
Cardinality Question
Let S be the set of real numbers r s.t. $\displaystyle \cos (r) \in Q\sqrt 2$.
$\displaystyle Q\sqrt 2 $ be the set of real numbers of the form $\displaystyle a+b\sqrt 2 $, where a,b is rational.
Find the cardinality of S.
I know 0,2pi,4pi.. etc are in S, so S is infinite
and $\displaystyle Q\sqrt 2 $ 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 $\displaystyle x\in\mathbb{R}$, the set $\displaystyle \{r\in\mathbb{R}\mid\cos(r)=x\}$ 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, $\displaystyle \{r\in\mathbb{R}\mid\cos(r)\in\mathbb{Q}(\sqrt{2}) \}=\bigcup_{x\in\mathbb{Q}(\sqrt{2})}\{r\in\mathbb {R}\mid\cos(r)=x\}$, so...