Important Inverse Trigonometric Functions Formulas for JEE and NEET

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.

3. Properties of Inverse Circular Functions:

P−1 (i)  sin (sin⁻¹ 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