Domain and Range
| Function | Domain | Range |
| y = sin-1x | [-1, 1] | [-π/2, π/2] |
| y = cos-1x | [-1, 1] | [0, π] |
| y = tan-1x | R | (-π/2, π/2) |
| y = sec-1x | R-{-1, 1} | [0, π] – {π/2} |
| y = cot-1x | R | (0, π) |
| y = cosec-1x | R-{-1, 1} | [-π/2, π/2] – {0} |
Elementary Properties of Inverse
Property I
(i) sin-1 (sin θ) = θ; θ ∈ [-π/2, π/2]
(ii) cos-1 (cos θ) = θ; θ ∈ [0, π]
(iii) tan-1 (tan θ) = θ; θ ∈ (-π/2, π/2)
(iv) cosec-1 (cosec θ) = θ; θ ∈ [-π/2, π/2]- {0}
(v) sec-1 (sec θ) = θ; θ ∈ [0, π] – {π/2}
(vi) cot-1 (cot θ) = θ; θ ∈ (0, π)
Property II
(i) sin(sin-1 x) = x; x ∈ [-1, 1]
(ii) cos(cos-1 x) = x; x ∈ [-1, 1]
(iii) tan(tan-1 x) = x; x ∈ R
(iv) cosec(cosec-1 x) = x; x ∈ (-∞, -1]∪[1, ∞)
(v) sec(sec-1 x) = x; x ∈ (-∞, -1]∪[1, ∞)
(vi) cot(cot-1 x) = x; x ∈ R
Property III
| (i) sin-1 (-x) = – sin-1 (x) | (iv) cos-1 (-x) = π – cos-1 (x) |
| (ii) tan-1 (-x) = – tan-1 (x) | (v) sec-1 (-x) = π – sec-1 (x) |
| (iii) cosec-1 (-x) = – cosec-1 (x) | (vi) cot-1 (-x) = π – cot-1 (x) |
Property IV
(i) sin-1 (1/x) = cosec-1 (x)
(ii) cos-1 (1/x) = sec-1 (x)
(iii) tan-1 (1/x) = \begin{cases}cot^{-1}x & if\ x > 0\\-\pi+cot^{-1}x & if\ x < 0\end{cases}
Property V
(i) sin-1 x+cos-1 x = π/2; x ∈ [-1, 1]
(ii) tan-1 x+cot-1 x = π/2; x ∈ R
(iii) sec-1 x+cosec-1 x = π/2; x ∈ (-∞, -1]∪[1, ∞)
Property VI
(i) 2sin-1 x = sin-1 (2x \sqrt{1-x^2} )
(ii) 2cos-1 x = cos-1 (2x2 – 1)
(iii) 2tan-1 x = tan-1 2x\over1-x^2
(iv) 2tan-1 x = sin-1 2x\over1+x^2
(v) 2tan-1 x = cos-1 1-x^2\over1+x^2
Property VII
(i) 3sin-1 x = sin-1 (3x – 4x3)
(ii) 3cos-1 x = cos-1 (4x3 – 3x)
(iii) 3tan-1 x = tan-1 3x - x^3\over 1-3x^2
Property VIII
| (i) sin-1 x | = cos-1 \sqrt{1-x^2} |
| = tan-1 x\over \sqrt{1-x^2} | |
| = cot-1 \sqrt{1-x^2}\over x | |
| = sec-1 1\over \sqrt{1-x^2} | |
| = cosec-1 1\over x | |
| (ii) cos-1 x | = sin-1 \sqrt{1-x^2} |
| = tan-1 \sqrt{1-x^2}\over x | |
| = cot-1 x\over \sqrt{1-x^2} | |
| = cosec-1 1\over \sqrt{1-x^2} | |
| = sec-1 1\over x | |
| (iii) tan-1 x | = sin-1 x\over \sqrt{1+x^2} |
| = cos-1 1\over \sqrt{1+x^2} | |
| = cosec-1 \sqrt{1+x^2} \over x | |
| = sec-1 \sqrt{1+x^2} | |
| = cot-1 1\over x |
Property IX
(i) sin-1x+sin-1y=sin-1({x\sqrt{1-y^2}+y\sqrt{1-x^2}})
(ii) sin-1x-sin-1y=sin-1({x\sqrt{1-y^2}-y\sqrt{1-x^2}})
(iii) cos-1x+cos-1y=cos-1({xy - \sqrt{1-x^2}\sqrt{1-y^2}})
(iv) cos-1 x-cos-1 y=cos-1(xy + \sqrt{1-x^2}\sqrt{1-y^2})
(v) tan-1x+tan-1y=tan-1(x+y\over 1-xy)
(vi) tan-1x-tan-1y=tan-1(x-y\over 1+xy)
