MHT CET · Maths · Trigonometric Ratios & Identities
The number of solutions of \(\cos 2 \theta=\sin \theta\) in \((0,2 \pi)\) are
- A 3
- B 2
- C 4
- D 1
Answer & Solution
Correct Answer
(A) 3
Step-by-step Solution
Detailed explanation
\(
\begin{aligned}
& \cos 2 \theta=\sin \theta \\
& \therefore 1-2 \sin ^2 \theta=\sin \theta \Rightarrow 2 \sin ^2 \theta+\sin \theta-1=0 \\
& \therefore(2 \sin \theta-1)(\sin \theta+1)=0 \Rightarrow \sin \theta=\frac{1}{2},-1
\end{aligned}
\)
We have \(\theta \in(0,2 \pi)\)
\(\therefore\) Possible values of \(\theta\) are \(\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{3 \pi}{6}\)
\begin{aligned}
& \cos 2 \theta=\sin \theta \\
& \therefore 1-2 \sin ^2 \theta=\sin \theta \Rightarrow 2 \sin ^2 \theta+\sin \theta-1=0 \\
& \therefore(2 \sin \theta-1)(\sin \theta+1)=0 \Rightarrow \sin \theta=\frac{1}{2},-1
\end{aligned}
\)
We have \(\theta \in(0,2 \pi)\)
\(\therefore\) Possible values of \(\theta\) are \(\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{3 \pi}{6}\)
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