Trigonometric Identities Question 134
Question: If $ \sin \theta =\frac{-4}{5} $ and $ \theta $ lies in the third quadrant, then $ \cos \frac{\theta }{2}= $
Options:
A) $ \frac{1}{\sqrt{5}} $
B) $ -\frac{1}{\sqrt{5}} $
C) $ \sqrt{\frac{2}{5}} $
D) $ -\sqrt{\frac{2}{5}} $
Show Answer
Answer:
Correct Answer: B
Solution:
Given that $ \sin \theta =-\frac{4}{5} $ and $ \theta $ lies in the III quadrant.
$ \Rightarrow \cos \theta =\sqrt{1-\frac{16}{25}}=\pm \frac{3}{5} $ $ \cos \frac{\theta }{2}=\pm \sqrt{\frac{1+\cos \theta }{2}}=\sqrt{\frac{1-3/5}{2}}=\pm \sqrt{\frac{1}{5}} $ But $ \cos \frac{\theta }{2}=-\frac{1}{\sqrt{5}}. $ since $ \frac{\theta }{2} $ will be in II quadrant. Hence $ \cos \frac{\theta }{2}=-\frac{1}{\sqrt{5}} $ .
BETA
