What is the distance between the following polar coordinates?: # (2,(13pi)/4), (7,(-3pi)/8) #

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Answer 1 · 2017-01-10T16:46:45

# sqrt(53-28sqrt((1-1/sqrt2)/2))=6.503,# nearly.

Let the points be #P(2, 13/4pi) and Q(7, -3/5pi)#.

The position-vector #vec(OP)#direction #theta =13/4pi=4pi-3/4pi#.

This is same as is the same as #theta = -3/4pi#.

So, as far as the position is concerned, the first point P is (1, -3/4pi).

Now, the #angle POQ= (-3/8pi)-(-3/4pi)=3/8pi#,

The lengths-relation for the sides of the #triangle POQ# is

#PQ = sqrt( OP^2+OQ^2-2 (OP)(OP) cos angle POQ) #

#= sqrt(2^2+7^2-2(2)(7)cos(3/8pi))#

#cos (3/8pi)=sin(pi/2-3/8pi)=sin (pi/8)=sqrt((1-cos(2(pi/8)))/2)#

#=sqrt(((1-1/sqrt2)/2)#. And so,

#PQ= sqrt(53-28sqrt((1-1/sqrt2)/2))=6.503,# nearly.