Question 451184
<pre><b>
{{{sin(expr(1/2)a)}}}

Start with the identity:

{{{Cos(2alpha) = Cos^2alpha - Sin^2alpha}}}

{{{Cos(2alpha) = (1-Sin^2alpha) - Sin^2alpha}}}

{{{Cos(2alpha) = 1-Sin^2alpha - Sin^2alpha}}}

{{{Cos(2alpha) = 1-2Sin^2alpha}}} <-- sometimes this is given. If so you can start here

{{{2Sin^2alpha = 1 - Cos(2alpha)}}}

{{{Sin^2alpha = (1 - Cos(2alpha))/2}}}

{{{Sin(alpha) = "" +- sqrt((1 - Cos(2alpha))/2)}}}

Now substitute {{{alpha=expr(1/2)a}}}

{{{Sin(expr(1/2)a) = "" +- sqrt((1 - Cos(2expr(1/2)a))/2)}}}

Do one cancellation on the right:

{{{Sin(expr(1/2)a) = "" +- sqrt((1 - Cos(cross(2)expr(1/cross(2))a))/2)}}}

{{{Sin(expr(1/2)a) = "" +- sqrt((1 - Cos(a))/2)}}}

The sign to use depends on the quadrant
in which the terminal side of {{{expr(1/2)a}}} lies. 

If the terminal side of {{{expr(1/2)a}}} lies in the 
first or second quadrants, the identity uses the 
positive sign.

If the terminal side of {{{expr(1/2)a}}} lies in the 
third or fourth quadrants, the identity uses the 
negative sign.

Edwin</pre>