Trigonometrical Equations 2 Question 2

2. The set of values of $\theta$ satisfying the inequation $2 \sin ^{2} \theta-5 \sin \theta+2>0$, where $0<\theta<2 \pi$, is

(2006, 3M)

(a) $0, \frac{\pi}{6} \cup \frac{5 \pi}{6}, 2 \pi$

(b) $0, \frac{\pi}{6} \cup \frac{5 \pi}{6}, 2 \pi$

(c) $0, \frac{\pi}{3} \cup \frac{2 \pi}{3}, 2 \pi$

(d) None of the above

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Answer:

Correct Answer: 2. (a)

Solution:

  1. Since, $2 \sin ^{2} \theta-5 \sin \theta+2>0$

$\Rightarrow \quad(2 \sin \theta-1)(\sin \theta-2)>0$

[where, $(\sin \theta-2)<0, \forall \theta \in R$ ]

$$ \therefore \quad(2 \sin \theta-1)<0 $$

$\Rightarrow \quad \sin \theta<\frac{1}{2}$

$\therefore$ From the graph, $\theta \in 0, \frac{\pi}{6} \cup \frac{5 \pi}{6}, 2 \pi$



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