In cartesian coordinates #(x,y)#, the distance #(d)# between two points #(x_1,y_1)# and #(x_2,y_2)# is given by:
#d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2#.
Now let us transform that to polar coordinates. In this new system, we know that #x = rhocostheta# and #y = rhosintheta#. Then:
#d^2 = (rho_2costheta_2 - rho_1costheta_1)^2 + (rho_2sintheta_2 -rho_1sintheta_1)^2#.
With a few algebric steps, we can rewrite the above expression as:
#d^2 = rho_2^2(cos^2theta_2 + sin^2theta_2) + rho_1^2(cos^2theta_1 + sin^2theta_1) - 2rho_1rho_2(costheta_1costheta_2 + sintheta_1sintheta_2)# .
Now, by using the trigonometric relations, we obtain the following expression for #d#:
#d = sqrt(rho_1^2 + rho_2^2 - 2rho_1rho_2cos(theta_2 - theta_1)#.
Then, applying our points #(rho_1,theta_1) = (6,pi/3)# and #(rho_2,theta_2) = (0,pi/2)#:
#d = sqrt(36 + 0 - 0) = 6#.
