How to find the exact value of cos 54 ?

Question

How to find the exact value of             cos      54      ?

GreerceViemkpy2 · Accepted Answer

cos          54                o            =              cos              (                  90          o                -                  36          o                )              =                            sin          36                o                  Let       A    =          36      o       hence       5    A    =          180      o      or       3    A    =          180      o        -    2    A   and hence            sin      3        A    =          sin              (                  180          o                -        2        A        )              =          sin      2        A   and adding to it, we obtain      3          sin      A        -    4                  sin        3            A        =    2          sin      A              cos      A      as             sin      A        ≠    0  , dividing by             sin      A      , we get      3    -    4                  sin        2            A        =    2          cos      A       or      3    -    4          (      1      -                        cos          2                A            )        -    2          cos      A        =    0   or      4                  cos        2            A        -    2          cos      A        -    1    =    0  and             cos      A        =                  2        ±                                            2              2                        -            4            ⋅            4            ⋅                          (              -              1              )                                                  2        ⋅        4              =                  2        ±                  20                    8        =                  1        ±                  5                    4      As             cos      A       cannot be negative,             cos      A        =                  1        +                  5                    4       and             sin      A        =                  1        -                                            (              1              +                              5                            )                        2                    16                      =                                        16            -                          (              6              +              2                              5                            )                                16                    =                                           10            -            2                          5                                16                      =                            10          -          2                      5                              4      Therefore                               cos          54                o            =                                10          -          2                      5                              4

How to find the exact value of cos54?

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