Inverse Trigonometric Functions Class 12th Mathematics Part I CBSE Solution

Class 12th Mathematics Part I CBSE Solution
Exercise 2.1
  1. sin^-1 (- 1/2) Find the principal values of the following:
  2. cos^-1 (root 3/2) Find the principal values of the following:
  3. cosec-1 (2) Find the principal values of the following:
  4. tan^-1 (- root 3) Find the principal values of the following:
  5. cos^-1 (- 1/2) Find the principal values of the following:
  6. tan-1 (-1) Find the principal values of the following:
  7. sec^-1 (2/root 3) Find the principal values of the following:
  8. cot^-1 (- root 3) Find the principal values of the following:
  9. cos^-1 (- 1/root 2) Find the principal values of the following:
  10. cosec^-1 (- root 2) Find the principal values of the following:
  11. Find the values of the following:
  12. cos^-1 (1/2) + 2sin^-1 (1/2) Find the values of the following:
  13. sin-1 x = y, then Find the values of the following:A. 0 ≤ y ≤ π B. - pi /2 less…
  14. tan^-1root 3-sec^-1 (-2) is equal to Find the values of the following:A. π B. -…
Exercise 2.2
  1. 3sin^-1x = sin^-1 (3x-4x^3) , x in[- 1/2 , 1/2] Prove the following:…
  2. 3cos^-1x = cos^-1 (4x^3 - 3x) , x in[1/2 , 1] Prove the following:…
  3. tan^-1 2/11 + tan^-1 7/24 = tan^-1 1/2 Prove the following:
  4. 2tan^-1 1/2 + tan^-1 1/7 = tan^-1 31/17 Prove the following:
  5. tan^-1 root 1+x^2 -1/x , x not equal 0 Write the following functions in the…
  6. tan^-1 1/root x^2 - 1 , |x|1 Write the following functions in the simplest form:…
  7. Write the following functions in the simplest form:
  8. tan^-1 (cosx-sinx/cosx+sinx) , 0x pi Write the following functions in the…
  9. tan^-1 x/root a^2 -x^2 , |x|a Write the following functions in the simplest…
  10. tan^-1 (3a^3x-x^3/a^3 - 3ax^2) , a0 -a/root 3 x a/root 3 Write the following…
  11. tan^-1 [2cos (2sin^-1 1/2)] Find the values of each of the following:…
  12. cot (tan-1a + cot-1a) Find the values of each of the following:
  13. tan 1/2 [sin^-1 2x/1+x^2 + cos^-1 1-y^2/1+y^2] , |x|1 , y = 0 xy1 Find the…
  14. If sin (sin^-1 1/5 + cos^-1x) = 1 , then find the value of x Find the values of…
  15. tan^-1 x-1/x-2 + tan^-1 x+1/x+2 = pi /4 , then find the value of x. Find the…
  16. sin^-1 (sin 2 pi /3) Find the values of each of the expression
  17. tan^-1 (tan 3 pi /4) Find the values of each of the expression
  18. tan (sin^-1 3/5 +cot^-1 3/2) Find the values of each of the expression…
  19. cos^-1 (cos 7 pi /6) is equal to Find the values of each of the expressionA. 7…
  20. sin (pi /3 - sin^-1 (- 1/2)) is equal to Find the values of each of the…
  21. tan^-1root 3-cot^-1 (- root 3) is equal to Find the values of each of the…
Miscellaneous Exercise
  1. cos^-1 (cos 13 pi /6) Find the value of the following:
  2. tan^-1 (tan 7 pi /6) Find the value of the following:
  3. Prove that
  4. sin^-1 8/17 + sin^-1 3/5 = tan^-1 77/36 Prove that
  5. cos^-1 4/5 + cos^-1 12/13 = cos^-1 33/65 Prove that
  6. cos^-1 12/13 + sin^-1 3/5 = sin^-1 56/65 Prove that
  7. tan^-1 63/16 = sin^-1 5/13 + cos^-1 3/5 Prove that
  8. tan^-1 1/5 + tan^-1 1/7 + tan^-1 1/3 + tan^-1 1/8 = pi /4 Prove that…
  9. tan^-1root x = 1/2 cos^-1 (1-x/1+x) , x in[0 , 1] Prove that
  10. cot^-1 (root 1+sinx+ root 1-sinx/root 1+sinx- root 1-sinx) = x/2 , x in (0 , pi…
  11. Prove that tan^-1 (root 1+x + root 1-x/root 1+x - root 1-x) = pi /4 - 1/2…
  12. 9 pi /8 - 9/4 sin^-1 1/3 = 9/4 sin^-1 2 root 2/3 Prove that
  13. 2tan^-1 (cosx) = tan^-1 (2cosecx) Solve the following equations:
  14. tan^-1 1-x/1+x = 1/2 tan^-1x , (x0) Solve the following equations:…
  15. sin (tan^-1x) , |x|1 is equal to Solve the following equations:A. x/root 1+x^2…
  16. sin^-1 (1-x) -2sin^-1x = pi /2 , then x is equal to Solve the following…
  17. tan^-1 (x/y) - tan^-1 x-y/x+y is equal to Solve the following equations:A. pi…

