What is the distance between the following polar coordinates?: ( - 4 , 3 π 4 ) , ( 5 , 3 π 8 )

Question

What is the distance between the following polar coordinates?:             (      -      4      ,                        3          π                4            )        ,          (      5      ,                        3          π                8            )

luisjoseblash2 · Accepted Answer

Step 1Let&#039;s solve a general case; say we have two points of Polar coordinates       A          (              r        1            ,              φ        1            )       and       B          (              r        2            ,              φ        2            )      Let l be the distance between A and B:      l    =    A    B  We might convert A and B to Cartesian coordinates, for which we have a formula, but instead let&#039;s keep everything in Polar.In the triangle       Δ    A    O    B   , we know the lenght of the sides OA, OB and the angle between the;       ∠    A    O    B  As such, we can apply the cosine law:      A          B      2        =    O          A      2        +    O          B      2        -    2    O    A    ⋅    O    B          cos              (        ∠        A        O        B        )            Substituting everything into Polar coordinates:            l      2        =                  r        1        2              +                  r        2        2              -    2          r      1              r      2              cos              (                  φ          1                -                  φ          2                )            Finally, we can solve our particular case:      A          (      -      4      ,                        3          π                4            )                    and              B          (      5      ,                        3          π                8            )            ∴          l      2        =                  (        -        4        )            2        +          5      2        -    2          (      -      4      )        ⋅    5          cos              (                              3            π                    4                -                              3            π                    8                )                  ∴          l      2        =    16    +    25    +    40          cos              (                              3            π                    4                )            Notice how                     3        π            4        =    π    -          π      4       , hence            cos              (                              3            π                    4                )              =    -          cos              (                  π          4                )              =    -                  2            2            ∴          l      2        =    41    -    40    ⋅                  2            2        =    41    -    20          2            ∴    l    =                  41        -        20                  2

What is the distance between the following polar coordinates?: <mstyle displaystyle="true">

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