# Given cos A= 2/5 for those value for A below, what is sin (A/2)?

## cos A=2/5 ......... (3(pi))/(2) < A <2pi

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Jim G. Share
Feb 22, 2018

#### Answer:

$\sin \left(\frac{A}{2}\right) = + \frac{\sqrt{30}}{10}$

#### Explanation:

$\text{using the "color(blue)"half angle formula}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\sin \left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 - \cos A}{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{since "(3pi)/2< A<2pi" then A is in the fourth quadrant}$

$\text{where } \sin A < 0$

$\Rightarrow \frac{3 \pi}{4} < \frac{A}{2} < \pi$

$\Rightarrow \sin \left(\frac{A}{2}\right) = + \sqrt{\frac{1 - \frac{2}{5}}{2}}$

$\textcolor{w h i t e}{\Rightarrow \sin \left(\frac{A}{2}\right)} = + \sqrt{\frac{3}{10}} = + \frac{\sqrt{3}}{\sqrt{10}} = + \frac{\sqrt{30}}{10}$

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