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^{−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 = tan−1 where x > 0, y > 0 & xy < 1
= π + tan^{−1} where x > 0, y > 0 & xy > 1
tan^{−1} x − tan^{−1}y = tan^{−1} 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 = 1 where x > 0, y > 0
(iv) cos^{−1} x + cos^{−1} y = cos^{−1} where x ≥ 0, y ≥ 0
P7
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
P8
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
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Trigonometric Equations Video  05:10 min 
Test: Measurement of Angles Test  20 ques 
1. What are inverse trigonometric functions? 
2. How do inverse trigonometric functions relate to trigonometric functions? 
3. What are the common inverse trigonometric functions? 
4. How are inverse trigonometric functions represented? 
5. What is the range of inverse trigonometric functions? 
Trigonometric Equations Video  05:10 min 
Test: Measurement of Angles Test  20 ques 

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