get on a nearby hill or rock and kill ones you don't want as they usually spawn in 2s or 3s (recommend an argentavis). then start shooting the mammoth with tranquillisers. when it torpor runs finish it off, they are very slow so very easy to hit. (protect from dire wolves, yutiranus, etc....)
Very easy to tame. They are just a bit slower than a human so you could just walk back and tranq. Run away once in a while because it does catch up while you are walking. Once your mammoth of choice is knocked just run away for 10 seconds and the rest will lose aggro. Mine just took down a low level alpha Rex and took about 1000 damage after I put 250 armour hat. It's got a good health pool anyways. It's better than the beaver in that it can defend itself and my base, it can hold much more wood. It also collects berries extremely well. Thatch is also not a problem with this. And on mobile it's water spray can knockback even a rex incase I gotta bail... Did I mention that it's faster than a rex? Only thing is that it doesn't get as much wood but in the Arctic there's plenty of wood. Just spend a few more minutes collecting wood is the solution...
Best way to tame is have a crossbow with butt load of tranq arrows walk backwards tranqing it as you go every now and then you need to turn around to get a bit further away. I know this works because i have three mammoths that i used this exact stratigy on
Playing on Genesis in the Arctic biome, and I built me a simple 3x3 trap out of stone foundations, stone stairs converted into ramps, and stone doorways. It might not want to get up the ramp, initially, and it's kinda slow, but it does hit hard, if you're just in fur. This one was lv90. Gonna try starve taming. But yeah, once it's in there, just keep popping it in the head with tranqs. It's big ol fat head sticks out above the trap, so you shouldn't have a problem with headshots. Hope this helps.