1. ## Cardinality Question

Let S be the set of real numbers r s.t. $\cos (r) \in Q\sqrt 2$.

$Q\sqrt 2$ be the set of real numbers of the form $a+b\sqrt 2$, where a,b is rational.

Find the cardinality of S.

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

2. For each number $x\in\mathbb{R}$, the set $\{r\in\mathbb{R}\mid\cos(r)=x\}$ 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, $\{r\in\mathbb{R}\mid\cos(r)\in\mathbb{Q}(\sqrt{2}) \}=\bigcup_{x\in\mathbb{Q}(\sqrt{2})}\{r\in\mathbb {R}\mid\cos(r)=x\}$, so...