Sin 2x = 0.836. Find x?

That's a dead end.

First, find the angle(s) whose sine(s) = 0.836. Then take ½ of that. Incidentally, since you haven't specified a finite range of values for x, there will be infinitely many values, periodically spaced.

sin(2x) = 0.836

2x = {sin⁻¹(0.836), π - sin⁻¹(0.836)} + 2nπ

x = {½sin⁻¹(0.836), ½π - ½sin⁻¹(0.836)} + nπ

x = {0.494976..., 1.07582...)} + nπ

where n = any integer (positive, negative, or zero).

Update:

In that case, just choose the values of x that fall within that range.

There are 4 of them; in degrees, they are:

x =

28º.3600545276589...

61º.6399454723411...

208º.3600545276589...

241º.6399454723411...

sin 2x = .836

2x = Arc sin .836 = 56.72010906 degrees

x = 28.36005453 degrees

2x = sin^-1 ( .0836)

x = [sin^-1 (.0836)]/2