Table of contents  
1. Definition (s)  
2. Principal Values and Domains of Inverse Circular Functions :  
3. Properties of Inverse Circular Functions:  
Inverse Trigonometric Functions: 
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.
(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.
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^{−1}y = where x > 0, y > 0
P−6 (i) sin^{−1} x + sin^{−1} y = sin^{−1} where x ≥ 0, y ≥ 0 & (x^{2} + y^{2}) ≤ 1
Note that : x^{2} + y^{2} ≤ 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 & x^{2} + y^{2} > 1
Note that : x^{2} + y^{2} > 1 ⇒ π/2 < sin^{−1} x + sin^{−1} y < π
(iii) sin^{–1}x – sin^{–1}y = 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}
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
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1. What are the principal values and domains of inverse circular functions? 
2. What are the properties of inverse circular functions? 
3. What are some important formulas for inverse trigonometric functions in JEE exams? 
4. How can inverse trigonometric functions be used to solve trigonometric equations? 
5. What are some frequently asked questions related to inverse trigonometric functions in exams? 
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