Cardinality Question

Let S be the set of real numbers r s.t. .

be the set of real numbers of the form , where a,b is rational.

Find the cardinality of S.

I know 0,2pi,4pi.. etc are in S, so S is infinite
and is countably infinite (dont know if it matters)
then im not sure how to go on...any hints would be appreciated.

Thanks in advance

For each number , the set is at most countable. Picture the (co)sine wave that is crossed by a horizontal line; there are at most countably many intersection points.

Now, , so...