sin67.5^@
The half angle formula for sin(theta/2) = +-sqrt((1-costheta)/2).
The +- sign depends on which quadrant your angle is in. Since 67.5^@ is in the 1st quadrant and sine is positive in that quadrant, we know that it is positive.
67.5^@ * 2 = 135^@, so:
sin(theta/2) = sqrt((1- cos135^@)/2)
quadquadquadquadquadquadquad=sqrt((1-(-sqrt2/2))/2)
quadquadquadquadquadquadquad=sqrt((2/2+sqrt2/2)/2)
quadquadquadquadquadquadquad=sqrt(((2+sqrt2)/2)/2)
quadquadquadquadquadquadquad=sqrt((2+sqrt2)/4)
quadquadquadquadquadquadquad=sqrt(2+sqrt2)/2
Hope this helps!