Given $x \in(0,2 \pi]$, and the six trigonometric values of $x$
$$
\sin x 、 \cos x 、 \tan x 、 \cot x 、 \sec x 、 \csc x
$$

are exactly the six edge lengths of a tetrahedron. Find the range of $x$.
Obviously, $x$ is an acute angle, and the six trigonometric function values are symmetric about $\frac{\pi}{4}$. Therefore, we only need to consider $x \in\left(0, \frac{\pi}{4}\right]$.
$$
\text { When } \frac{\pi}{6}<x \leqslant \frac{\pi}{4}
$$

At this time, it can be verified that the six trigonometric function values arranged as shown in Figure 4 satisfy the conditions.
$$
\begin{array}{l}
\text { When } 0<x \leqslant \frac{\pi}{6} \text {, } \\
2 \leqslant \csc x, \sin x<\tan x \leqslant \frac{\sqrt{3}}{3}, \\
\frac{\sqrt{3}}{2} \leqslant \cos x, \sqrt{3} \leqslant \cot x, \sec x \leqslant \frac{2 \sqrt{3}}{3} .
\end{array}
$$

Then $\sin x, \tan x$ must be one adjacent to $\csc x$ and the other opposite to $\csc x$. Without loss of generality, assume $\sin x$ is adjacent to $\csc x$ (the other case is similar).

As shown in Figure 4, let $B C=\csc x, C A=\sin x, A D=$ $\tan x$. Then in $\triangle A B C$, we have $A B=\cot x$.
Thus, in $\triangle A B D$, we have
$$
\begin{array}{l}
\cos x+\tan x<\sec x+\tan x \\
\leqslant \frac{2 \sqrt{3}}{3}+\frac{\sqrt{3}}{3}=\sqrt{3} \leqslant \cot x,
\end{array}
$$

Therefore, it cannot form a triangle, meaning that $x$ does not satisfy the conditions at this time.
Symmetrically, we get that $\frac{\pi}{4} \leqslant x<\frac{\pi}{3}$ satisfies.
Thus, $x \in\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$.