What is the distance between #(2 , (7 pi)/4 )# and #(8 , (3 pi )/2 )#?

#2 sqrt{17 - 4 sqrt 2}#

Explanation:

#u = (2 cos frac{7 pi}{4}, 2 sin frac{7 pi}{4}) #

#u = (2 cos frac{pi}{4}, -2 sin frac{pi}{4}) = (sqrt 2, -sqrt 2)#

#v = (8 cos frac{3 pi}{2}, 8 sin frac{3 pi}{2})#

#v = (0, -8)#

dist#(u, v) = sqrt {2 + (8 - sqrt 2)^2}#

#= sqrt{2 + 64 + 2 - 16 sqrt 2}#

But by your text, it could be different.