A transcendental curve written entirely with addition, subtraction, multiplication and division.

The Taylor series expansion of sin(x) represents the sine function as an infinite polynomial built from powers of x: sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + …. Each additional term improves the approximation, especially near zero, and enough terms can reproduce sine with remarkable accuracy over wider intervals. The expansion replaces a transcendental function with ordinary arithmetic, revealing how smooth functions can be reconstructed from their derivatives at a single point. It is both a computational tool and a window into the structure of analytic functions. It is used primarily in numerical analysis, physics, and engineering.