LetZ1=(8+i)sin⁡θ+(7+4i)cos⁡θ and Z2=(1+8i)sin⁡θ +(4+7i)cos⁡θ are two complex numbers If Z1⋅Z2=a+i b where a,b∈R If M is the greatest value of (a + b)∀θ∈R then the value of M1/3 is____________

Z1=(8sin⁡θ+7cos⁡θ)+i(sin⁡θ+4cos⁡θ)Z2=(sin⁡θ+4cos⁡θ)+i(8sin⁡θ+7cos⁡θ)Hence,Z1=x+iy and Z2=y+ixWhere x=(8sin⁡θ+7cos⁡θ) and y=(sin⁡θ+4cos⁡θ)Z1⋅Z2=(xy−xy)+ix2+y2=ix2+y2=a+ib⇒ a=0;b=x2+y2Now, x2+y2=(8sin⁡θ+7cos⁡θ)2+(sin⁡θ+4cos⁡θ)2=65sin2⁡θ+65cos2⁡θ+120sin⁡θ⋅cos⁡θ=65+60sin⁡2θ⇒ Z1⋅Z2max=125