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Tangent Function

Plots

Plots

Plots

Alternate forms

sin(x)/cos(x)

(i (e^(-i x) - e^(i x)))/(e^(-i x) + e^(i x))

Alternate form assuming x is real

sin(2 x)/(cos(2 x) + 1)

Roots

x = π n, n element Z

Integer root

x = 0

Properties as a real function

{x element R : x/π + 1/2 not element Z}

R (all real numbers)

periodic in x with period π

surjective onto R

odd

Series expansion at x = 0

x + x^3/3 + (2 x^5)/15 + O(x^6)
(Taylor series)

Derivative

d/dx(tan(x)) = sec^2(x)

Indefinite integral

integral tan(x) dx = -log(cos(x)) + constant
(assuming a complex-valued logarithm)

Identities

tan(x) = tan(m π + x) for m element Z

tan(x) = -cot(2 x) + csc(2 x)

tan(x) = (1 - cos(2 x)) csc(2 x)

tan(x) = sin(2 x)/(1 + cos(2 x))

tan(x) = -(2 tan(x/2))/(-1 + tan^2(x/2))

tan(x) = (2 tan(x/2))/(1 - tan^2(x/2))

tan(x) = (x sqrt(-tan^2(x)))/sqrt(-x^2)

tan(x) = (sec(x/3) sin(x))/(-1 + 2 cos((2 x)/3))

Alternative representations

tan(x) = 1/cot(x)

tan(x) = cot(π/2 - x)

tan(x) = -cot(π/2 + x)

Series representations

tan(x) = i + 2 i sum_(k=1)^∞ (-1)^k q^(2 k) for q = e^(i x)

tan(x) = i sum_(k=-∞)^∞ (-1)^k e^(2 i k x) sgn(k)

tan(x) = -i + 2 i sum_(k=0)^∞ (-1)^k e^(-2 i (1 + k) x) for Im(x)<0

Integral representations

tan(x) = integral_0^x sec^2(t) dt

tan(x) = 2/π integral_0^∞ (-1 + t^((2 x)/π))/(-1 + t^2) dt for 0<Re(x)<π/2

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