Understanding rotation matrices

1. How does $ {\sqrt 2 \over 2} = \cos (45^\circ)$?

2. Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function represent? What I mean is, the blue triangle is the pre-rotated block, the green is the post-rotated block, and the purple is the rotated change ($45^\circ$) between them.

3. How do these trig functions in this matrix represent a clockwise rotation? (Like, why does "$-\sin \theta $ " in the bottom left mean clockwise rotation... and "$- \sin \theta $ " in the upper right mean counter clockwise? Why not "$-\cos \theta $ "? $$\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta & \cos \theta \end{bmatrix}$$

Any help in understanding the trig representations of a rotation would be extremely helpful! Thanks