Question 237746
Whenever a Trig problem refers to "exact" values, it implies that an angle which is a multiple of 30 or 45 is involved since these are the only angles for which exact values can be found.<br>
So in your problem we need to "connect" sin(22.5) with an angle that is a multiple of 30 or 45. This is not difficult since 22.5 = 45/2. So we'll use the half-angle formula formula for sin:
{{{sin((1/2)x) = 0 +- sqrt((1-cos(x))/2)}}}
Substituting 45 in for x:
{{{sin((1/2)45) = 0 +- sqrt((1-cos(45))/2)}}}
{{{sin(22.5) = 0 +- sqrt((1- sqrt(2)/2)/2)}}}
Since 22.5 is in the first quadrant and sin is positive in the first quadrant, we will use the positive square root:
{{{sin(22.5) = sqrt((1- sqrt(2)/2)/2)}}}
The only thing left to do is simplify the expression within the outer square root:
{{{sin(22.5) = sqrt((2/2- sqrt(2)/2)/2)}}}
{{{sin(22.5) = sqrt(((2 - sqrt(2))/2)/2)}}}
{{{sin(22.5) = sqrt((((2 - sqrt(2))/2)/2)(2/2))}}}
{{{sin(22.5) = sqrt((((2 - sqrt(2))/cross(2))/2)(cross(2)/2))}}}
{{{sin(22.5) = sqrt((2 - sqrt(2))/4)}}}
{{{sin(22.5) = sqrt(2 - sqrt(2))/sqrt(4)}}}
{{{sin(22.5) = sqrt(2 - sqrt(2))/2}}}