`

Ex. 4.14 | Q 2.06 | Page 115

`2sin^-1  3/5-tan^-1  17/31=pi/4`

Ex. 4.14 | Q 2.07 | Page 115

`2tan^-1  1/5+tan^-1  1/8=tan^-1  4/7`

Ex. 4.14 | Q 2.08 | Page 115

`2tan^-1  3/4-tan^-1  17/31=pi/4`

Ex. 4.14 | Q 2.09 | Page 115

`2tan^-1(1/2)+tan^-1(1/7)=tan^-1(31/17)`

Ex. 4.14 | Q 2.1 | Page 115

`4tan^-1  1/5-tan^-1  1/239=pi/4`

Ex. 4.14 | Q 3 | Page 115

If `sin^-1  (2a)/(1+a^2)-cos^-1  (1-b^2)/(1+b^2)=tan^-1  (2x)/(1-x^2)`, then prove that `x=(a-b)/(1+ab)`

Ex. 4.14 | Q 4.1 | Page 115

Prove that

`tan^-1((1-x^2)/(2x))+cot^-1((1-x^2)/(2x))=pi/2`

Ex. 4.14 | Q 4.2 | Page 115

Prove that

`sin{tan^-1  (1-x^2)/(2x)+cos^-1  (1-x^2)/(2x)}=1`

Ex. 4.14 | Q 5 | Page 115

If `sin^-1  (2a)/(1+a^2)+sin^-1  (2b)/(1+b^2)=2tan^-1x,` Prove that  `x=(a+b)/(1-ab).`

Ex. 4.14 | Q 6 | Page 115

Show that `2tan^-1x+sin^-1  (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.

Ex. 4.14 | Q 7.1 | Page 116

Find the value of the following:

`tan^-1{2cos(2sin^-1  1/2)}`

Ex. 4.14 | Q 7.2 | Page 116

Find the value of the following:

`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1

Ex. 4.14 | Q 8.1 | Page 116

Solve the following equation for x:

`tan^-1  1/4+2tan^-1  1/5+tan^-1  1/6+tan^-1  1/x=pi/4`

Ex. 4.14 | Q 8.2 | Page 116

Solve the following equation for x:

`3sin^-1  (2x)/(1+x^2)-4cos^-1  (1-x^2)/(1+x^2)+2tan^-1  (2x)/(1-x^2)=pi/3`

Ex. 4.14 | Q 8.3 | Page 116

Solve the following equation for x:

`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`

Ex. 4.14 | Q 8.4 | Page 116

Solve the following equation for x:

`2tan^-1(sinx)=tan^-1(2sinx),x!=pi/2`

Ex. 4.14 | Q 8.5 | Page 116

Solve the following equation for x:

`cos^-1((x^2-1)/(x^2+1))+1/2tan^-1((2x)/(1-x^2))=(2x)/3`

Ex. 4.14 | Q 8.6 | Page 116

Solve the following equation for x:

`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`

Ex. 4.14 | Q 9 | Page 116

Prove that `2tan^-1(sqrt((a-b)/(a+b))tan  theta/2)=cos^-1((a costheta+b)/(a+b costheta))`

Ex. 4.14 | Q 10 | Page 116

Prove that:

`tan^-1  (2ab)/(a^2-b^2)+tan^-1  (2xy)/(x^2-y^2)=tan^-1  (2alphabeta)/(alpha^2-beta^2),`   where `alpha=ax-by  and  beta=ay+bx.`

Ex. 4.14 | Q 11 | Page 116

For any a, b, x, y > 0, prove that:

`2/3tan^-1((3ab^2-a^3)/(b^3-3a^2b))+2/3tan^-1((3xy^2-x^3)/(y^3-3x^2y))=tan^-1  (2alphabeta)/(alpha^2-beta^2)`

`where  alpha =-ax+by, beta=bx+ay`

Chapter 4: Inverse Trigonometric Functions Exercise 4.15 solutions [Pages 116 - 119]

Ex. 4.15 | Q 1 | Page 116

Write the value of `sin^-1((-sqrt3)/2)+cos^-1((-1)/2)`

Ex. 4.15 | Q 2 | Page 116

Write the difference between maximum and minimum values of  sin−1 x for x ∈ [− 1, 1].

Ex. 4.15 | Q 3 | Page 116

If `sin^-1x+sin^-1y+sin^-1z=(3pi)/2,`  then write the value of x + y + z.

Ex. 4.15 | Q 4 | Page 117

If x > 1, then write the value of sin−1 `((2x)/(1+x^2))` in terms of tan−1 x.