Exercise 2.1
Question 1.

Find the principal values of the following:



Answer:

Let us take  = x

Therefore, 


We know that principle value range of sin-1 is 


And,


Therefore principle value of  is 


Question 2.

Find the principal values of the following:



Answer:

Let us take 


Then, 


We know that principle value range of cos-1 is [0, π]


And 


Therefore, principle value of  is 



Question 3.

Find the principal values of the following:

cosec–1 (2 )


Answer:

Let cosec-1 2 = x

Therefore, 


And 


We know that principle value range of cosec-1 is 


Therefore, principle value of cosec-1(2) is  .



Question 4.

Find the principal values of the following:



Answer:

Let us take 


Then we get,



And 


We know that principle value range of tan-1 is 


Therefore, principle value of tan-1 is.



Question 5.

Find the principal values of the following:



Answer:

Let us take 


Then we will get,



And 


We know that principle value range of cos-1 is [0, π]


Therefore principle value of cos-1is 



Question 6.

Find the principal values of the following:

tan–1 (–1)


Answer:

Let us take tan-1(-1) = x then we get,




And 


We know that principle value range of tan-1 is 


Therefore, principle value of tan-1(-1) is 



Question 7.

Find the principal values of the following:



Answer:

Let 

Then,



We know that range of the principle value branch of sec-1 is [0, π] 


And 


Therefore principle value of  is 



Question 8.

Find the principal values of the following:



Answer:

Let us consider 


Then we get,



We know that the range of the principal value branch of cot-1 is [0, π].


And, 


Therefore, the principle value of is .


Question 9.

Find the principal values of the following:



Answer:

Let 


Therefore,



We know that range of the principle value branch of cos-1 is [0, π]


And 


Therefore principle value of  is 



Question 10.

Find the principal values of the following:



Answer:

Let us take the values of 


Then,



We know that range of the principle value branch of cosec-1 is 


And 


Therefore principle value of  is 



Question 11.

Find the values of the following:


Answer:

Let us consider tan-1(1) = x then we get


We know that range of the principle value branch of tan-1 is 


Therefore, 


Let 



We know that range of the principle value branch of cos-1 is [0, π]


Therefore, 


Let 



We know that range of the principle value branch of sin-1 is 


Therefore 


Now,




Question 12.

Find the values of the following:



Answer:

Let 


Then, we get,



We know that range of the principle value branch of cos-1 is [0, π]


Therefore 


Let 


We know that range of the principle value branch of sin-1 is 


Therefore 


Now,




Question 13.

Find the values of the following:

sin–1 x = y, then
A. 0 ≤ y ≤ π

B. 

C. 0 < y < π

D. 


