What is the distance between the following polar coordinates?: # (1,(3pi)/4), (5,(5pi)/8) #

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Answer 1 · 2018-03-21T05:58:52

Distance#~~4.094#

Call the two points A and B:

#A=(r_a, theta_a) = (1, (3pi)/4)#

#B=(r_b, theta_b) = (5, (5pi)/8)#

Convert them to rectangular form:

#A=(x_a, y_a)#

#x_a = r_acos(theta_a) = 1*cos((3pi)/4) = -sqrt(2)/2#

#y_a = r_asin(theta_a) = 1*sin((3pi)/4) = sqrt(2)/2#

#A=(-sqrt(2)/2, sqrt(2)/2)~~(-0.707,0.707)#

#B=(x_b, y_b)#

#x_b = r_bcos(theta_b) = 5*cos((5pi)/8) = -5sqrt(2 - sqrt(2))/2#

#y_b = r_bsin(theta_b) = 5*sin((5pi)/8) = 5sqrt(2+sqrt(2))/2#

#B=(-5sqrt(2-sqrt(2))/2, 5sqrt(2+sqrt(2))/2)~~(-1.913,4.619)#

Now apply the Pythagorean Theorem to find the length of the line segment #bar(AB)#.

#||bar(AB)|| = sqrt((x_b-x_a)^2+(y_b-y_a)^2)#

#||bar(AB)|| ~~sqrt(((-1.913)-(-0.707))^2+(4.619-0.707)^2)#

#||bar(AB)|| ~~sqrt((-1.206)^2+(3.912)^2)=sqrt(1.455+15.306)#

#||bar(AB)|| ~~sqrt(16.761)~~4.094#