I'm considering the set $$\left\{\sin(k)\mid k\in\Bbb Z\backslash \left\{0\right\}\right\}.$$ All of its members are transcendental numbers, but together they don't form the complete set of all transcendental numbers between $-1$ and $1$.
Does this set of numbers belong to, and form on $\left[-1,1\right]$ completely, a differently named class of numbers? What would that name be?
Edit due to comment. Or, if not, would these numbers belong to (but not completely form between $-1$ and $1$) a differently named subclass of transcendental numbers?
there is no name for this class. You will end up with $\aleph_0$ items in the set which is dense in $(-1,1)\cap\mathbb{R}$. In fact, proving that this is dense is going to be hard.