Question

Find $\theta \in [\frac{\pi }{2},\frac{3\pi }{2}]$ satisfying $2{\cos }^2\theta +\sin \theta \le 2.$

• A. $[\frac{\pi }{2},\frac{3\pi }{2}]\cap [\pi ,\frac{3\pi }{2}]$
• B. $[\frac{\pi }{2},\frac{5\pi }{6}]\cup [\pi ,\frac{3\pi }{2}]$
• C. $[\frac{\pi }{2},\frac{4\pi }{3}]\cap [\pi ,\frac{4\pi }{2}]$
• D. $[\frac{\pi }{2},\frac{2\pi }{3}]\cap [\pi ,\frac{2\pi }{2}]$

Answer 1

Answer: (B)

$[\frac{\pi }{2},\frac{5\pi }{6}]\cup [\pi ,\frac{3\pi }{2}]$

$2{\cos }^2\theta +sin\theta \le 2$

${\cos }^2\theta +si{n}^2\theta =1$

$=2(1-{\sin }^2\theta )+sin\theta \le 2$

$=2-2{\sin }^2\theta +sin\theta \le 2$

$=2{\sin }^2\theta -sin\theta \ge 0$

$=2t^2-t\ge 0$

let $\sin \theta =t$

$t(2t-1)\ge 0$

$\left(t\right)(2t-1)\ge 0$

$tϵ(-\infty ,0]$and$[1/2,-\infty )$

$t=\sin \theta$and $\sin \theta ϵ[-1,-1]$

so

$\sin \theta ϵ[-1,0]$ and$[1/2,1]$