Answer:

sin–1 x = y


We know that range of the principle value branch of sin-1 is 


Therefore,


Hence, the option (B) is correct.


Question 14.

Find the values of the following:

 is equal to
A. π

B. 

C. 

D. 


Answer:

Let us take


Then we get,



We know that range of the principle value branch of tan-1 is 


Therefore, 


Let Then we get,



We know that range of the principle value branch of sec-1 is [0, π] 


Therefore, 


Now,



Hence, the option (B) is correct.



Exercise 2.2
Question 1.

Prove the following:



Answer:

Let 


We have,


R.H.S =

Now, we know that,

sin 3x = 3sin x - 4 sin3x

Therefore,

= sin-1(sin (3θ))


= 3 θ


= 3sin-1x


= L.H.S


Hence Proved


Question 2.

Prove the following:



Answer:

Let x = cos θ

Then, Cos-1 = θ


Now, R.H.S. = cos-1(4x3 - 3x)


= cos-1(4cos3θ - 3cos θ)


= cos-1(cos3θ)


= 3θ


= 3cos-1x


= L.H.S.


Hence Proved



Question 3.

Prove the following:



Answer:

L.H.S. 






= R.H.S.


Hence Proved.



Question 4.

Prove the following:



Answer:

L.H.S. = 







= R.H.S.


Hence Proved.



Question 5.

Write the following functions in the simplest form:



Answer:


Now, Put x = tanθ ⇒ θ =tan-1x


Therefore, 








Therefore, 


Question 6.

Write the following functions in the simplest form:



Answer:

Let us take, 


[We have done this substitution on the bases of identity sec2θ – 1 = tan2θ]


Therefore, 

Now we know that, cosec2θ – 1 = cot2θ

Therefore,


Question 7.

Write the following functions in the simplest form:


Answer:

Hence, 


Question 8.

Write the following functions in the simplest form:



Answer:


Dividing by cos x,





As we know tan(Π/4) = 1




Hence, 


Question 9.

Write the following functions in the simplest form:



Answer:

We will solve this problem on the bases of the identity 1 - sin2θ = cos2θ


So, for a2 - x2, we can substitute x = a sinθ or x = a cosθ


Now, Let us put x = a sinθ




Therefore,




Hence, 


Question 10.

Write the following functions in the simplest form:



Answer:


Put 


Now,






= 3θ




Question 11.

Find the values of each of the following:



Answer:


We will solve the inner bracket first.


So, we will first find the principal value of 


We know that, 


Therefore,





= tan-11


= π/4


Hence,


The value of 



Question 12.

Find the values of each of the following:

cot (tan–1a + cot–1a)


Answer:



= 0


Hence, the value of cot(tan-1a + cot-1 a) = 0



Question 13.

Find the values of each of the following:



Answer:


We will solve this problem by expressing sin2θ and cos2θ in terms of tanθ


Now let us put, x = tanθ. Then we will have,


θ = tan-1x



Now again, Let’s put, y = tan∅. Then we will have,


∅ = tan-1y



Now,







Hence, the value of




Question 14.

Find the values of each of the following:

If , then find the value of x


Answer:






On comparing the co-efficient on both sides we get,




Question 15.

Find the values of each of the following:

, then find the value of x.


Answer:













Question 16.

Find the values of each of the expression



Answer:


(For  type of problem we have to always check whether the angle is in the principal range or not. This angle must be in the principal range


So here, 


Now, can be written as,







Hence, .


Question 17.

Find the values of each of the expression



Answer:


(For  type of problem we have to always check whether the angle is in the principal range or not. This angle must be in the principal range.)


So here, 


Now, can be written as,






Hence, 



Question 18.

Find the values of each of the expression



Answer:

Let  and

 , so all ratio of y are positive and


and 


Also,


as 


So, 





Hence, 



Question 19.

Find the values of each of the expression

 is equal to
A. 

B. 

C. 

D. 


