Important Formulas: Inverse Trigonometric Functions

# Important Inverse Trigonometric Functions Formulas for JEE and NEET

 Table of contents 1. Definition (s) 2. Principal Values and Domains of Inverse Circular Functions : 3. Properties of Inverse Circular Functions: Inverse Trigonometric Functions:

## 1. Definition (s)

sin−1 x , cos−1 x , tan−1 x  etc.  denote angles or real  numbers whose sine is  x, whose cosine is x and whose tangent is x,  provided that the  answers given are numerically smallest available. These are also written as arc sinx, arc cosx etc .

If there are two angles one positive & the other negative having same numerical value, then positive angle should be taken.

## 2. Principal Values and Domains of Inverse Circular Functions :

(i) y = sin−1 x where −1 ≤ x ≤ 1 ;  and sin y = x.

(ii) y = cos−1 x where −1 ≤ x ≤ 1 ; 0 ≤ y ≤ π and cos y = x.

(iii) y = tan−1 x where x ∈ R;  and  tan y = x.

(iv) y = cosec−1 x  where  x ≤ −1 or  x ≥ 1 ;  y ≠ 0 and cosec y = x

(v) y = sec−1 x  where  x ≤ −1  or  x ≥ 1  ;  0 ≤ y ≤ π  ;  y ≠ π/2 and sec y = x.

(vi) y = cot−1 x  where  x ∈ R ,  0 < y < π  and  cot y = x.

NOTE  THAT : (a) 1st  quadrant  is  common  to  all  the  inverse  functions.

(b) 3rd  quadrant  is  not  used  in  inverse  functions.

(c) 4th  quadrant is used  in  the  CLOCKWISE DIRECTION  i.e. − π/2 ≤ y ≤ 0.

Question for Important Formulas: Inverse Trigonometric Functions
Try yourself:
Which of the following is the principal value of y for the inverse function y = sec^(-1) x where x = -2?

## 3. Properties of Inverse Circular Functions:

P−1 (i)  sin (sin1 x) =  x,  −1 ≤ x ≤ 1

(ii) cos (cos−1 x) = x ,  −1 ≤ x ≤ 1

(iii) tan (tan−1 x) = x,  x ∈ R

(iv) sin−1 (sin x) = x,

(v) cos−1 (cos x) = x  ;  0 ≤ x ≤ π

(vi) tan−1 (tan x) = x  ;

P−2 (i)  cosec−1 x = sin−1(1/x) ; x ≤ −1 ,  x ≥ 1

(ii)  sec−1 x = cos−1(1/x) ; x ≤ −1,  x ≥ 1

(iii)  cot−1 x = tan−1 (1/x) ; x > 0 = π + tan−1(1/x) ; x < 0

P−3 (i) sin−1 (−x) = − sin−1 x , −1 ≤ x ≤ 1

(ii) tan−1 (−x) = − tan−1 x , x ∈ R

(iii) cos−1 (−x) = π − cos−1 x , −1 ≤ x ≤ 1

(iv) cot−1 (−x) = π − cot−1 x, x ∈ R

P−4 (i)  sin−1 x + cos−1 x = π/2 −1 ≤ x ≤ 1

(ii) tan−1 x + cot−1 x = π/2 x ∈ R

(iii)  cosec−1 x + sec−1 x = π/2 |x| ≥ 1

P−5 tan−1 x + tan−1 y =  where x > 0, y > 0 & xy < 1

= π + tan−1 where  x > 0,  y > 0 & xy > 1

tan−1 x − tan−1y =  where x > 0,  y > 0

P−6 (i) sin−1 x + sin−1 y = sin−1 where x ≥ 0, y ≥ 0 & (x2 + y2) ≤ 1

Note that : x2 + y2 ≤ 1 ⇒ 0 ≤ sin−1 x + sin−1 y ≤ π/2

(ii) sin−1 x + sin−1 y = π − sin−1 where x ≥ 0, y ≥ 0  & x2 + y2 > 1

Note that : x2 + y2 > 1 ⇒ π/2 < sin−1 x + sin−1 y < π

(iii) sin–1x – sin–1y = sin–1 where x > 0 , y > 0

(iv) cos−1 x ± cos−1 y = cos−1 where  x ≥ 0 ,  y ≥ 0

P−7 If  tan−1 x + tan−1 y + tan−1 z =  tan−1 if, x > 0, y > 0, z > 0 & xy + yz + zx < 1

