To solve this trigonometric equation, we first need to rewrite it in terms of only one trigonometric function.

Using the double angle formula for sine:

sin(2x) = 2sinxcosx

Therefore, the equation becomes:

-\sqrt{3}cosx = 2sinxcosx - 5\pi

Combining like terms:

-\sqrt{3}cosx - 2sinxcosx = -5\pi

Factor out the common factor of cosx:

cosx(-\sqrt{3} - 2sinx) = -5\pi

Now, we have two possibilities:

Let's solve these two possibilities:

cosx = 0
Since cosx = 0, x = nπ + (π/2) where n is an integer.

-\sqrt{3} - 2sinx = -5\pi
Rearrange the equation:
2sinx = 5pi + sqrt{3}

sinx = (5π + √3) / 2

x = arcsin((5π + √3) / 2)

So the solution to the trigonometric equation is:

x = nπ + (π/2) for n as an integer, and x = arcsin((5π + √3) / 2)