1 If y = sinx, thenx = sin-1 y , similarly for other inverse T-function.
2 Domain and Range of Inverse T-function:
| Function | Domain (D) | Range(R) |
| sin-1 x | -1 £ x £ 1 | -( p/2) £ q £ (p/2) |
| cos-1 x | -1 £ x £ 1 | 0 £ q £ p |
| tan-1 x | - ¥ < x < ¥ | -( p/2) < q < (p/2) |
| cot-1 x | - ¥ < x < ¥ | 0< q < p |
| sec-1 x | x £ -1, x ³ 1 | 0 £ q £ p, q ¹ (p/2) |
| cosec-1 x | x £ -1, x ³ 1 | -( p/2) £ q £ (p/2), q ¹ 0 |
3 Properties of Inverse T-function:
(i) sin-1(sin q) = q provided – (q/2) £ q £ (p/2)
Cos-1 (cos q) = q provided q £ q £ p
Tan-1(tan q) = q provided – (p/2) < q < (p/2)
Cot-1 (cot q) = q provided 0 < q < p
Sec-1(sec q) = q provided 0 £ q < (p/2) or (p/2) < q £ p
Cosec-1 (cosec q) = q provided – (p/2) £ q < 0 or 0 < q £ (p/2)
(ii) sin (sin-1 x ) = x provided -1 £ x £ 1
Cos(cos-1 x) = x provided – 1 £ x £ 1
Tan (tan-1 x ) = x provided - ¥ < x < ¥
Cot (cot-1 x) = x provided - ¥ < x < ¥
Sec(sec-1 x) = x provided - ¥ < x £ -1 or 1 £ -1
Cosec (cosec-1 x) = x provided - ¥ < x £ -1 or 1 £ x < ¥
(iii) sin-1 (-x) = -sin-1 x ,
Cos-1 (-x) = - p - cos-1 x
Tan-1 (-x)=p - tan-1 x
cot-1 (-x) = p - cot-1 x
cosec-1 (-x) = - cosec-1 x
sec-1 (-x)=p - sec-1 x
(iv) sin-1 x + cos-1 x = (p/2) , " x Î [-1,1]
Tan-1 x + cot-1 x = (p/2) , " x Î R
Sec-1 x + cosec-1 x = (p/2), " x Î (-¥, -1) È (1, ¥)
4 value of one inverse function in terms of another inverse function :
(iv) sin-1 (1/x) = cosec-1 x, " x Î (-¥, 1) È (1, ¥)
(v) cos-1 (1/x) = sec-1 x, " x Î¥> ,1) È (1, ¥)
5 Formulae foe sum and difference of inverse trigonometric function :
(i) tan-1x + tan-1y = tan-1  ; if x > 0, y > 0 xy < 1
(ii) tan-1x + tan-1y=p + tan-1  ; if x > 0, y > 0, xy > 1
(iii) tan-1x - tan-1y = tan-1  ; if xy > -1
(iv) tan-1x tan-1y = p + tan-1  ; if x > 0, y<0, xy < -1
(v) tan-1x +tan-1y + tan-1z = tan-1
(vi) sin-1x ± sin-1y = sin-1  if x,y ³ 0 & x2 + y2 £ 1
(vii) sin-1x ± sin-1y = p - sin-1  if x,y ³ 0 & x2 + y2 > 1
(viii) cos-1x ± cos-1y = cos-1  if x,y > 0 & x2 + y2 £ 1
(ix) cos-1x ± cos-1y = p - cos-1  if x,y > 0 & x2 + y2 > 1
6 Inverse trigonometric rations of multiple angles
(i) 2sin-1x = sin-1 (2x Ö(1-x2)) , if -1 £ x £ 1
(ii) 2cos-1x = cos-1 (22x2-1), if -1 £ x £ 1
(iii) 2tan-1x = tan-1 
(iv) 3sin-1x = sin-1 (3x-4x3)
(v) 3cos-1x = cos-1 (4x3 - 3x)
(vi) 3tan-1 = tan-1 
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