Given the hyperbolic metric d s 2 = d x 2 + d y 2 x 2 on the half plane x &gt; 0, find the length of the arc of the circle x 2 + y 2 = 1 from ( cos ⁡ α , sin ⁡ α ) to ( cos ⁡ β , sin ⁡ β )I found that d s 2 = d θ 2 cos 2 ⁡ θ but when I try to plug in π / 3 , − π / 3, which should give me the arc length of 2 π / 3,I get 4 π / 3 = ( π / 3 − ( − π / 3 ) ) 2 c o s 2 ( π / 3 ) I feel like I'm making a simple mistake but I cant place it

Question

Given the hyperbolic metric   d      s    2    =            d              x        2            +      d              y        2                    x      2       on the half plane   x  &amp;gt;  0, find the length of the arc of the circle       x    2    +      y    2    =  1 from   (  cos  ⁡  α  ,  sin  ⁡  α  ) to   (  cos  ⁡  β  ,  sin  ⁡  β  )I found that   d      s    2    =                    d                  θ          2                                      cos          2                ⁡        θ             but when I try to plug in   π      /    3  ,  −  π      /    3, which should give me the arc length of   2  π      /    3,I get   4  π      /    3  =                                          (            π                          /                        3            −            (            −            π                          /                        3            )            )                    2                          c          o                      s            2                                (            π                          /                        3                    )                    I feel like I&#039;m making a simple mistake but I cant place it

thatuglygirlyu · Accepted Answer

The circle       x    2    +      y    2    =  1 can be parametrised by   (  cos  ⁡  θ  ,  sin  ⁡  θ  ). If   x  (  θ  )  =  cos  ⁡  θ and   y  (  θ  )  =  sin  ⁡  θ then  d      s    2    =            d              x        2            +      d              y        2                    x      2        =            (              sin        2            ⁡      θ      +              cos        2            ⁡      θ      )            d              θ        2                            cos        2            ⁡      θ        =      sec    2    ⁡  θ    d      θ    2    .The arc-length that you are interested in is given by:  s  =  ∫      d          s      2        =      ∫          α              β            |    sec  ⁡  θ      |      d  θ    .

Given the hyperbolic metric d s 2 </msup> = d

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