Compute the dot product of the two vectors:
#veca*vecb = cos(theta)sin(theta)+ sin(theta)cos(theta)#
#veca*vecb = 2sin(theta)cos(theta)#
Substitute #2sin(theta)cos(theta) = sin(2theta)#
#veca*vecb = sin(2theta)#
Use the other form of the dot product:
#veca*vecb = |veca||vecb|cos(alpha)#
The identity #cos^2(theta)+sin^2(theta) = 1# makes it obvious that the magnitudes of both vectors are 1:
#veca*vecb = cos(alpha)#
Substitute #veca*vecb = sin(2theta)#:
#sin(2theta) = cos(alpha)#
Use the inverse cosine on both sides:
#alpha = cos^-1(sin(2theta))#
Use the identity #cos^-1(x) = pi/2-sin^-1(x)# where #x = sin(2theta)#:
#alpha = pi/2-sin^-1(sin(2theta))#
The inverse sine of the sine leaves only the argument:
#alpha = pi/2-2theta#
