Finally converted cheerful Muslim (5)
I believe the answer is:
sunni
'muslim' is the definition.
(I know that Sunni is a type of muslim)
I don't understand how the rest of the clue works.
Can you help me to learn more?
(Other definitions for sunni that I've seen before include "Form of religion" , "Member of another Muslim sect" , "One branch of Islam" , "Islamic faction" , "Moslem" .)