MHT CET · Maths · Trigonometric Equations
The Solution set of the equation \(\sin ^2 \theta-\cos \theta=\frac{1}{4}\) in the interval \([0,2 \pi]\) is
- A \(\left\{\frac{\pi}{6}, \frac{5 \pi}{6}\right\}\)
- B \(\left\{\frac{\pi}{3}, \frac{5 \pi}{3}\right\}\)
- C \(\left\{\frac{\pi}{3}, \frac{2 \pi}{3}\right\}\)
- D \(\left\{\frac{2 \pi}{3}, \frac{4 \pi}{3}\right\}\)
Answer & Solution
Correct Answer
(B) \(\left\{\frac{\pi}{3}, \frac{5 \pi}{3}\right\}\)
Step-by-step Solution
Detailed explanation
\(\sin ^2 \theta-\cos \theta=\frac{1}{4} \)
\( \left(1-\cos ^2 \theta\right)-\cos \theta=\frac{1}{4} \)
\( 4-4 \cos ^2 \theta-4 \cos \theta-1=0 \)
\( \Rightarrow 4 \cos ^2 \theta+4 \cos \theta-3=0 \)
\( \Rightarrow \cos \theta=\frac{-1}{2} \text { or } \cos \theta=\frac{1}{2} \)
\( \therefore \theta=2 \pi-\frac{\pi}{3}, \frac{\pi}{3} \)
\( \therefore \theta=\left\{\frac{5 \pi}{3}, \frac{\pi}{3}\right\}\)
\( \left(1-\cos ^2 \theta\right)-\cos \theta=\frac{1}{4} \)
\( 4-4 \cos ^2 \theta-4 \cos \theta-1=0 \)
\( \Rightarrow 4 \cos ^2 \theta+4 \cos \theta-3=0 \)
\( \Rightarrow \cos \theta=\frac{-1}{2} \text { or } \cos \theta=\frac{1}{2} \)
\( \therefore \theta=2 \pi-\frac{\pi}{3}, \frac{\pi}{3} \)
\( \therefore \theta=\left\{\frac{5 \pi}{3}, \frac{\pi}{3}\right\}\)
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