Finally converted cheerful Muslim (5)

I believe the answer is:

sunni

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'muslim' is the definition.
(I know that Sunni is a type of muslim)

I don't understand how the rest of the clue works.

(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" .)

I've seen this clue in The Times.

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