What is the distance between the following polar coordinates?: # (4,pi), (5,pi) #

2 Answers
Dec 26, 2015

#1#

Explanation:

The distance formula for polar coordinates is

#d=sqrt(r_1^2+r_2^2-2r_1r_2Cos(theta_1-theta_2)#
Where #d# is the distance between the two points, #r_1#, and #theta_1# are the polar coordinates of one point and #r_2# and #theta_2# are the polar coordinates of another point.
Let #(r_1,theta_1)# represent #(4,pi)# and #(r_2,theta_2)# represent #(5,pi)#.
#implies d=sqrt(4^2+5^2-2*4*5Cos(pi-pi)#
#implies d=sqrt(16+25-40Cos(0)#
#implies d=sqrt(41-40*1)=sqrt(41-40)=sqrt(1)=1#
#implies d=1#
Hence the distance between the given points is #1#.

Dec 29, 2015

#1#

Explanation:

(this is an attempt to restore my original answer)

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Using common insight rather than applying the Pythagorean Theorem and #cos# conversions:

The distance between any two polar coordinates with the same angle is the difference in their radii.