Taylor Series Approximation
Approximate sin/cos/eˣ/ln(1+x) via Taylor polynomial of degree n.
Result
T_5(0.5)
0.47942708
Exact: 0.47942554 · error: 1.545e-6.
- Functionsin
- x0.5
- Order5
- Approximation0.4794270833
- Exact0.4794255386
- Error1.5447e-6
Step-by-step
- Maclaurin series sum from k=0 to 5.
- Approximation = 0.479427.
- Error = |approx − exact| = 1.545e-6.
How to use this calculator
- Pick function, x, and polynomial order.
About this calculator
Taylor series approximates a function as a polynomial around a point. Maclaurin: around 0. sin(x) ≈ x − x³/6 + x⁵/120 − …; cos(x) ≈ 1 − x²/2 + x⁴/24 − …; eˣ ≈ 1 + x + x²/2 + x³/6 + …. More terms = better approximation. Used in calculator firmware (chip can't natively compute trig), physics (small-angle approx sin x ≈ x), and numerical analysis. Source: NIST DLMF Ch. 1.
Frequently asked
Polynomials are easy to compute (just multiplications + additions). Most functions can be approximated by them in some range.
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