Ex. 4.15 | Q 5 | Page 117

If x < 0, then write the value of cos−1 `((1-x^2)/(1+x^2))` in terms of tan−1 x.

Ex. 4.15 | Q 6 | Page 117

Write the value of tan1x + tan−1 `(1/x)`for x > 0.

Ex. 4.15 | Q 7 | Page 117

Write the value of tan1 x + tan−1 `(1/x)`  for x < 0.

Ex. 4.15 | Q 8 | Page 117

What is the value of cos−1 `(cos  (2x)/3)+sin^-1(sin  (2x)/3)?`

Ex. 4.15 | Q 9 | Page 117

If −1 < x < 0, then write the value of `sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))`

Ex. 4.15 | Q 10 | Page 117

Write the value of sin (cot−1 x).

Ex. 4.15 | Q 11 | Page 117

Write the value of

\[\cos^{- 1} \left( \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\].

Ex. 4.15 | Q 12 | Page 117

Write the range of tan−1 x.

Ex. 4.15 | Q 13 | Page 117

Write the value of cos−1 (cos 1540°).

Ex. 4.15 | Q 14 | Page 117

Write the value of sin−1

\[\left( \sin( -{600}°) \right)\].

 

 

Ex. 4.15 | Q 15 | Page 117

Write the value of cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]

Ex. 4.15 | Q 16 | Page 117

Write the value of sin1 (sin 1550°).

Ex. 4.15 | Q 17 | Page 117

Evaluate sin

\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]

Ex. 4.15 | Q 18 | Page 117

Evaluate sin \[\left( \tan^{- 1} \frac{3}{4} \right)\]

Ex. 4.15 | Q 19 | Page 117

Write the value of cos−1 \[\left( \tan\frac{3\pi}{4} \right)\]

Ex. 4.15 | Q 20 | Page 117

Write the value of cos \[\left( 2 \sin^{- 1} \frac{1}{2} \right)\]

Ex. 4.15 | Q 21 | Page 117

Write the value of cos1 (cos 350°) − sin−1 (sin 350°)

Ex. 4.15 | Q 22 | Page 117

Write the value of cos\[\left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right)\]

Ex. 4.15 | Q 23 | Page 117

If tan−1 x + tan−1 y = `pi/4`,  then write the value of x + y + xy.

Ex. 4.15 | Q 24 | Page 117

Write the value of cos−1 (cos 6).

Ex. 4.15 | Q 25 | Page 117

Write the value of sin−1 \[\left( \cos\frac{\pi}{9} \right)\]

Ex. 4.15 | Q 26 | Page 117

Write the value of sin \[\left\{ \frac{\pi}{3} - \sin^{- 1} \left( - \frac{1}{2} \right) \right\}\]

Ex. 4.15 | Q 27 | Page 117

Write the value of tan1\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]

Ex. 4.15 | Q 28 | Page 118

Write the value ofWrite the value of \[2 \sin^{- 1} \frac{1}{2} + \cos^{- 1} \left( - \frac{1}{2} \right)\]

Ex. 4.15 | Q 29 | Page 118

Write the value of \[\tan^{- 1} \frac{a}{b} - \tan^{- 1} \left( \frac{a - b}{a + b} \right)\]

Ex. 4.15 | Q 30 | Page 118

Write the value of cos−1 \[\left( \cos\frac{5\pi}{4} \right)\]

Ex. 4.15 | Q 31 | Page 118

Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]

Ex. 4.15 | Q 32 | Page 118

Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]

Ex. 4.15 | Q 33 | Page 118

If \[\tan^{- 1} (\sqrt{3}) + \cot^{- 1} x = \frac{\pi}{2},\] find x.

Ex. 4.15 | Q 34 | Page 118

If \[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2},\] then find x.

 

Ex. 4.15 | Q 35 | Page 118

Write the value of \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right)\]

Ex. 4.15 | Q 36 | Page 118

If 4 sin−1 x + cos−1 x = π, then what is the value of x?

Ex. 4.15 | Q 37 | Page 118

If x < 0, y < 0 such that xy = 1, then write the value of tan1 x + tan−1 y.

