What is the distance between the following polar coordinates?: ( 1 , 3 π 4 ) , ( 5 , 5 π 8 )

Question

What is the distance between the following polar coordinates?:             (      1      ,                        3          π                4            )        ,          (      5      ,                        5          π                8            )

Madalynn Acosta · Accepted Answer

Step 1Call the two points A and B:      A    =          (              r        a            ,              θ        a            )        =          (      1      ,                        3          π                4            )            B    =          (              r        b            ,              θ        b            )        =          (      5      ,                        5          π                8            )      Convert them to rectangular form:      A    =          (              x        a            ,              y        a            )                  x      a        =          r      a              cos              (                  θ          a                )              =    1    ⋅          cos              (                              3            π                    4                )              =    -                            2                    2                  y      a        =          r      a              sin              (                  θ          a                )              =    1    ⋅          sin              (                              3            π                    4                )              =                            2                    2            A    =          (      -                                    2                          2            ,                                    2                          2            )        ≈          (      -      0.707      ,      0.707      )            B    =          (              x        b            ,              y        b            )                  x      b        =          r      b              cos              (                  θ          b                )              =    5    ⋅          cos              (                              5            π                    8                )              =    -    5                            2          -                                    2                                          2                  y      b        =          r      b              sin              (                  θ          b                )              =    5    ⋅          sin              (                              5            π                    8                )              =    5                            2          +                                    2                                          2            B    =          (      -      5                                    2            -                                          2                                                    2            ,      5                                    2            +                                          2                                                    2            )        ≈          (      -      1.913      ,      4.619      )      Now apply the Pythagorean Theorem to find the length of the line segment                     A        B            ¯                  |              |                              A            B                    ¯                |            |        =                                        (                          x              b                        -                          x              a                        )                    2                +                              (                          y              b                        -                          y              a                        )                    2                                |              |                              A            B                    ¯                |            |        ≈                                        (                          (              -              1.913              )                        -                          (              -              0.707              )                        )                    2                +                              (            4.619            -            0.707            )                    2                                |              |                              A            B                    ¯                |            |        ≈                                        (            -            1.206            )                    2                +                              (            3.912            )                    2                      =                  1.455        +        15.306                        |              |                              A            B                    ¯                |            |        ≈                  16.761              ≈    4.094

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

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