Question

The number of solutions of $ \cos 2\theta =\sin \theta $ in $ (0,2\pi ) $ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$
• E. $0$

Answer 1

Answer: (C)

We have $ \cos 2\theta =\sin \theta $
$ \Rightarrow $ $ \cos 2\theta =\cos \left( \frac{\pi }{2}-\theta \right) $
$ \Rightarrow $ $ 2\theta =2n\pi \pm \left( \frac{\pi }{2}-\theta \right),n\in Z $
Taking + sign, we have $ \theta =\frac{2n\pi }{3}+\frac{\pi }{6},n\in Z $
$ \Rightarrow $ $ \theta =\frac{\pi }{6}+\frac{5\pi }{6}\in (0,2\pi ) $
Taking - sign, we have $ \theta =2n\pi -\frac{\pi }{2},n\in Z $ $ \theta =\frac{3\pi }{2} $
$ \Rightarrow $ $ \theta =\frac{3\pi }{2}\in (0,2\pi ) $
Hence, there are three solutions.