When is it correct to use "The" in front of a family member?

“Ann is Bill’s sister” is a simple statement; Bill may have more sisters but we do not know.

“Ann is a sister of Bill”; using the indefinite article “a” suggests that Ann is one of several sisters.

“Ann is the sister of Bill”; using the definite article “the” suggests that Bill has only one sister, who is Ann. This is not synonymous with the first simple statement above.

“Jean is the mother of John” is similarly correct as regards usage, but it contains redundancy, because John can only have one mother. It is therefore synonymous with the simpler statement “Jean is John’s mother”. Use either.