arctan1

arctan(1) refers to the value of the inverse tangent function at the input 1. The tangent function is inverted on the interval (-π/2, π/2), so arctan returns the unique angle θ in this range whose tangent equals 1. Since tan(π/4) = 1, arctan(1) = π/4. In degrees, this is 45°. The numerical value is approximately 0.7853981634 radians.

Geometrically, arctan(1) corresponds to a right triangle with equal legs, where the opposite and adjacent sides are the same length. In such a triangle, the slope or tangent of the non-right angle is 1, yielding an angle of 45 degrees (π/4 radians). On the unit circle, this angle also has sin and cos equal in magnitude, both equal to √2/2.

In general, the equation tan θ = 1 has infinitely many solutions θ = π/4 + kπ for integers k, due to the periodicity of the tangent function. However, arctan is defined to return the principal value, which is π/4 for the argument 1. The value arctan(1) is commonly used in mathematical contexts as the exact representation of π/4 and appears in various formulas for computing π, such as Machin-like identities, where arctan values are combined to express π.