Question

If sin x = $\frac{\sqrt{5}}{3}$ and x lies in IInd quadrant, find the values of $cos\frac{x}{2},andsin\frac{x}{2}$ and $tan\frac{x}{2}$.

Answer 1

$\therefore$ x lies in $I{I}^{nd}$ quad.

$\Rightarrow \frac{\pi }{2}<x<\pi \Rightarrow \frac{\pi }{4}<x\frac{x}{2}<\frac{\pi }{2}$

Which means $\frac{x}{2}$ lies in first quad.

Now, $sinx=\frac{\sqrt{5}}{3}=\frac{p}{h}$

$\Rightarrow p=\sqrt{5}\Rightarrow b=2$

h = 3

So, $cosx\frac{b}{h}=\frac{-2}{3}$

(-ve due to $I{I}^{nd}$ quad)

Thus,

$cos\frac{x}{2}=\sqrt{\frac{1+cosx}{2}}=\sqrt{\frac{1-\frac{2}{3}}{2}}=\frac{1}{\sqrt{6}}sin\frac{x}{2}=\sqrt{\frac{1-cosx}{2}}=\sqrt{\frac{1+\frac{2}{3}}{2}}=\sqrt{\frac{5}{6}}tan\frac{x}{2}=\frac{sin\frac{x}{2}}{cos\frac{x}{2}}=\frac{\sqrt{\frac{5}{6}}}{\frac{1}{\sqrt{6}}}=\sqrt{5}$