You're kind of up the creek. A trap is going to end up catching a dog or a cat just as easy as a coyote. If you want to save you cat bring it in a night. Once the coyotes recognize your area as a food source they are going to clean out all the cats and small dogs they can.
+1
Also realize that there is more than one coyote, and once you eliminate one, by what ever means, it will become more difficult to eliminate the others. Coyotes are intelligent, highly adaptive, and they thrive in urban areas.
I suggest that you don't break the law dealing with this problem, and depending on what the laws are in your area even trying to live trap the animal may be illegal. If I was in your situation I would call, and try to get your neighbors to call animal control and report the coyotes every time you see them.
Honestly my money is on the coyotes. You may be able to thin the numbers down, and drive them out of your area for a while, but they will recover and come back to snack.