Need to get familiar with the unit circle.
From 0 to 2pi, there are two angles whose sine is 0.5; they occur in the first and second quadrants. Because you're going from -2pi to 2pi, that means the equation will have four solutions.
If sin(x)=a/b , that gives you all the info you need to construct a right triangle. Remember that sine is a ratio of the opposite leg to the hypotenuse in a right triangle. So a=opposite leg and b=hypotenuse. If you just see a number, meaning not a fraction that means that b=1. You probably already know this, but it's important to these problems.
You need to know in which quadrants each trig function is positive or negative. In quadrant I all of them are positive but in quadrant II, the leg along the x-axis of the triangle is negative so cos(x) becomes negative. The hypotenuse is always considered positive.
This Wikipedia page covers a lot of good concepts on this subject. It really requires combining various properties so that you can use the given information to make other conclusions. It's not easy to sum up in a paragraph.
Trigonometric functions - Wikipedia, the free encyclopedia
Example: Let's look at sin(x)=0.5. Using whole numbers, the vertical leg is 1 and the hypotenuse is 2. Thus the horizontal leg is the square root of 3. This is found by the Pythagorean Theorem. This is one of the special triangles you need to memorize and has angles of 30deg, 60deg and 90deg. Thus x=30deg.