Search any question & find its solution
Question:
Answered & Verified by Expert
If $\sin \theta=\frac{-4}{5}$ and $\theta$ lies in the third quadrant, then $\cos \frac{\theta}{2}=$
Options:
Solution:
2566 Upvotes
Verified Answer
The correct answer is:
$-\frac{1}{\sqrt{5}}$
Given that $\sin \theta=-\frac{4}{5}$ and $\theta$ lies in the III quadrant.
$\begin{aligned}
& \Rightarrow \cos \theta=\sqrt{1-\frac{16}{25}}= \pm \frac{3}{5} \\
& \cos \frac{\theta}{2}= \pm \sqrt{\frac{1+\cos \theta}{2}}=\sqrt{\frac{1-3 / 5}{2}}= \pm \sqrt{\frac{1}{5}} \\
& \text { But } \cos \frac{\theta}{2}=-\frac{1}{\sqrt{5}} \text {. since } \frac{\theta}{2} \text { will be in II quadrant. } \\
& Hence \cos \frac{\theta}{2}=-\frac{1}{\sqrt{5}} \\
&
\end{aligned}$
$\begin{aligned}
& \Rightarrow \cos \theta=\sqrt{1-\frac{16}{25}}= \pm \frac{3}{5} \\
& \cos \frac{\theta}{2}= \pm \sqrt{\frac{1+\cos \theta}{2}}=\sqrt{\frac{1-3 / 5}{2}}= \pm \sqrt{\frac{1}{5}} \\
& \text { But } \cos \frac{\theta}{2}=-\frac{1}{\sqrt{5}} \text {. since } \frac{\theta}{2} \text { will be in II quadrant. } \\
& Hence \cos \frac{\theta}{2}=-\frac{1}{\sqrt{5}} \\
&
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.