Trigonometric Identities Question 281
Question: If $ \cos 7\theta =\cos \theta -\sin 4\theta , $ then the general value of $ \theta $ is
Options:
A) $ \frac{n\pi }{6},\frac{n\pi }{3}+{{(-1)}^{n}}\frac{\pi }{18} $
B) $ \frac{n\pi }{3},\frac{n\pi }{3}+{{(-1)}^{n}}\frac{\pi }{18} $
C) $ \frac{n\pi }{4},\frac{n\pi }{3}\pm \frac{\pi }{18} $
D) $ \frac{n\pi }{4},\frac{n\pi }{3}+{{(-1)}^{n}}\frac{\pi }{18} $
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Answer:
Correct Answer: D
Solution:
$ \cos 7\theta =\cos \theta -\sin 4\theta \Rightarrow \sin 4\theta =\cos \theta -\cos 7\theta $
$ \Rightarrow ,\sin 4\theta =2\sin 4\theta \sin 3\theta $
$ \Rightarrow \sin 4\theta (1-2\sin 3\theta )=0 $
$ \therefore ,\sin 4\theta =0 $ or $ \sin 3\theta =\frac{1}{2} $
$ \Rightarrow ,4\theta =n\pi $ or $ 3\theta =n\pi +{{(-1)}^{n}}\frac{\pi }{6} $
$ \Rightarrow ,\theta =\frac{n\pi }{4} $ or $ \frac{n\pi }{3}+{{(-1)}^{n}}\frac{\pi }{18} $
BETA
