For f ( t ) = ( sin 2 t , t π - 2 ) what is the distance between f ( π 4 ) and f ( π ) ?

Question

For             f              (        t        )              =          (                        sin          2                t            ,              t        π            -      2      )       what is the distance between             f              (                  π          4                )             and             f              (        π        )             ?

Lara Alvarez · Accepted Answer

Step 1Find the two points by plugging in             π      4       and       π   for t in the point             f              (        t        )             :            f              (                  π          4                )              =          (                        sin          2                          (                      π            4                    )                    ,                                    π            4                          π            -      2      )                                    f                      (                          π              4                        )                                =          (                        (                                    2                        2                    )                2            ,              1        4            -      2      )                                    f                      (                          π              4                        )                                =          (              2        4            ,              1        4            -              8        4            )                                    f                      (                          π              4                        )                                =          (              1        2            ,      -              7        4            )      And            f              (        π        )              =          (                        sin          2                          (          π          )                    ,              π        π            -      2      )                                    f                      (            π            )                                =          (              0        2            ,      1      -      2      )                                    f                      (            π            )                                =          (      0      ,      -      1      )      So, we need to find the distance between the points             (              1        2            ,      -              7        4            )       and             (      0      ,      -      1      )       using the distance formula, which states that the distance between             (              x        1            ,              y        1            )       and             (              x        2            ,              y        2            )       is      d    =                                        (                          x              2                        -                          x              1                        )                    2                +                              (                          y              2                        -                          y              1                        )                    2                    Thus the distance we want is      d    =                                        (            0            -                          1              2                        )                    2                +                              (            -            1            -                          (              -                              7                4                            )                        )                    2                                d        =                                        (            -                          1              2                        )                    2                +                              (            -            1            +                          7              4                        )                    2                                d        =                                        (            -                          1              2                        )                    2                +                              (                          3              4                        )                    2                                d        =                            1          4                +                  9          16                                d        =                            13          16                                d        =                  13            4

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