Your calculator doesn't *know* sin(0.7). It looks up a polynomial — a Taylor series — and does a few multiplications. Polynomials approximate every nice function in the universe, given enough terms.
Any 'nice' function can be approximated near a point by a polynomial — a Taylor series. The more terms you add, the better the fit.
Numerical algorithms, physics approximations (small-angle sin θ ≈ θ), AI gradient methods, signal processing, GPU shader maths — all rely on Taylor expansions instead of exact functions.
Famous Taylor series (around 0)
- eˣ = 1 + x + x²/2! + x³/3! + … (everywhere)
- sin x = x − x³/3! + x⁵/5! − … (everywhere)
- cos x = 1 − x²/2! + x⁴/4! − … (everywhere)
- 1/(1−x) = 1 + x + x² + x³ + … (only when |x| < 1)
Approximate eˣ near 0
1 + x + x²/2 + x³/6 + x⁴/24 + … (works for any x; converges everywhere)
Use the first three terms of cos x to estimate cos(0.1).
Taylor series only converge for some functions over their full domain. 1/(1−x) blows up at x = 1, and the series only works for |x| < 1. Always check the radius of convergence.
- Taylor series = polynomial approximation built from a function's derivatives.
- Closer to the centre point and more terms = better fit.
- Many physical formulas are *just* the first term or two of a Taylor expansion (e.g. small-angle sin θ ≈ θ).