Chapter 2. Inverse Trigonometric Functions

Important formula :

Class 12 Inverse trigonometry function

. The domains and Ranges of inverse trigonometric Function :

Function	Domain	Range
	[– 1 ,1 ]
	[– 1 , 1]

. Properties of Inverse Trigonometric Functions :

1. 

2. 

3.

4.  

5.  

6. 

7.   ,  or

8.  or

9.  or

10. or

11.    

12.  

13.  

14.  

15.  

16.  

17.  or  x≥1

18.   x≤1 or  x≥1 

19.  

20.   

21.  

22.  

23.  

24.   

25.   

27. 

28.   

29.  

30.  

31. 

32. 

31.    

32.  

33.   

34.  

35. 

36. 

37. 

38.

39.  

40.   

41.

42.

43.   

44.   

45. 

46. 

47.

Class 12 Chapter 2. Inverse Trigonometric Functions Exercise 2.1

Find the principal values of the following :

Q1.    

[ Note: ]

Solution:  We have, 

Q2.     

[ Note: ]

Solution: We have,

Q3.   

Solution: We have, 

[ Note: ]

Q4.    

Solution: We have,  

[ Note:  ]

Q5.

Solution: We have, 

[ Note:   ]

Q6.   

Solution: We have,

[ Note: ]

Q7.     

Solution: We have,

[Note: , or  ]

Q8.    

Solution: We have,

[ Note: ]

Q9.    

Solution: We have,

[ Note:  ]

Q10.   

Solution: We have,

[ Note:   ]

Find the values of the following :

Q11.       

 Solution: We have, 

Q12.

Solution:  We have, 

Q13. If  , then  

(A)        (B)     (C)       (D)    

Solution: We have,  ;

Correct Answer: (B) 

Q14.  is equal to

(A)      (B)      (C)       (D)  

Solution: We have,

Correct Answer: (B)

Class 12 Chapter 2. Inverse Trigonometric Functions Exercise 2.2

Prove the following :

Q1.    

Solution:   Let   and

We know that, 

Now,

  

  Proved.

Q2.

Solution:  Let   and

We know that,  

Now,   

      Proved.

Q3.

Solution:  LHS :

 

   R.H.S.  Proved.

Q4.

Solution:  LHS : 

Write the following functions in the simplest form :

Q5. 

Solution:  We have, 

Let    and

Q6.

Solution:  We have,

Let   and

Q7.

Solution: We have, 

 =

Q8.

Solution: We have,

Q9.

Solution:  We have,  

Let 

  and 

 

=

 

Q10.

Solution:  We have,

Let  and

 

Find the values of each of the following :

Q11.

Solution:  We have,

Q12.

Solution:  We have,

Q13. and

Solution:  We have,

Q14. If , then find the value of   .

Solution:  We have,

Therefore, the value of  is

Q15. If , then find the value of  .

Solution:   We have,

Find the values of each of the expressions in Exercises 16 to 18 .

Q16.

Solution:   We have,

Q17.

Solution:  We have,

Q18.

Solution:  We have, 

Q19. is equal to

(A)   (B)   (C)      (D) 

Solution:  (B)

[ We have,

]

Q20. is equal to

(A)     (B)     (C)     (D) 1

Solution:  (D)  1

[ We have,

  ]

Q21.   is equal to :

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

Solution:  (B)

We have,

Class 12 Chapter 2. Inverse Trigonometric Functions Miscellaneous Exercise on Chapter 2

Find the value of the following :

Q1.

Solution: We have,

Q2.

Solution:  We have,

Prove that  :

Q3.

Solution : LHS :  

Let

RHS  Proved .

Q4.

Solution:

[ Note :

;  ]

LHS : 

                                                                                                                                 

RHS  Proved.

Q5.

Solution:

[ Note : 

;   ]

LHS :

   RHS

LHS = RHS Proved.

Q6.

Solution:

[  Note:   ]

LHS :

[ Note :     ]

  RHS

LHS = RHS  Proved.

Q7.

Solution:  RHS :

Let 

Let   and 

   LHS  Proved.

Q8.

Solution: 

LHS :

  RHS

LHS = RHS   Proved

Prove that :

Q9.

Solution:

LHS :   

RHS

LHS = RHS  Proved. 

Q10.

Solution:  LHS :

  RHS

LHS = RHS  Proved.

Q11.

[Hint : Put  ]

Solution :

Let  and

LHS:

=

RHS

LHS = RHS   Proved .

Q12.

Solution:  LHS :  

Let   and

Now,

  RHS

LHS = RHS Proved

Solve the following equations :

Q13.

Solution:  We have,  

Q14.

Solution:    We have,

 

Q15.  is equal to

(A)     (B)     (C)       (D) 

Solution:  (D) 

[We have, 

Let 

Now,  

  ]

Q16. , then  is equal to

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

Solution:  (A)  0 ,

[ We have,

or 

     ]

Q17. is equal to :

(A)    (B)     (C)     (D)

Solution:   (C)  

[ We have,

=

    ]