All Inverse trigonometry functions and formulas or Identity list | pdf

 Inverse trigonometry functions and formulas

Inverse trigonometry

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Negative Property

sin-1 (-x) = -sin-1 x

cos-1 (-x) = π - cos-1 x

tan-1 (-x) = -tan-1 x

cot-1 (-x) = π - cot-1 x

sec-1 (-x) = π - sec-1 x

cosec-1 (-x) = -cosec-1 x

Complementary Property

sin-1 x + cos-1 x = \(\frac {π}{2}\)

tan-1 x + cot-1 x = \(\frac {π}{2}\)

sec-1 x + cosec-1 x = \(\frac {π}{2}\)

Properties of Inverse Circular function

sin-1 x = cosec-1 \(\frac {1}{x}\)

cosec-1 x = sin-1 \(\frac {1}{x}\)

cos-1 x = sec-1 \(\frac {1}{x}\)

sec-1 x = cos-1 \(\frac {1}{x}\)

tan-1 x = cot-1 \(\frac {1}{x}\)

cot-1 x = tan-1 \(\frac {1}{x}\)

Self Adjusting Property

sin-1 (sin x) = x   :   x ∈ \(\left [ \frac {-π}{2}, \frac {π}{2} \right ] \)

cos-1 (cos x) = x   :   x ∈ \(\left [ 0, π \right ] \)

tan-1 (tan x) = x   :   x ∈ \(\left ( \frac {-π}{2}, \frac {π}{2} \right ) \)

cot-1 (cot x) = x   :   x ∈ \( \left ( 0, π \right ) \)

sec-1 (sec x) = x   :   x ∈ [0, π] - \(  \left\{ \frac {π}{2} \right\} \)

cosec-1 (cosec x) = x   :   x ∈ \(\left [ \frac {-π}{2}, \frac {π}{2} \right ] - \left\{0\right\} \)

sin (sin-1 x) = x   :   x ∈ \(\left [ -1, 1 \right ] \)

cos (cos-1 x) = x   :   x ∈ \(\left [ -1, 1 \right ] \)

tan (tan-1 x) = x   :   x ∈ R

cot (cot-1 x) = x   :   x ∈ R

sec (sec-1 x) = x   :   x ∈ R - (-1, 1)

cosec (cosec-1 x) = x   :   x ∈ R - (-1, 1)

Addition and Subtraction formulas

sin-1 x + sin-1 y = sin-1 \(x\sqrt {1 - y^2} + y\sqrt {1 - x^2}\);  x ≥ 0, y ≥ 0 & x2 + y2 ≤ 1

sin-1 x + sin-1 y = π - sin-1 \(x\sqrt {1 - y^2} + y\sqrt {1 - x^2}\);  x ≥ 0, y ≥ 0 & x2 + y2 ≥ 1

sin-1 x - sin-1 y = sin-1 \(x\sqrt {1 - y^2} - y\sqrt {1 - x^2}\);  x,y ∈ [0, 1]

cos-1 x + cos-1 y = cos-1 \(xy - \sqrt {1 - x^2}\sqrt {1 - y^2}\);  x,y ∈ [0, 1]

cos-1 x - cos-1 y = cos-1 \(xy + \sqrt {1 - x^2}\sqrt {1 - y^2}\);  0 ≤ x < y ≤ 1

cos-1 x - cos-1 y = - cos-1 \(xy + \sqrt {1 - x^2}\sqrt {1 - y^2}\);  0 ≤ y < x ≤ 1

tan-1 x + tan-1 y = tan-1 \(\frac{x+y}{1 - xy}\);  x,y > 0 & xy < 1

tan-1 x + tan-1 y = π + tan-1 \(\frac{x+y}{1 - xy}\);  x,y > 0 & xy > 1

tan-1 x + tan-1 y = π/2 if x,y > 0 & xy = 1

tan-1 x + tan-1 y = -π/2 if x,y < 0 & xy = 1

tan-1 x - tan-1 y = tan-1 \(\frac{x-y}{1 + xy}\);  x,y ≥ 0 & xy > -1

2tan-1 x = sin-1 \(\frac {2x}{1+x^2}\);  -1 ≤ x ≤ 1

2tan-1 x = tan-1 \(\frac {2x}{1-x^2}\);   -1 < x < 1

2tan-1 x = cos-1 \(\frac {1-x^2}{1+x^2}\);   x ≥ 0

2sin-1 x = sin-1 \(2x\sqrt {1 - x^2}\)

3sin-1 x = sin-1 (3x - 4x3 )

2cos-1 x = cos-1 (2x2 - 1)

3cos-1 x = cos-1 (4x3 - 3x)

3tan-1 x = tan-1 \( \left ( \frac {3x - x^3}{1 - 3x^2} \right ) \)

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