Ex. 4.15 | Q 38 | Page 118

What is the principal value of `sin^-1(-sqrt3/2)?`

Ex. 4.15 | Q 39 | Page 118

Write the principal value of `sin^-1(-1/2)`

Ex. 4.15 | Q 40 | Page 118

Write the principal value of \[\cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right)\]

Ex. 4.15 | Q 41 | Page 118

Write the value of \[\tan\left( 2 \tan^{- 1} \frac{1}{5} \right)\]

Ex. 4.15 | Q 42 | Page 118

Write the principal value of \[\tan^{- 1} 1 + \cos^{- 1} \left( - \frac{1}{2} \right)\]

Ex. 4.15 | Q 43 | Page 118

Write the value of \[\tan^{- 1} \left\{ 2\sin\left( 2 \cos^{- 1} \frac{\sqrt{3}}{2} \right) \right\}\]

Ex. 4.15 | Q 44 | Page 118

Write the principal value of `tan^-1sqrt3+cot^-1sqrt3`

Ex. 4.15 | Q 45 | Page 118

Write the principal value of \[\cos^{- 1} \left( \cos680^\circ  \right)\]

Ex. 4.15 | Q 46 | Page 118

Write the value of \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]

Ex. 4.15 | Q 47 | Page 118

Write the value of \[\sec^{- 1} \left( \frac{1}{2} \right)\]

Ex. 4.15 | Q 48 | Page 118

Write the value of \[\cos^{- 1} \left( \cos\frac{14\pi}{3} \right)\]

Ex. 4.15 | Q 49 | Page 118

Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]

Ex. 4.15 | Q 50 | Page 118

Wnte the value of the expression \[\tan\left( \frac{\sin^{- 1} x + \cos^{- 1} x}{2} \right), \text { when } x = \frac{\sqrt{3}}{2}\]

Ex. 4.15 | Q 51 | Page 119

Write the principal value of \[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\}\]

Ex. 4.15 | Q 52 | Page 119

The set of values of `\text(cosec)^-1(sqrt3/2)`

Ex. 4.15 | Q 53 | Page 119

Write the value of  \[\tan^{- 1} \left( \frac{1}{x} \right)\]  for x < 0 in terms of `cot^-1x`

Ex. 4.15 | Q 54 | Page 119

Write the value of  `cot^-1(-x)`  for all `x in R` in terms of `cot^-1(x)`

Ex. 4.15 | Q 55 | Page 119

Wnte the value of\[\cos\left( \frac{\tan^{- 1} x + \cot^{- 1} x}{3} \right), \text{ when } x = - \frac{1}{\sqrt{3}}\]

Ex. 4.15 | Q 56 | Page 119

If \[\cos\left( \tan^{- 1} x + \cot^{- 1} \sqrt{3} \right) = 0\] , find the value of x.

 
Ex. 4.15 | Q 57 | Page 119

Find the value of \[2 \sec^{- 1} 2 + \sin^{- 1} \left( \frac{1}{2} \right)\]

Ex. 4.15 | Q 58 | Page 119

If \[\cos\left( \sin^{- 1} \frac{2}{5} + \cos^{- 1} x \right) = 0\], find the value of x.

 
Ex. 4.15 | Q 59 | Page 119

Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]

Ex. 4.15 | Q 60 | Page 119

Find the value of \[\tan^{- 1} \left( \tan\frac{9\pi}{8} \right)\]

Chapter 4: Inverse Trigonometric Functions Exercise 4.16 solutions [Pages 119 - 122]

Ex. 4.16 | Q 1 | Page 119

If \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}} \right)\]  = α, then x2 =



  • sin 2 α

  • sin α

  • cos 2 α

  • cos α

Ex. 4.16 | Q 2 | Page 120

The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is

 

  • `sqrt29/3`

  • `29/3`

  • `sqrt3/29`

  • `3/29`

Ex. 4.16 | Q 3 | Page 120

2 tan−1 {cosec (tan−1 x) − tan (cot1 x)} is equal to

  • cot−1 x

  • cot−1`1/x`

  • tan−1 x

  • none of these

Ex. 4.16 | Q 4 | Page 120

If  \[\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = \alpha, then\frac{x^2}{a^2} - \frac{2xy}{ab}\cos \alpha + \frac{y^2}{b^2} = \]

  • sin2 α

  • cos2 α

  • tan2 α

  • cot2 α

Ex. 4.16 | Q 5 | Page 120

The positive integral solution of the equation
\[\tan^{- 1} x + \cos^{- 1} \frac{y}{\sqrt{1 + y^2}} = \sin^{- 1} \frac{3}{\sqrt{10}}\text{ is }\]

  •  x = 1, y = 2

  •  x = 2, y = 1

  •  x = 3, y = 2

  • x = −2, y = −1

Ex. 4.16 | Q 6 | Page 120

If sin−1 − cos−1 x = `pi/6` , then x = 

 
  • `1/2`

  • `sqrt3/2`

  • `-1/2`

  • none of these

Ex. 4.16 | Q 7 | Page 120

sin\[\left[ \cot^{- 1} \left\{ \tan\left( \cos^{- 1} x \right) \right\} \right]\]  is equal to

