If A = \ θ: 2 ^ 2 θ + sin θ ≤ 2\ and B = \ θ: (π)/(2) ≤ θ ≤...

2 - 2\sin^{2} \theta + sin \theta \le 2$<br />$2\sin^{2} \theta - sin \theta \ge 0$<br />$sin \theta\left(2sin \theta - 1\right) \ge 0$<br />Case I:$sin \theta \ge 0, sin \theta \ge \dfrac{1}{2}$<br />$⇒ sin \theta \ge \dfrac{1}{2} ⇒ \theta \in \left[\dfrac{\pi}{2}, \dfrac{5\pi}{6}\right]$<br /><img src="https://cdn-question-pool.Quantrex.app/nta_abhyas/jee_main/images/345678iugfdfghjkzjzjzjz.JPG" style="max-width:100$sin \theta \le 0, sin \theta \le \dfrac{1}{2}$<br />$⇒ sin \theta \le 0$<br />$⇒ \theta \in \left[\pi, \dfrac{3\pi}{2}\right]$<br />$⇒ A \cap B = \left\{\theta: \dfrac{\pi}{2} \le \theta \le \dfrac{5\pi}{6}\right\}$or \left\{\theta: \pi \le \theta \le \dfrac{3\pi}{2}\right\}$