Trigonometric Equations Question 336
Question: If $ {{\sin }^{2}}\theta -2\cos \theta +\frac{1}{4}=0, $ then the general value of $ \theta $ is
[MP PET 1984]
Options:
A) $ n\pi \pm \frac{\pi }{3} $
B) $ 2n\pi \pm \frac{\pi }{3} $
C) $ 2n\pi \pm \frac{\pi }{6} $
D) $ n\pi \pm \frac{\pi }{6} $
Show Answer
Answer:
Correct Answer: B
Solution:
- $ 1-{{\cos }^{2}}\theta -2\cos \theta +\frac{1}{4}=0 $
$ \Rightarrow $ $ {{\cos }^{2}}\theta +2\cos \theta -\frac{5}{4}=0 $
$ \Rightarrow $ $ \cos \theta =\frac{-2\pm \sqrt{4+5}}{2}=-1\pm \frac{3}{2} $ Since $ |\cos \theta |,\le 1 $ , hence $ \cos \theta =-1-\frac{3}{2} $ is ruled out.
$ \Rightarrow $ $ \cos \theta =-1+\frac{3}{2}=\frac{1}{2}=\cos ( \frac{\pi }{3} ) $
$ \Rightarrow $ $ \theta =2n\pi \pm \frac{\pi }{3} $ .
BETA
