To solve the trigonometric equation sin^3(x) + 2cos^5(x) = sqrt(10), we can use trigonometric identities to simplify the expression.

First, we know that sin^2(x) + cos^2(x) = 1. We can rewrite the equation sin^3(x) + 2cos^5(x) = sqrt(10) as sin^2(x) • sin(x) + 2cos^2(x) • cos^3(x) = sqrt(10).

Solving for sin(x) and cos(x) using the Pythagorean identity, sin^2(x) = 1 - cos^2(x) and cos^2(x) = 1 - sin^2(x), we get sin(x) • (1 - cos^2(x)) + 2cos^2(x) • (1 - sin^2(x)) = sqrt(10).

Expanding the equation gives us sin(x) - sin(x) • cos^2(x) + 2cos^2(x) - 2sin^2(x) • cos^2(x) = sqrt(10).

Rearranging the terms gives us sin(x) + 2cos^2(x) - sin(x) • cos^2(x) - 2sin^2(x) • cos^2(x) = sqrt(10).

Now, we can use the double angle formulas to further simplify the equation:

sin(2x) = 2sin(x) • cos(x)
cos(2x) = cos^2(x) - sin^2(x)

Substitute these formulas into the equation to get:

(sin(x) + 2cos^2(x) - sin(2x)/2 - 2cos(2x)/2 = sqrt(10)

At this point, it would be hard to find an exact value for x, but you can approximate the solution using numerical methods or a calculator.