 

 
  • x

  • `sqrt(1-x^2`

  • `1/x`

  • none of these

     
Ex. 4.16 | Q 8 | Page 120

The number of solutions of the equation \[\tan^{- 1} 2x + \tan^{- 1} 3x = \frac{\pi}{4}\] is

 

  • 2

  • 3

  • 1

  • none of these

Ex. 4.16 | Q 9 | Page 120

If α = \[\tan^{- 1} \left( \tan\frac{5\pi}{4} \right) \text{ and }\beta = \tan^{- 1} \left( - \tan\frac{2\pi}{3} \right)\] , then

 
  • 4 α = 3 β

  • 3 α = 4 β

  • α − β = `(7pi)/12`

  • none of these

Ex. 4.16 | Q 10 | Page 120

The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]

  • 0

  • 1

  • 2

  • infinite

Ex. 4.16 | Q 11 | Page 120

If x < 0, y < 0 such that xy = 1, then tan−1 x + tan−1 y equals

 

  • `pi/2`

  • `-pi/2`

  • − π

  • none of these

Ex. 4.16 | Q 12 | Page 120

\[\text{ If } u = \cot^{- 1} \sqrt{\tan \theta} - \tan^{- 1} \sqrt{\tan \theta}\text{ then }, \tan\left( \frac{\pi}{4} - \frac{u}{2} \right) =\]

  • `sqrt(tantheta`

  • `sqrt(cottheta)`

  •  tan θ

  • cot θ

Ex. 4.16 | Q 13 | Page 120

\[\text{ If }\cos^{- 1} \frac{x}{3} + \cos^{- 1} \frac{y}{2} = \frac{\theta}{2}, \text{ then }4 x^2 - 12xy \cos\frac{\theta}{2} + 9 y^2 =\]

  • 36

  • 36 − 36 cos θ

  • 18 − 18 cos θ

  • 18 + 18 cos θ

Ex. 4.16 | Q 14 | Page 120

If α = \[\tan^{- 1} \left( \frac{\sqrt{3}x}{2y - x} \right), \beta = \tan^{- 1} \left( \frac{2x - y}{\sqrt{3}y} \right),\] 
 then α − β =

  • `pi/6`

  • `pi/3`

  • `pi/2`

  • `-pi/3`

Ex. 4.16 | Q 15 | Page 121

Let f (x) = `e^(cos^-1){sin(x+pi/3}.`
Then, f (8π/9) = 

  • e5π/18

  •  e13π/18

  • e−2π/18

  • none of these

Ex. 4.16 | Q 16 | Page 121

\[\tan^{- 1} \frac{1}{11} + \tan^{- 1} \frac{2}{11}\]  is equal to

 

 

  • 0

  • 1/2

  • − 1

  • none of these

Ex. 4.16 | Q 17 | Page 121

If \[\cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = \theta,\]  then 9x2 − 12xy cos θ + 4y2 is equal to

  • 36

  •  −36 sin2 θ

  • 36 sin2 θ

  • 36 cos2 θ

Ex. 4.16 | Q 18 | Page 121

If tan−1 3 + tan−1 x = tan−1 8, then x =

  • 5

  • 1/5

  • 5/14

  • 14/5

Ex. 4.16 | Q 19 | Page 121

The value of \[\sin^{- 1} \left( \cos\frac{33\pi}{5} \right)\] is 

 

  • `(3pi)/5`

  • `-pi/10`

  • `pi/10`

  • `(7pi)/5`

Ex. 4.16 | Q 20 | Page 121

The value of \[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right)\] is

 

  • `pi/2`

  • `(5pi)/3`

  • `(10pi)/3`

  • 0

Ex. 4.16 | Q 21 | Page 121

sin \[\left\{ 2 \cos^{- 1} \left( \frac{- 3}{5} \right) \right\}\]  is equal to

 

  • `6/25`

  • `24/25`

  • `4/5`

  • `-24/25`

Ex. 4.16 | Q 22 | Page 121

If θ = sin−1 {sin (−600°)}, then one of the possible values of θ is

 

  • `pi/3`

  • `pi/2`

  • `(2pi)/3`

  • `-(2pi)/3`

Ex. 4.16 | Q 23 | Page 121

If \[3\sin^{- 1} \left( \frac{2x}{1 + x^2} \right) - 4 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + 2 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) = \frac{\pi}{3}\] is equal to

 

