First, the range of the cosinus function is [-1;1]

#rarr# therefore the range of #4cos(X)# is [-4;4]

#rarr# and the range of #4cos(X)+2# is [-2;6]

Second, the period #P# of the cosinus function is defined as: #cos(X) = cos(X+P)# #rarr P = 2pi#.

#rarr# therefore:

#(3theta_2+3/2pi)-(3theta_1+3/2pi) = 3(theta_2-theta_1) = 2pi#

#rarr# the period of #4cos(3theta+3/2pi)+2# is #2/3pi#

Third, #cos(X)=1# if #X=0#

#rarr# here #X=3(theta+pi/2)#

#rarr# therefore #X=0# if #theta = -pi/2#

#rarr# therefore the phase shift is #-pi/2#