Does the number a: a=cos(a) (fixed point for cosine = 0.73908513321516...) have any other interesting properties

Question

Can it be expressed in terms that do not directly or indirectly require the cosine calculation?

Is there a way to compute it without iteration?

I presume it is irrational...

Answer 1

Such equations do not have closed-form solutions and the only way to calculate the root of it is by Numerical Analysis methods, such as iteration as you mentioned.

Take into account that there are numerous methods that converge faster or slower and each of them are chosen depending on the problem given.

Answer 2

The solution of the equation is the Dottie number.

As you will see, a quite good approximation is given by $$a \approx \left(\frac \pi {160} \right)^{1/13}\approx0.7390853722$$ Otherwise, you could approximate it using $[1,n]$ Padé approximants built around $a=\frac 1 {\sqrt 2}$ and get (nasty) explicit expressions.

For example, using $n=1$, you would get $$a=\frac 1 {\sqrt 2}+\frac{2 \left(\sqrt{2}+\sqrt{2} \sin \left(\frac{1}{\sqrt{2}}\right)-\sin \left(\sqrt{2}\right)-2 \cos \left(\frac{1}{\sqrt{2}}\right)\right)}{-7-8 \sin \left(\frac{1}{\sqrt{2}}\right)+\sqrt{2} \cos \left(\frac{1}{\sqrt{2}}\right)+\cos \left(\sqrt{2}\right)}$$

Just for the fun of it, I produced values of $a_{(n)}$ $$\left( \begin{array}{cc} n & a_{(n)} \\ 1 & \color{red} {0.73908}127514673336 \\ 2 & \color{red} {0.7390851}5691043600 \\ 3 & \color{red} {0.739085133}03909324 \\ 4 & \color{red} {0.739085133215}51992 \\ 5 & \color{red} {0.73908513321516}649 \\ 6 & \color{red} {0.739085133215160}52 \\ 7 & \color{red} {0.73908513321516064} \end{array} \right)$$

You could also perform one single iteration of high order iterations methods, starting with $a_0=\frac{1}{\sqrt{2}}$, and stil get explicit expressions. The corresponding numerical values would be $$\left( \begin{array}{cc} n & a_{(n)} \\ 1 & \color{red} {0.739}31860587975274 \\ 2 & \color{red} {0.7390}7951355898452 \\ 3 & \color{red} {0.739085}25658441687 \\ 4 & \color{red} {0.7390851}2998208803 \\ 5 & \color{red} {0.739085133}30261553 \\ 6 & \color{red} {0.73908513321}267735 \\ 7 & \color{red} {0.739085133215}23303 \\ 8 & \color{red} {0.7390851332151}5848 \end{array} \right)$$