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Inverse Cosine

Plots

Plots

Plots

Alternate forms

1/2 (π - 2 sin^(-1)(x))

π/2 + i log(sqrt(1 - x^2) + i x)

Root

x = 1

Properties as a real function

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

{y element R : 0<=y<=π}

injective (one-to-one)

Series expansion at x = -1

π - 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

π/2 - x - x^3/6 - (3 x^5)/40 + O(x^6)
(Taylor series)

Series expansion at x = 1

(-1)^ceiling(arg(x - 1)/(2 π)) (i sqrt(2) sqrt(x - 1) - (i (x - 1)^(3/2))/(6 sqrt(2)) + (3 i (x - 1)^(5/2))/(80 sqrt(2)) - (5 i (x - 1)^(7/2))/(448 sqrt(2)) + (35 i (x - 1)^(9/2))/(9216 sqrt(2)) + O((x - 1)^(11/2)))

Series expansion at x = ∞

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

Derivative

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

Indefinite integral

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

Global maximum

max{cos^(-1)(x)} = π at x = -1

Global minimum

min{cos^(-1)(x)} = 0 at x = 1

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