Trigonometric Identities Question 256
Question: If $ \cos \theta +\cos 2\theta +\cos 3\theta =0, $ then the general value of $ \theta $ is :
Options:
A) $ \theta =2m\pi \pm 2\pi /3 $
B) $ \theta =2m\pi \pm \pi /4 $
C) $ \theta =m\pi +{{(-1)}^{n}}2\pi /3 $
D) $ \theta =m\pi +{{(-1)}^{n}}\pi /3 $
Show Answer
Answer:
Correct Answer: A
Solution:
Given $ \cos \theta +\cos 2\theta +\cos 3\theta =0 $
$ \Rightarrow (cos3\theta +cos\theta )+cos2\theta =0 $
$ \Rightarrow 2\cos 2\theta .\cos \theta +\cos 2\theta =0 $
$ \Rightarrow \cos 2\theta .(2\cos \theta +1)=0 $ we have, $ \cos \theta =\cos \alpha \Rightarrow \theta =2n\pi \pm \alpha $
$ \therefore $ For general value of $ \theta , $ $ \cos 2\theta =0 $
$ \Rightarrow ,\cos 2\theta =\cos \frac{\pi }{2}\Rightarrow ,2\theta =2n\pi \pm \frac{\pi }{2} $
$ \Rightarrow \theta =m\pi \pm \frac{\pi }{4} $ or $ 2\cos \theta +1=0; $
$ \Rightarrow ,\cos \theta =\frac{-1}{2}\Rightarrow \cos \theta =\cos \frac{2\pi }{3} $ So, $ \theta =2m\pi \pm \frac{2\pi }{3} $
BETA
