If sin = - 4 5 and lies in the third quadrant, then cos 2 is equal to

Question

If $\sin θ = - \frac{4}{5}$ and $θ$ lies in the third quadrant, then $\cos \frac{θ}{2}$ is equal to

Smart Achievers · Accepted Answer

(2)

Given that, $\sin θ = - \frac{4}{5}$ and $θ$ lies in the 3rd quadrant.

$∴$ $\cos θ = - \sqrt{1 - \frac{16}{25}} = - \frac{3}{5}$

Now, $\cos \frac{θ}{2} = \pm \sqrt{\frac{1 + \cos θ}{2}}$

$= \pm \sqrt{\frac{1 - \frac{3}{5}}{2}} = \pm \sqrt{\frac{1}{5}}$

But we take $\cos \frac{θ}{2} = - \frac{1}{\sqrt{5}} .$

Since, if $θ$ lies in the 3rd quadrant, then $\frac{θ}{2}$ will be in the 2nd quadrant.

Hence, $\cos \frac{θ}{2} = - \frac{1}{\sqrt{5}}$

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