Note : (i) If  tan−1 x + tan−1 y + tan−1 z =  π then   x + y + z = xyz

(ii) If  tan−1 x + tan−1 y + tan−1 z = π/2 then  xy + yz + zx = 1

P−8 2 tan−1 x =  sin−1 Note very carefully that :

REMEMBER THAT : (i) sin−1 x + sin−1 y + sin−1 z = 3π/2 ⇒ x = y = z = 1

(ii) cos−1 x + cos−1 y + cos−1 z = 3π ⇒ x = y = z = −1

(iii) tan−1 1 + tan−1 2 + tan−1 3 = π and tan−1 1 + tan−1

## Inverse Trigonometric Functions:

### SOME USEFUL GRAPHS

1. y = sin−1 x , |x| ≤ 1, y ∈

2. y  = cos−1 x, |x| ≤ 1 , y ∈ [0, π]

3. y = tan−1 x , x ∈ R,

4. y = cot−1 x, x ∈ R , y ∈ (0 , π)

5. y = sec−1 x , |x| ≥ 1 , y

6. y = cosec−1 x , |x| ≥ 1,

7. (a) y = sin−1 (sin x) , x ∈ R ,

Periodic with period  2 π

(b) y = sin (sin−1 x),  = x x ∈ [−1, 1], y ∈ [−1, 1], y is aperiodic

8. (a) y = cos−1(cos x), x ∈ R, y ∈[0, π], = x periodic with period 2 π

(b) y = cos (cos−1 x), =  x

x ∈ [− 1, 1],  y ∈ [−1, 1], y is aperiodic

9. (a) y = tan (tan−1 x),  x ∈ R,  y ∈ R,  y is aperiodic = x

(b) y = tan−1 (tan x), =  x

10. (a) y = cot−1 (cot x), =  x

(b) y = cot (cot −1 x), = x

11. (a) y = cosec −1 (cosec x), =  x

(b) y =  cosec (cosec−1 x), = x

|x| ≥ 1 , |y| ≥ 1, y is aperiodic

12. (a) y = sec −1 (sec x), = x

y is periodic with period 2π ;

(b) y = sec (sec−1 x), = x

|x| ≥ 1 ; |y| ≥ 1], y is aperiodic

The document Important Inverse Trigonometric Functions Formulas for JEE and NEET is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on Important Inverse Trigonometric Functions Formulas for JEE and NEET

 1. What are the principal values and domains of inverse circular functions?
Ans. The principal values of inverse circular functions are restricted to specific ranges to ensure unique values for each input. The domains of inverse circular functions depend on the original circular function being inverted, such as sine, cosine, tangent, etc.
 2. What are the properties of inverse circular functions?
Ans. Some properties of inverse circular functions include the fact that they undo the actions of their respective circular functions, they have restricted domains to ensure uniqueness, and they are useful in solving trigonometric equations.
 3. What are some important formulas for inverse trigonometric functions in JEE exams?
Ans. Some important formulas for inverse trigonometric functions in JEE exams include the relationships between different trigonometric functions, the properties of inverse trigonometric functions, and their applications in solving trigonometric equations.
 4. How can inverse trigonometric functions be used to solve trigonometric equations?
Ans. Inverse trigonometric functions can be used to solve trigonometric equations by applying the properties and formulas associated with them, such as using the inverse trigonometric functions to find the values of angles that satisfy the given equation.
 5. What are some frequently asked questions related to inverse trigonometric functions in exams?
Ans. Some frequently asked questions related to inverse trigonometric functions in exams include determining the principal values of inverse trigonometric functions, finding the domains of inverse circular functions, and applying inverse trigonometric functions to solve trigonometric equations.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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