# Taylor Sine

Wave · y = f(x, t)

`y = Σ_(k=0)^(floor(mod(t, 12))) (−1)^k · x^(2k+1)/factorial(2k+1)`

[Open in the app](https://www.wavelace.com/app#p=119) · [This page](https://www.wavelace.com/presets/taylor-sine)

### What it draws

The Taylor polynomial of `sin(x)`, gaining a term a second. The first is `x` itself, the straight line every small-angle approximation uses. The next subtracts `x³/6`, the next adds `x⁵/120`, and the alternating signs and factorials continue to twelve terms before the count resets.

Near the origin each new term matters less, because `x^(2k+1)` shrinks faster than `(2k+1)!` grows. Away from it the opposite holds for a while, which is why the early curves swing so far.

### Where it stops working

Watch the right-hand edge, at `x = 4`, where `sin(4) = −0.76`. One term puts the curve at `4.00`, two overcorrect to `−6.67`, three to `1.87`, five to `−0.66`, and by eight it has settled on `−0.76`. Each polynomial is exact at the origin and drifts further the further out it is read.

The useful range grows with the term count rather than the accuracy at a point. Five terms track the real sine within `0.01` out to `x ≈ 3.25`, and past that they leave. This series does converge everywhere in the end, unlike many, but only in the limit, and a polynomial of any fixed length is eventually beaten by its own highest power.

### Try

- Raise `Span of x` to 12. The curve tracks the sine across the middle and every partial sum still leaves it, now far enough out to run off the top.
- Freeze it at three terms with `sum((−1)^k · x^(2k+1)/factorial(2k+1), k, 0, 2)`. The overshoot at the edges stops moving and can be read off.
- Swap in the cosine series, `sum((−1)^k · x^(2k)/factorial(2k), k, 0, floor(mod(t, 12)))`. Even powers, and it starts from 1 rather than 0.
- Try `sum(x^k/factorial(k), k, 0, floor(mod(t, 12)))`. Without the alternating sign the series builds `e^x`, which climbs rather than waves.

### Read more

- [Taylor series](https://en.wikipedia.org/wiki/Taylor_series)
- [Taylor's theorem](https://en.wikipedia.org/wiki/Taylor%27s_theorem)
- [Radius of convergence](https://en.wikipedia.org/wiki/Radius_of_convergence)
- [Small-angle approximation](https://en.wikipedia.org/wiki/Small-angle_approximation)