What is the distance between the following polar coordinates?: # (4,(-7pi)/12), (2,(pi)/8) #

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Answer 1 · 2017-07-01T16:07:41

#D~~5.4535#

The distance formula for polar coordinates can be derived from the distance formula for rectangular coordinates

#D=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

Instead of using #x# and #y# values, though, we would just plug in their polar equivalents

#x=rcos(theta)#

#y=rsin(theta)#

Plugging those and using a couple of trigonometric identities, you get the following in purely polar coordinates

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

Plugging in the polar coordinates you have been given, we get

#D=sqrt((4)^2+(2)^2-2(4)(2)cos(-(7pi)/12-pi/8))#

#D=sqrt(16+4-16cos(-(17pi)/24))#

#D~~sqrt(20-16(-0.60876))#

#D~~5.4535#