Trigonometric Equations Question 359
Question: If $ \cot \theta +\cot ( \frac{\pi }{4}+\theta )=2 $ , then the general value of $ \theta $ is
Options:
A) $ 2n\pi \pm \frac{\pi }{6} $
B) $ 2n\pi \pm \frac{\pi }{3} $
C) $ n\pi \pm \frac{\pi }{3} $
D) $ n\pi \pm \frac{\pi }{6} $
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Answer:
Correct Answer: D
Solution:
- $ \cot \theta +\cot ( \frac{\pi }{4}+\theta )=2\Rightarrow \frac{\cos \theta }{\sin \theta }+\frac{\cos {(\pi /4)+\theta }}{\sin {(\pi /4)+\theta }}=2 $
$ \Rightarrow $ $ \sin ( \frac{\pi }{4}+2\theta )=2\sin \theta \sin ( \frac{\pi }{4}+\theta ) $
$ \Rightarrow $ $ \sin ( \frac{\pi }{4}+2\theta )+\cos ( \frac{\pi }{4}+2\theta )=\frac{1}{\sqrt{2}} $
$ \Rightarrow $ $ \cos 2\theta =\frac{1}{2}\Rightarrow 2\theta =2n\pi \pm \frac{\pi }{3}\Rightarrow \theta =n\pi \pm \frac{\pi }{6} $ .
BETA
