I want to know what is the best way to teach simple trigonometric equations, such as $$\sin x=0$$. Should I use the trigonometric circle or the sine graph? Which is better?
Since the students have already been exposed to the unit circle and the graph, I would give them the equation and ask for discussion of how to figure it out. If the students come up with the answers themselves they are most likely to remember it far better than if you tell them.
If they only come up with one way, present the other way as well. If they have no idea, present both ways as tools and ask which would help.
I believe self-discovery is best.
Start with the unit circle to give them an intuition for what it means (& see this answer for some diagrams that might help). Then teach students how to plot a function from input-output pairs (if they don't know that already), and put that skill to work so that they are deriving the graph from the intuition. This will permit a deeper understanding than presenting the graph first, and help students be able to figure out key points (like, "does this one start at 0 or 1?") when they've forgotten memorized graph specifics but retained the original intuition of where the curves comes from.