What is the formula for the distance between two polar coordinates?

2 Answers
Aug 21, 2014

Answer:

#sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2)#

Explanation:

The distance is #sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2)# if we are given #P_1=(r_1, theta_1)# and #P_2=(r_2, theta_2)#.

This is an application of the cosine law. Taking the difference between #theta_1# and #theta_2# gives us the angle between side #r_1# and side #r_2#. And the cosine law gives us the length of the #3^(rd)# side.

Jun 13, 2017

Answer:

See below.

Explanation:

Given in cartesian coordinates.

#P_1=(x_1,y_1)# and #P_2= (x_2,y_2)#

the transition formulas

#{(x=r cos theta),(y=r sin theta):}#

then

#(x_1,y_1) rArr (r_1 cos theta_1, r_1 sin theta_1)#
#(x_2,y_2) rArr (r_2 cos theta_2, r_2 sin theta_2)#

so

#d = sqrt((x_1-x_2)^2+(y_1-y_2)^2) rArr sqrt((r_1 costheta_1-r_2 cos theta_2)^2+(r_1 sin theta_1-r_2 sin theta_2)^2)#

then

#d = sqrt(r_1^2+r_2^2-2r_1r_2(cos theta_1 cos theta_2+sin theta_1 sin theta_2)) = sqrt(r_1^2+r_2^2-2r_1r_2cos (theta_1 -theta_2))#