Tim, sorry if it seemed like I was criticizing, but coyotes serve a purpose in nature. As Cecil pointed out they do help control the populations of rodents, coons, cats, rabbits, etc. I don't have a problem with hunting or trapping, but I don't think you'll ever be able to kill enough coyotes to stop the problem.
Using dogs or donkeys seems like a good approach, but if you don't want to do that for some reason there's still something you can try.
Taste aversion has some advantages over other methods. Taste aversion happens when an animal eats something and shortly after gets sick. Vets use apomorphine for this purpose when training puppies not to chew on shoes etc. It's tasteless and odorless but will cause the animal to vomit after eating it.
If you were to inject apomorphine into the carcass of a freshly killed goat, the coyotes would ingest it and then get sick. They will learn that goat isn't good to eat. The biggest advantage with this method is that the coyotes will teach their young to avoid that type of prey also.
You can't buy or use apomorphine yourself, but maybe you could work with a veterinarin.
