What is the distance between #(2 ,(5 pi)/6 )# and #(4 , (23 pi )/12 )#?

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Answer 1 · 2016-09-06T14:22:19

#sqrt(5+sqrt 2(sqrt 3 +1))#

If O is the pole, A is #(2, 5/6pi)# and B is #(4, 33/12pi)#, then AB is

#sqrt(OA^2+OB^2-2XOAXOBXcosangleAOB)#

#=sqrt(2^2+4^2-(2)(2)(4)cos(23/12pi-5/6pi)#

#=sqrt(20-16 cos(13/12pi))#

#=2sqrt (5-4coss(pi+pi/12)#

#=2 sqrt(5+4cos 15^o)#

Here, #cos 15^o=cos(45^o-30^o)#

#=cos 45^o cos 30^o + sin 45^o sin 30^o#

#=(sqrt 3+1)/(2sqrt 2)#.

So, #AB = sqrt(5+4((sqrt3+1)/(2sqrt2))#

#=sqrt(5+sqrt 2(sqrt 3 +1))#