What is the distance between the following polar coordinates?: ( 2 , 3 π 4 ) , ( 1 , 15 π 8 )

Question

What is the distance between the following polar coordinates?:            (      2      ,                        3          π                4            )        ,          (      1      ,                        15          π                8            )

Bruce Bridges · Accepted Answer

Step 1            Distance between Polar Co-ordinates:        A          (              r        1            ,              θ        1            )                    and              B          (              r        2            ,              θ        2            )       is                    D        =                                                            r                1                2                                      +                                          r                2                2                                      -            2                          r              1                                      r              2                                      cos                              (                                  θ                  1                                -                                  θ                  2                                )                                                    ...        →                  (          I          )                    We have,             P      1              (      2      ,                        3          π                4            )                    and                    P      2              (      1      ,                        15          π                8            )      So,             r      1        =    2    ,          r      2        =    1    ,          θ      1        =                  3        π            4                    and                    θ      2        =                  15        π            8            ⇒          θ      1        -          θ      2        =                  3        π            4        -                  15        π            8        =                  6        π        -        15        π            8        =                  -        9        π            8      Step 2      ⇒          cos              (                  θ          1                -                  θ          2                )              =          cos              (                              -            9            π                    8                )                  ⇒          cos              (                  θ          1                -                  θ          2                )              =          cos              (                              9            π                    8                )              →          [      ∵              cos                  (          -          θ          )                    =              cos        θ            ]            ⇒          cos              (                  θ          1                -                  θ          2                )              =          cos              (                              9            π                    8                )              →          [      ∵              cos                  (          -          θ          )                    =              cos        θ            ]            ⇒          cos              (                  θ          1                -                  θ          2                )              =          cos              (        π        +                  π          8                )              =    -          cos              (                  π          8                )            Using: (I) we get      D    =                            2          2                +                  1          2                -        2                  (          2          )                          (          1          )                          (          -                      cos                          (                              π                8                            )                                )                          ⇒    D    =                  4        +        1        +        4        ⋅                  cos                      (                          π              8                        )                                    ⇒    D    =                  5        +        4                  cos                      (                          π              8                        )                                    ⇒    D    ≈    2.9488

Jude Hunt · Accepted Answer

Step 1It is nontrivial to determine polar distances (since drawing a line in polar coordinates isn&#039;t very easy), so we should convert to Cartesian coordinates.The first point is moderately easy:      x    =    r          cos      θ        =    2    ⋅    -                            2                    2        =    -                  2                  y    =    r          sin      θ        =    2    ⋅                            2                    2        =                  2            The second point is a little harder, but we only need to use half angle identities. Let      θ    =                  15        π            4        ≡    -                  π            4       . Therefore,             cos              (        θ        )              =                            2                    2       and            cos              (                  θ          2                )              =    ±                                        1            +                          cos              θ                                2                      =    ±                            2          +                      2                              2                  sin              (                  θ          2                )              =    ±                            1          -                      cos            θ                              2        =    ±                            2          -                      2                              2      Since                     15        π            8       s in the 4th quadrant, its cosine is positive and its sine is negative. Therefore,      x    =    r          cos              (                              15            π                    8                )              =                            2          +                      2                              2            y    =    r          sin              (                              15            π                    8                )              =                            2          -                      2                              2      This allows us to use Pythagorean theorem in order to find the distance between the two points:      Δ    x    =                            2          +                      2                              2        +          2            Δ    y    =                            2          -                      2                              2        -          2      Pulling out a factor of                     2             from each, then squaring and adding them,      d          s      2        =    2          [      1      +                        2                2            +      1      +      1      -                        2                2            +      1      +      2                        1          +                                    2                        2                              -      2                        1          -                                    2                        2                              ]            =    2          [      4      +      2              (                              1            +                                          2                            2                                      -                              1            -                                          2                            2                                      )            ]      Using the cosine sum formula with             π      8       and             π      4       , we get      =    8          (      1      +              cos                  (                                    3              π                        8                    )                    )        =    4                  cos        2                    (                              3            π                    16                )            Therefore the total distance is      D    =    2          cos              (                              3            π                    16                )              =                  2        +                              2            -                                          2                                                          ≈    1.66

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

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