Thetabeta

Thetabeta is a fictional mathematical constant used in this article to illustrate how constants defined by equations are described and analyzed in mathematics. It combines theta and beta as notational inspiration and is not part of established literature.

A common definition defines thetabeta as the unique real root t in the interval (0, π/2) of the equation sin t = e^{-t}. This provides a concrete, well-posed constant. Alternative definitions may be used in other contexts.

The equation is transcendental, with no closed-form solution. Thetabeta is irrational. Numerical methods yield t ≈ 0.5879.

Newton-Raphson iteration can be used to approximate the value. For f(t) = sin t − e^{-t} and f′(t) = cos t + e^{-t}, starting from an initial guess t0 = 0.5, the method converges rapidly to t ≈ 0.5879.

In teaching and exposition, thetabeta demonstrates how to define a constant via an equation, prove existence and uniqueness, and report approximate values for use in further computations. It also reflects how constants are often named by merging symbols from the underlying variables, such as theta and beta, to convey its origin while remaining a self-contained mathematical object. See also transcendental equations and numerical root-finding methods. Notes: as a fictional construct, thetabeta serves primarily as an educational example rather than a reference to an established constant.