Indexed Family of Sets Union and Intersection checking my answer

Question

Let

$A_x = \{ y \in \Bbb R: \sin x \le y \le \cos x \}\;$ for $ x \in [-\frac{\pi}{4},\frac{\pi}{4}]\}$

Find

$\bigcup_{z \in \Bbb R} \bigcap_{x \ge z} A_x\;$ and $\;\bigcap_{z \in \Bbb R} \bigcup_{x \ge z} A_x$

My answer:

$\bigcup_{z \in \Bbb R} \bigcap_{x \ge z} A_x = \{{\frac{\sqrt2}{2}\}}$

$\bigcap_{z \in \Bbb R} \bigcup_{x \ge z} A_x = (-\frac{\sqrt2}{2},1)$