Question

What is the distance between the following polar coordinates?: `(6,(7pi)/12), (3,(-5pi)/8)`

Answer 1

graph{((x+3/2sqrt(2+sqrt(2)))^2+(y-3/2sqrt(2-sqrt(2)))^2-.01)((x+3sqrt(2+sqrt(3)))^2+(y+3sqrt(2-sqrt(3)))^2-.01)=0 [-10.71, 5.1, -2.67, 5.23]}

`d=3sqrt(5-sqrt(8+2(sqrt6-sqrt2)))`

Explanation:

`cos(pi/12)=sqrt(2+sqrt(3))/2`
`sin(pi/12)=sqrt(2-sqrt(3))/2`
`cos(pi/8)=sqrt(2+sqrt(2))/2`
`sin(pi/8)=sqrt(2-sqrt(2))/2`

so

`P_1=(-3sqrt(2+sqrt(3));-3sqrt(2-sqrt(3)))`
`P_2=(-3/2sqrt(2+sqrt(2));3/2sqrt(2-sqrt(2)))`

`bar(P_1P_2)=sqrt((-3sqrt(2+sqrt(3))+3/2sqrt(2+sqrt(2)))^2+(3sqrt(2-sqrt(3))+3/2sqrt(2-sqrt(2)))^2)=`
`=sqrt(18+cancel(9sqrt3)+9/2+cancel(9/4sqrt2)-9sqrt((2+sqrt3)(2+sqrt2))+18-cancel(9sqrt3)+9/2-cancel(9/4sqrt2)+9sqrt((2-sqrt2)(2-sqrt3)))=`
`=sqrt(45-9(sqrt(4+sqrt6+2(sqrt2+sqrt3))-sqrt(4+sqrt6-2(sqrt2+sqrt3))))`
`=3sqrt(5-(sqrt(4+sqrt6+2sqrt(5+2sqrt6))-sqrt(4+sqrt6-2sqrt(5+2sqrt6))))`

Now put
`a=8+2sqrt(6)`
and
`b=8`
then
`a^2-b=80+32sqrt(6)`
so using
`sqrt(a-sqrtb)=sqrt((a+sqrt(a^2-b))/2)-sqrt((a+sqrt(a^2-b))/2)`
`sqrt(8+2sqrt(6)-sqrt(8))=`
`=sqrt((8+2sqrt(6)+4sqrt(5+2sqrt6))/2)-sqrt((8+2sqrt(6)-4sqrt(5+2sqrt6))/2)`
`=sqrt(4+sqrt(6)+2sqrt(5+2sqrt6))-sqrt(4+sqrt(6)-2sqrt(5+2sqrt6))`

So

`bar(P_1P_2)=3sqrt(5-sqrt(8+2sqrt(6)-sqrt(8)))=3sqrt(5-sqrt(8+2(sqrt(6)-sqrt(2))))`