If \( \sin\theta=-\frac45 \) and \( \theta \) Lies in the Third Quadrant, Find \( \cos\frac{\theta}{2} \)
Question
If
\[ \sin\theta=-\frac45 \]
and \(\theta\) lies in the third quadrant, then the value of
\[ \cos\frac{\theta}{2} \]
is
(a) \(\frac15\)
(b) \(\frac1{\sqrt{10}}\)
(c) \(\frac1{\sqrt5}\)
(d) \(\frac{3}{\sqrt{10}}\)
Solution
Given
\[ \sin\theta=-\frac45 \]
Since \(\theta\) is in the third quadrant, \(\cos\theta\) is also negative.
Using
\[ \sin^2\theta+\cos^2\theta=1 \]
\[ \cos\theta = -\sqrt{1-\left(\frac45\right)^2} = -\sqrt{\frac{9}{25}} = -\frac35 \]
Now use the half-angle identity:
\[ \cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}} \]
Since \(\theta\) is in the third quadrant,
\[ 180^\circ<\theta<270^\circ \]
Therefore,
\[ 90^\circ<\frac{\theta}{2}<135^\circ \]
So \(\frac{\theta}{2}\) lies in the second quadrant, where cosine is negative.
Hence,
\[ \cos\frac{\theta}{2} = -\sqrt{\frac{1-\frac35}{2}} \]
\[ = -\sqrt{\frac{\frac25}{2}} = -\sqrt{\frac15} \]
\[ = -\frac1{\sqrt5} \]
Final Answer
\[ \boxed{-\frac1{\sqrt5}} \]
The exact value is \(\boxed{-\frac1{\sqrt5}}\). Since this is not listed among the given options, the question appears to have a sign error in the options. The magnitude corresponds to (c) \(\frac1{\sqrt5}\).