arctan0

Arctan(0) refers to the value of the inverse tangent function at the input zero. The arctangent function, denoted arctan x, is the inverse of the tangent function when restricted to the interval (-pi/2, pi/2). Since tan 0 = 0, there is a unique y in (-pi/2, pi/2) with tan y = 0, namely y = 0. Therefore arctan(0) = 0. In degrees, this is 0°.

Arctan is an odd function, meaning arctan(-x) = -arctan(x). It is continuous and strictly increasing on the real line, with range (-pi/2, pi/2). The derivative of arctan x is 1/(1 + x^2), so arctan′(0) = 1. The principal value is used to define arctan as the inverse of tan on the specified interval.

Near zero, arctan x has the Maclaurin series arctan x = x − x^3/3 + x^5/5 − ..., which shows that arctan x behaves like x for small x. Consequently, the limit lim_{x→0} arctan x / x = 1. Graphically, the function passes through the origin and approaches ±pi/2 as x → ±∞, giving horizontal asymptotes at y = ±pi/2.

In summary, arctan(0) equals 0, reflecting the principal value of the inverse tangent, with 0 being its unique zero within the principal branch.