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Inverse Sine

Plots

Plots

Plots

Alternate form

-i log(sqrt(1 - x^2) + i x)

Root

x = 0

Properties as a real function

{x element R : -1<=x<=1}

{y element R : -π/2<=y<=π/2}

injective (one-to-one)

odd

Series expansion at x = -1

-π/2 + sqrt(2) sqrt(x + 1) + (x + 1)^(3/2)/(6 sqrt(2)) + (3 (x + 1)^(5/2))/(80 sqrt(2)) + (5 (x + 1)^(7/2))/(448 sqrt(2)) + (35 (x + 1)^(9/2))/(9216 sqrt(2)) + O((x + 1)^5)
(Puiseux series)

Series expansion at x = 0

x + x^3/6 + (3 x^5)/40 + O(x^6)
(Taylor series)

Series expansion at x = 1

1/2 (π + (-1)^ceiling(arg(x - 1)/(2 π)) (-2 i sqrt(2) sqrt(x - 1) + (i (x - 1)^(3/2))/(3 sqrt(2)) - (3 i (x - 1)^(5/2))/(40 sqrt(2)) + (5 i (x - 1)^(7/2))/(224 sqrt(2)) - (35 i (x - 1)^(9/2))/(4608 sqrt(2)) + O((x - 1)^(11/2))))

Series expansion at x = ∞

1/2 (-2 i log(x) + π - i log(4)) + i/(4 x^2) + (3 i)/(32 x^4) + O((1/x)^6)
(generalized Puiseux series)

Derivative

d/dx(sin^(-1)(x)) = 1/sqrt(1 - x^2)

Indefinite integral

integral sin^(-1)(x) dx = sqrt(1 - x^2) + x sin^(-1)(x) + constant

Global maximum

max{sin^(-1)(x)} = π/2 at x = 1

Global minimum

min{sin^(-1)(x)} = -π/2 at x = -1

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