How to find sin ⁡ ( − 11 π 12 )

Question

How to find   sin  ⁡  (  −            11      π        12    )

redclick53 · Accepted Answer

Since the sine function is odd and fulfills

\sin (t) = \sin (π - t)

we have:

\sin (- \frac{11 π}{12}) = - \sin (\frac{11 π}{12}) = - \sin \frac{π}{12}

and since

\frac{π}{12}

is an acute angle, by the half-angle formula

\sin \frac{t}{2} = \sqrt{\frac{1 - \cos t}{2}}

we get:

\sin (- \frac{11 π}{12}) = - \sqrt{\frac{1 - \cos \frac{π}{6}}{2}} = - \frac{1}{2} \sqrt{2 - \sqrt{3}} .

rs450nigglix2 · Accepted Answer

Since   sin  ⁡  (  −  x  )  =  −  sin  ⁡  x  sin  ⁡      (    −                  11        π            12        )    =  −  sin  ⁡      (                  11        π            12        )  We can compute   sin  ⁡      (                  11        π            12        )   using the sum of angles formula for sine.                    sin        ⁡                  (                                    11              π                        12                    )                                    =        sin        ⁡                  (                                    8              π                        12                    +                                    3              π                        12                    )                                                  =        sin        ⁡                  (                                    2              π                        3                    +                      π            4                    )                                                  =        sin        ⁡                  (                                    2              π                        3                    )                cos        ⁡                  (                      π            4                    )                +        cos        ⁡                  (                                    2              π                        3                    )                sin        ⁡                  (                      π            4                    )                                                  =                  (                                    3                        2                    )                          (                                    2                        2                    )                +                  (          −                      1            2                    )                          (                                    2                        2                    )                                                  =                              6                    4                −                              2                    4                                                  =                                            6                        −                          2                                4                    Hence,  sin  ⁡      (    −                  11        π            12        )    =  −  sin  ⁡      (                  11        π            12        )    =  −      (                            6                −                  2                    4        )    =                    2            −              6              4

How to find sin ⁡

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