You are trying to persuade your communication professor to let you take a makeup exam. What interpersonal goal are you trying to meet?

You are trying to achieve an instrumental goal.

Explanation:

We have an instrumental goal when we try to accomplish a specific objective or task through a particular interpersonal encounter. In the case we were given here, your instrumental goal is quite clear: through the interpersonal encounter with your professor, your objective is to convince him/her to let you take a makeup exam. The interaction, at least to you, has a very specific purpose. Therefore, you have an instrumental goal.