Answer:


(For cos-1(cos x) type of problem we have to always check whether the angle is in the principal range or not. This angle must be in the principal range.)


So here, 




Now, can be written as,




where [since, cos(π + x) = -cosx]


 as cos-1(-x) = π – cos-1



Hence, .


Question 20.

Find the values of each of the expression

is equal to
A. 

B. 

C. 

D. 1


Answer:

as 


 as 


We all know that the principal value branch of sin-1 is



Therefore,


Hence, the value of 


Question 21.

Find the values of each of the expression

is equal to
A. π

B. 

C. 0

D. 


Answer:







Miscellaneous Exercise
Question 1.

Find the value of the following:



Answer:


(Forcos-1(cosx) type of problem we have to always check whether the angle is in the principal range or not. This angle must be in the principal range.)


So here, 


Now, can be written as,




where 



Hence, 



Question 2.

Find the value of the following:



Answer:


(For tan-1(tanx) type of problem we have to always check whether the angle is in the principal range or not. This angle must be in the principal range.)


So here, 


Now, can be written as,




where,  [since, tan(π+x) = tanx]



Hence, 



Question 3.

Prove that


Answer:

Taking LHS

Let 


Then,


Therefore,



 ……. (1)


and 

As, LHS = RHS

Hence Proved !


Question 4.

Prove that



Answer:














 ………………………. (1)


Let


Then, 




 ………………………. (2)


Now,


L.H.S.


Putting the value from equation (1) and (2))





=R.H.S.


Hence Proved.


Question 5.

Prove that



Answer:


Let


Then,




 ……………………….(1)


Let 


Then, 




 ……………(2)


Let


Then, 




 …………… (3)


Now,


L.H.S. 


Putting the value from the equation (1) And (2))




 ……….by equation (3)


=R.H.S.


Hence Proved.



Question 6.

Prove that



Answer:

We can also solve this problem by using the identity Sin(A + B) = sinA cosB + cosA sinB


Letand 

So,

sin A = 3/5 and cos B = 12/13

Therefore,

cos A = 4/5 and sin B = 5/13

As R.H.S. is sin-1 we will use sin (A+B)






= R.H.S.


Hence Proved.


Question 7.

Prove that



Answer:


Let Then, 





LetThen,





Now,


R.H.S. =


Putting the value from equation (1) and (2))





=L.H.S.


Hence Proved.



Question 8.

Prove that



Answer:

L.H.S.









= R.H.S.


Hence Proved.



Question 9.

Prove that



Answer:


Let Then, 




So now putting the value, we get,


R.H.S. 




= L.H.S.



Question 10.

Prove that



Answer:

Consider, 


On Rationalizing, we get,




Now,


L.H.S.


 = R.H.S.


Hence Proved


Question 11.

Prove that



Answer:


Letso that


Now,


L.H.S. 









= R.H.S.


Hence Proved.



Question 12.

Prove that



Answer:


Now, L.H.S. 




Now, Let




L.H.S.


= R.H.S.


Hence Proved



Question 13.

Solve the following equations:



Answer:





Hence, 


Question 14.

Solve the following equations:



Answer:



As we know,



We know,



So,







Hence,

 
Question 15.

Solve the following equations:

 is equal to
A. 

B. 

C. 

D. 


Answer:

Let then 






Question 16.

Solve the following equations:

, then x is equal to
A. 

B. 

C. 0

D. 


Answer:


Now we will put Now, we will put x= siny in the given equation, and we get,






⟹ 1 – siny = cos2y as


⟹ 1 - cos2y = sin y


⟹ 2sin2y = sin y


⟹ siny(2siny - 1) = 0


⟹ siny = 0 or 1/2


∴ x = 0 or 1/2


Now, if we put then we will see that,


L.H.S. =




R.H.S


Hence,  is not the solution of the given equation.


Thus, x = 0


Question 17.

Solve the following equations:

 is equal to
A. 

B. 

C. 

D. 


Answer:






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