# 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 .