Cos(x) = x

1. Apr 1, 2005

Antiphon

The solution to this equation is approximately 0.739085.

Does anyone know how to express the solution exactly
in terms of contants like pi, e, phi, etc?

(phi = golden ratio = 1/2 + sqrt(5)/2)

2. Apr 1, 2005

Data

In all likelyhood it's impossible to do so in a simple way.

3. Apr 1, 2005

dextercioby

Sure it is.If "x" is a solution to the equation,then be can expressed as

$$x=\frac{x}{\pi e\varphi} \pi e\varphi$$

Daniel.

4. Apr 1, 2005

Data

You're right, of course. I took some license in my interpretation of his question. I'll be more precise:

It's very likely impossible to express the solution in terms of a finite number of products, extractions of roots, additions, exponentiations, and divisions of elements of the set $$\{e, \pi, \phi\} \cup \mathbb{Z}$$

~

Last edited: Apr 1, 2005
5. Apr 1, 2005

dextercioby

Let's tell Antiphon that not all transcendental numbers can be written using only $e$ and $\pi$ and the set of algebraic numbers...

Daniel.

6. Apr 4, 2005

Antiphon

I suspected this, but I asked the question assuming it was possible.

So then you think it's impossible or you're not sure in this case?

Perhaps then I should assign it a greek letter!

7. Apr 4, 2005

dextercioby

1.It is impossible.

2.You should.

Daniel.

8. Jan 3, 2007

Ali 2

The solution of the equation cos (x) = x can be given as applying the cosine function infinite nubmer of times to a starting point ..

x = cos cos cos ...... cos (a)

In other words , the solution can be expressed as :

$$x = \lim _ { n \to \infty } \cos ^ { \circ n } ( a )$$

That came from the Contraction Mapping Theorem .