If sin theta = 3/4, then theta = sin^-1(3/4) = 0.84806207898148...

By the identity sin theta = sin(pi - theta), theta can also be 2.29353057460831...

Even though neither answer can be written as a ratio of two integers (and hence are not rational), they are still real numbers because they can be labeled on a number line.  So yes, sin theta can be 3/4 with theta being a real number.