Inverse Trigonometric Functions Formula

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)

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