When Doing the security missions on the first island, you will have to build an ACU center. When it's built, press A on it and select the helicopter and take control of it. I stayed outside the original pen a good distance and shot until I hit the dinosaur. You do not have to full tranquilize it for the achievement. You just need to hit it once from that distance. There is quite a large bullet drop when you are that far back. If it's too hard to see where the dart lands, just move forward some more. You have unlimited shots, so just keep trying until you get a hit.
You can also fly up and shoot 125m down and have less trouble with the bullet drop. (Thanks: MAST3R SHAKE04)
Shoot it two more times for this achievement:
Hit one while it's running for this achievement:
While you are at it, shoot a dinosaur in the head for this achievement: