# Viete Product

Wave · y = f(x, t)

`y = ∏_(k=1)^(1 + floor(mod(t, 12))) cos(x/2^k)`

[Open in the app](https://www.wavelace.com/app#p=120) · [This page](https://www.wavelace.com/presets/viete-product)

### What it draws

A product rather than a sum. Each factor is `cos(x/2^k)`, so the first halves the angle, the second quarters it, the third takes an eighth, and each one is nearer to 1 than the last. The clock multiplies in a new factor every second, up to twelve, then starts again from one.

One factor alone is `cos(x/2)`, a slow cosine crossing zero at `x = π`. As factors accumulate the curve pulls down into a single central hump with smaller ripples either side, and stops changing: at `x = 3` it runs `0.0707`, then `0.0518`, then `0.0473`, settling on `0.04704`.

### What it converges to

That limit is `sin(x)/x`, and the identity behind it is one line of algebra repeated. Since `sin(x) = 2 sin(x/2) cos(x/2)`, halving again and again leaves `sin(x)` as a run of cosines multiplied by `sin(x/2ⁿ)`, and that last factor tends to `x/2ⁿ` as the angle shrinks. Divide through and the product is `sin(x)/x`.

### History

François Viète published this in 1593, the first infinite product ever written down and the first formula to give `π` as an endless sequence of operations rather than a construction. He wrote it at `x = π/2`, where `sin(x)/x` is `2/π` and every cosine unrolls into a nested square root of twos. Twenty factors here give `0.636619772`, which is `2/π` to nine places.

### Try

- Compare against the limit by typing `sin(x)/x`. From about eight factors the two curves are the same line.
- Hold two factors with `prod(cos(x/2^k), k, 1, 2)`. A product of two cosines, and the ripples are still visibly wrong at the edges.
- Start the halving later with `prod(cos(x/2^k), k, 2, 12)`. Dropping the first factor leaves `sin(x)/x` divided by `cos(x/2)`. At `x = π` that cosine is zero, but so is the sine, and the curve passes calmly through `2/π ≈ 0.637`.
- Raise `Span of x` to 30. More ripples appear, each smaller than the last, decaying as `1/x`.

### Read more

- [Viète's formula](https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula)
- [François Viète](https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te)
- [Sinc function](https://en.wikipedia.org/wiki/Sinc_function)
- [Infinite product](https://en.wikipedia.org/wiki/Infinite_product)