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

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

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