I figured out how to do it. Here's how:

1. I had to group the cone to an object I don't care about, such as a null object.

2. I then took this null object, orient constrained it to the joint so it lined up with the joint.

3. I then wrote down the rotational values that the joint caused the null object to rotate.

4. I unconstrained the null object from the joint.

5. Then I rotated the null object the same number of degrees the joint caused it to rotate when it was constrained.

6. Then I orient constrained the joint to the null object, this caused the joint to be constrained but it did not have to rotate to get to the correct orientation.

7. Then I took my cone and made it the parent of the null object, so when I rotate the cone the joint moves as well.