What is the cause of your hypothyroidism? is it Hashi?, Rheumatoid arthritis is an autoimmune disease and Hypothyroidism is more prevalent in RA than in the general population, if you have one autoimmune disease you can have another. What tests have you done? Rheumatoid factor (RF)? ANA's?
I have seen a number of links that related hypothyroidism and arthritis. Here is one from the Mayo Clinic.
http://www.mayoclinic.com/health/hypothyroidism/AN00069
Has the Synthroid been enough to get you to a euthyroid state? What tests have been done and what were the actual results by which your doc changes your meds.?
The most important thyroid test is for free T3, followed by free T4 and TSH. Free T3 is the active component that largely regulates metabolism. Apparently you have been previously diagnosed as hypo t. For people who are hypo t, the best end result is to get the free T3 into the upper third of its range and TSH into the lower end of its range.