Trigonometric Identities Question 31
Question: If $ \sin \theta +\sin 2\theta +\sin 3\theta =\sin \alpha $ and $ \cos \theta +\cos 2\theta +\cos 3\theta =\cos \alpha $ , then q is equal to
[AMU 2001]
Options:
A) $ \alpha /2 $
B) $ \alpha $
C) $ 2\alpha $
D) $ \alpha /6 $
Show Answer
Answer:
Correct Answer: A
Solution:
$ \sin \theta +\sin ,3\theta +\sin ,2\theta =\sin ,\alpha $
Þ $ 2\sin 2\theta \cos \theta +\sin 2\theta =\sin \alpha $
Þ $ \sin 2\theta (2\cos \theta +1)=\sin \alpha $ ?..(i) Now $ \cos \theta +\cos 3\theta +\cos 2\theta =\cos \alpha $ $ 2\cos 2,\theta \cos ,\theta +\cos 2\theta =\cos \alpha $ $ \cos 2\theta ,(2\cos \theta +1)=\cos \alpha $ ?..(ii) From (i) and (ii), $ \tan 2\theta =\tan \alpha $
Þ $ 2\theta =\alpha $
Þ $ \theta =\alpha /2 $ .
BETA
