Formulas: Inverse Trigonometric Functions | Mathematics (Maths) Class 11 - Commerce PDF Download

General Definition(s): 1. 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 are sinx , are 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 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. .

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⁻¹x, −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 = tan−1  where x > 0, y > 0 & xy < 1
= π + tan⁻¹ where x > 0,  y > 0 & xy > 1
tan⁻¹ x − tan⁻¹y = tan⁻¹ 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 = 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 =  Note very carefully that :

REMEMBER  THAT :
(i) sin⁻¹ x + sin⁻¹ y + sin⁻¹ z =  ⇒ 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,

6. y = cosec⁻¹ x, |x| ≥ 1,

7. (a) y = sin⁻¹ (sin x), x ∈ R,
Periodic with period 2π aperiodic

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

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

(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
x ∈ R − {n π}, y ∈ (0, π), periodic with π

(b) y = cot (cot⁻¹ x), = x
x ∈ R,  y ∈ R, y is aperiodic

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