The set {x in R: |cos x| geq sin x} cap left[ | vedclass

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(A) We need to find the intersection of the set $\{x \in R: |\cos x| \geq \sin x\}$ with the interval $\left[0, \frac{3 \pi}{2}\right]$.Consider the g

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$\left[0, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]$

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(A) We need to find the intersection of the set $\{x \in R: |\cos x| \geq \sin x\}$ with the interval $\left[0, \frac{3 \pi}{2}\right]$.Consider the graphs of $y = |\cos x|$ and $y = \sin x$ in the interval $\left[0, \frac{3 \pi}{2}\right]$.$1$. In the interval $\left[0, \frac{\pi}{4}\right]$, $\cos x \geq \sin x$, so $|\cos x| \geq \sin x$ holds.$2$. In the interval $\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$, $\sin x > \cos x$, so $|\cos x| $3$. In the interval $\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]$, $\cos x$ is negative, so $|\cos x| = -\cos x$. Since $\cos x \leq 0$ and $\sin x$ can be negative, we check the condition $-\cos x \geq \sin x$, which is $\cos x + \sin x \leq 0$. This holds for $x \in \left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]$.Thus, the solution is $\left[0, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]$.

The set $\{x \in R: |\cos x| \geq \sin x\} \cap \left[0, \frac{3 \pi}{2}\right]$ is equal to

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