Your calculator has no button-shaped brain for sine — when you type sin(0.5), it adds up a Taylor series behind the scenes. A Taylor series rebuilds a function from its derivatives at a single point: f(x) = f(a) + f′(a)(x−a) + f′′(a)(x−a)²/2! + …, with each term's size set by the product of a derivative value and a power term, divided by a factorial. More terms mean a tighter fit — five terms nail sin(0.5) to six decimal places, which is why the chip stops there.

Engineers approximate eˣ, ln(1+x), and cos(x) in embedded systems too small for full math libraries, and physics students linearize pendulums with the first two terms. Each series term is a product at heart: type the derivative value fⁿ(a) in the Variable A box and the power term (x−a)ⁿ/n! in the Variable B box, and the Result box gives that term's contribution. For e⁰·¹'s second term, 1 × (0.01/2) = 0.005 — add the terms up and the approximation emerges.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Taylor Series Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.