A second way to find the damping ratio is to build the characteristic equation of the system directly from the given exponent and frequency, instead of matching formulas.
The response $x(t) = Ae^{-5t}\sin(10t+\phi)$ tells us the roots of the system's characteristic equation are complex conjugates of the form:
\[ s = -5 \pm 10i \]because the real part of the root gives the decay rate in the exponential ($e^{-5t}$) and the imaginary part gives the oscillation frequency (the $10t$ inside the sine).
For a standard second-order system, the characteristic equation is $s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$, whose roots are $s = -\zeta\omega_n \pm i\omega_n\sqrt{1-\zeta^2}$. Comparing with $s = -5 \pm 10i$ gives the same two conditions as before, but let's instead use the magnitude of the root directly.
The magnitude of the complex root equals $\omega_n$ (a standard property: on the s-plane, the distance of a root from the origin is the natural frequency):
\[ \omega_n = |s| = \sqrt{5^2 + 10^2} = \sqrt{25+100} = \sqrt{125} = 11.180 \]The real part of the root equals $-\zeta\omega_n$, so:
\[ \zeta = \frac{5}{\omega_n} = \frac{5}{\sqrt{125}} = \frac{5}{11.180} = 0.4472 \]Rounded to 3 decimal places, $\zeta = 0.447$, matching the earlier method.
\[ \boxed{\zeta \approx 0.447} \]