JEE Advanced · Mathematics · 8. Trigonometric Equations
The set of values of \(\theta\) satisfying the inequation \(2 \sin ^2 \theta-5 \sin \theta+2>0\), where \(0 < \theta < 2 \pi\), is
- A \(\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)\)
- B \(\left[0, \frac{\pi}{6}\right] \cup\left[\frac{5 \pi}{6}, 2 \pi\right]\)
- C \(\left[0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, 2 \pi\right]\)
- D None of these
Answer & Solution
Correct Answer
(A) \(\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)\)
Step-by-step Solution
Detailed explanation
As, \(2 \sin ^2 \theta-5 \sin \theta+2>0 \)
\( \Rightarrow(2 \sin \theta-1)(\sin \theta-2)>0 \)
\( [\text {where, }(\sin \theta-2) < 0 \text { for all } \theta \in R] \)
\(\therefore (2 \sin \theta-1) < 0\)
\(\Rightarrow \sin \theta < \frac{1}{2}, \text { shown as } \)
\( \therefore \theta \in\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)\)
\( \Rightarrow(2 \sin \theta-1)(\sin \theta-2)>0 \)
\( [\text {where, }(\sin \theta-2) < 0 \text { for all } \theta \in R] \)
\(\therefore (2 \sin \theta-1) < 0\)
\(\Rightarrow \sin \theta < \frac{1}{2}, \text { shown as } \)
\( \therefore \theta \in\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)\)
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