In nonlinear optical switching devices based on dye-doped polymer systems, the spatial orientation of the dye molecules in the polymer is an important parameter. These devices are generally constructed by orienting dye molecules with a large dipole moment using an electric field. Imagine placing a vector along the molecular dipole moment such that the molecular orientation can be described by the orientation of this vector in space relative to the applied field (z direction) as illustrated in the text. 

For random molecular orientation about the z axis, the probability distribution describing molecular orientation along the z axis is given by:

Orientation is quantified using moments of cos θ.

a. Determine Œ©cosθŒª for this probability distribution.

b. Determine Œ©cos2θŒª for this probability distribution

d = r(sin@) %3D sine de P(@) : sine de