  • `1/sqrt3`

  • `-1/sqrt3`

  • `sqrt3`

  • `-sqrt3/4`

Ex. 4.16 | Q 24 | Page 121

If 4 cos−1 x + sin−1 x = π, then the value of x is

 

  • `2/3`

  • `1/sqrt2`

  • `sqrt3/2`

  • `2/sqrt3`

Ex. 4.16 | Q 25 | Page 121

It \[\tan^{- 1} \frac{x + 1}{x - 1} + \tan^{- 1} \frac{x - 1}{x} = \tan^{- 1}\]   (−7), then the value of x is

 

  • 0

  • −2

  • 1

  • 2

Ex. 4.16 | Q 26 | Page 121

If \[\cos^{- 1} x > \sin^{- 1} x\], then

  • \[\frac{1}{\sqrt{2}} < x \leq 1\]

  •  \[0 \leq x < \frac{1}{\sqrt{2}}\]

  •  \[- 1 \leq x < \frac{1}{\sqrt{2}}\]

  •  x > 0

Ex. 4.16 | Q 27 | Page 121

In a ∆ ABC, if C is a right angle, then
\[\tan^{- 1} \left( \frac{a}{b + c} \right) + \tan^{- 1} \left( \frac{b}{c + a} \right) =\]

 

 

  • `pi/3`

  • `pi/4`

  • `(5x)/2`

  • `pi/6`

Ex. 4.16 | Q 28 | Page 121

The value of sin \[\left( \frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8} \right)\] is

 

  • `1/sqrt2`

  • `1/sqrt3`

  • `1/(2sqrt2)`

  • `1/(3sqrt3)`

Ex. 4.16 | Q 29 | Page 122

\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\] 

 

  • 7

  • 6

  • 5

  • none of these

Ex. 4.16 | Q 30 | Page 122

If tan−1 (cot θ) = 2 θ, then θ =

 

  • `+-pi/3`

  • `+-pi/4`

  • `+-pi/6`

  • none of these

Ex. 4.16 | Q 31 | Page 122

If \[\sin^{- 1} \left( \frac{2a}{1 - a^2} \right) + \cos^{- 1} \left( \frac{1 - a^2}{1 + a^2} \right) = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right),\text{ where }a, x \in \left( 0, 1 \right)\] , then, the value of x is

 

  • 0

  • `a/2`

  •  a

  • `(2a)/(1-a^2)`

Ex. 4.16 | Q 32 | Page 122

The value of  \[\sin\left( 2\left( \tan^{- 1} 0 . 75 \right) \right)\] is equal to

 

  • 0.75

  • 1.5

  • 0.96

  • `sin^-1 1.5`

Ex. 4.16 | Q 33 | Page 122

If > 1, then \[2 \tan^{- 1} x + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] is equal to

 

  • `4tan^-1x`

  • 0

  • `pi/2`

     

  •  π

Ex. 4.16 | Q 34 | Page 122

The domain of  \[\cos^{- 1} \left( x^2 - 4 \right)\] is

 

  • [3, 5]

  • [−1, 1]

  •  \[\left[ - \sqrt{5}, - \sqrt{3} \right] \cup \left[ \sqrt{3}, \sqrt{5} \right]\]

  •  \[\left[ - \sqrt{5}, - \sqrt{3} \right] \cap \left[ \sqrt{3}, \sqrt{5} \right]\]

Ex. 4.16 | Q 35 | Page 122

The value of \[\tan\left( \cos^{- 1} \frac{3}{5} + \tan^{- 1} \frac{1}{4} \right)\]

 

  • `19/8`

  • `8/19`

  • `19/12`

  • `3/4`

Chapter 4: Inverse Trigonometric Functions

Ex. 4.1Ex. 4.2Ex. 4.3Ex. 4.4Ex. 4.5Ex. 4.50Ex. 4.6Ex. 4.7Ex. 4.8Ex. 4.9Ex. 4.10Ex. 4.11Ex. 4.12Ex. 4.13Ex. 4.14Ex. 4.15Ex. 4.16

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) - Shaalaa.com

RD Sharma solutions for Class 12 Mathematics chapter 4 - Inverse Trigonometric Functions

RD Sharma solutions for Class 12 Maths chapter 4 (Inverse Trigonometric Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 4 Inverse Trigonometric Functions are Inverse Trigonometric Functions (Simplification and Examples), Properties of Inverse Trigonometric Functions, Graphs of Inverse Trigonometric Functions, Inverse Trigonometric Functions - Principal Value Branch, Basic Concepts of Trigonometric Functions.

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