Viète's formula for pi
The number π has many surprising and interesting properties. It is defined as the ratio between the perimeter and the diameter of a circle. There are many interesting identities of this number; in the present post I will introduce the Viète's formula, a very beautiful formula involving π:
![vietes-formula.png](https://steemitimages.com/DQmRazBpVHVLYJgTkq7CqM2smvyL54282Lv4B3X6V9YR92x/vietes-formula.png)
I few days ago I wrote about the Wallis product, other beautiful formula that express π as an infinite product of rational numbers.
Viète's formula was the first identity in history to express the number π as a limit, and it was published 1593 by the French mathematician François Viète (1540-1603). Using his formula he calculated π with a precision of nine digits.
Here, I will present Euler's proof of the formula. It requieres basic knowledge of trigonometry and limits.
The formula for the sin of the angle double is:
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/84/ql_4984ba08ce1c772b7d7d2d398e54ff84_l3.png)
Iterating it (or using induction) we have:
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/d7/ql_90ff007232471df54fb6a6133d5fc1d7_l3.png)
On the other hand, from the well-known limit expression
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/f7/ql_90f370d6d2d678c5e929866ebef3fcf7_l3.png)
we conclude
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/06/ql_fedeae784bd1f7b425dfba9e2d524606_l3.png)
Therefore
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/16/ql_d8fcf92d6314d23ba646c233096a8516_l3.png)
This identity is due to Euler.
We recall the formula of the cosine of the half angle:
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/7a/ql_0f80cdbf9d0e37dff9bc708b6456c67a_l3.png)
Let x=π/2, we have:
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/9b/ql_4b496442b3123f0ceaebcaee2110e99b_l3.png)
Hence
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/a8/ql_4191ab99e6a2e5da8fc8b65c0fa36aa8_l3.png)
Substituting these expressions into (1), we get
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/28/ql_f9314699465ea8f9fcbd9a8256fa9128_l3.png)
or equivalently
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/23/ql_0ebc17aa9d136de0bdaae80e470a6b23_l3.png)
where
![](https://steemitimages.com/640x0/http://quicklatex.com/cache3/a5/ql_0176b5511f5f2cc1aa26ae4472757ba5_l3.png)
References:
https://en.wikipedia.org/wiki/Viète%27s_formula
http://www.theoremoftheday.org/GeometryAndTrigonometry/Viete/TotDViete.pdf
Hola @nenio, bastante interesante esta fórmula.
Precísamente ayer hice un post hablando de un cálculo que hice de PI hace varios años y el resultado aunque no es el mismo es bastante similar a esta fórmula de Viete.
Te invito a leer mi post aquí.
Un saludo