How to use an appropriate Half-Angle Formula to find the exact value of the expression sin ( 13 π 12 ) ?

Question

How to use an appropriate Half-Angle Formula to find the exact value of the expression             sin              (                              13            π                    12                )            ?

Karina Weeks · Accepted Answer

The appropriate Half-Angle Formula is: since we want a sine            sin              (                  x          2                )              -    ±                                        1            -                          cos              x                                2                    We note that                     13        π            12        &amp;gt;    π    =                  12        π            12       but       &amp;lt;                  3        π            2        =                  18        π            12      So it is in the third quadrant and has negative sine.To find       x  , we ask oursolves &quot;&quot;Ok, so                     13        π            12       is half of what?&quot;&quot;We could solve             1      2        x    =                  13        π            12      , but it is pretty clear that             1      2        ⋅                  13        π            6        =                  13        π            12      , so       x    =                  13        π            6      (Just double the number we&#039;re interested in.)Finally, we note that             cos              (                              13            π                    6                )              =                  3            2       So we get:            sin              (                  1          2                          (                                    13              π                        6                    )                )              =    -                                        1            -                          cos                              (                                                      13                    π                                    6                                )                                              2                          =    -                                        1            -                                          3                            2                                2                          =    -                                        2            -                          3                                4                          =    -                            2          -                      3                              2

How to use an appropriate Half-Angle Formula to find the exact value of the expression sin ((13pi)/12)?

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