Find the exact value of $x$ if $\frac{\sin x}x =\sqrt x$

Question

It is quite easy to find that x is approximately .802. Is there any way in which we can solve this equation to find the exact value of x?

$\frac{\sin x}{x} = x^{1/2}$

Answer 1

The exact value, no way but good approximations (remember that $x=\cos(x)$ does not show explicit solutions).

Plotting the function, we can see that the root is close to $\frac \pi 4$. So, to make expressions "simple", let us use $[1,n]$ Padé approximants for $\frac{\sin x}{x} - x^{1/2}$. They will write $$\frac{\sin x}{x} - x^{1/2}=\frac {\frac{4 \sqrt{2}-\pi ^{3/2}}{2 \pi }+a_1^{(n)} \left(x-\frac \pi 4 \right)}{1+\sum_{k=1}^n b_k \left(x-\frac \pi 4 \right)^k }$$ from which the approximate solution $$x_{(n)}=\frac \pi 4 +\frac{\pi ^{3/2}-4 \sqrt{2}}{2 \pi a_1^{(n)}}$$ This would give quite nasty formulae since $$a_1^{(0)}=\frac{-8 \sqrt{2}+2 \sqrt{2} \pi -\pi ^{3/2}}{\pi ^2}$$ $$a_1^{(1)}=\frac{64-24 \pi +16 \sqrt{2} \pi ^{3/2}-3 \pi ^2+\sqrt{2} \pi ^{5/2}-60 \sqrt{2 \pi }}{2 \pi \left(8 \sqrt{2}-2 \sqrt{2} \pi +\pi ^{3/2}\right)}$$ $$\left( \begin{array}{cc} n & x_{(n)} \approx \\ 0 & 0.8027883593 \\ 1 & 0.8028042932 \\ 2 & 0.8028037219 \\ 3 & 0.8028037319 \\ 4 & 0.8028037317 \end{array} \right)$$ which is the solution for ten significant figures.

If you use Newton method, it would be much less tedious and you could get as many decimal places as you wish $$\left( \begin{array}{cc} n & x_n \\ 0 & 0.785398163397448309615660845819875721049292350 \\ 1 & 0.802788359292669370099497626707352560682643311 \\ 2 & 0.802803731726750434568551610417466777052560631 \\ 3 & 0.802803731737889315511829476566222402429256954 \\ 4 & 0.802803731737889315511835324604000441222668911 \end{array} \right)$$