How to find the exact value of csc ( 5 π 6 ) using the half angle formula?

Question

How to find the exact value of             csc              (                              5            π                    6                )             using the half angle formula?

buenasvistasm9xy · Accepted Answer

Half-angle formula for             sin             is :             sin              (                  θ          2                )              =    ±                                        1            -                          cos              θ                                2                      ,   where sign is to be taken properly.Putting,       θ    =    5          π      3      , we get,            sin              {                              5                          π              3                                2                }              =          sin              (        5                  π          6                )              =    ±                                        1            -                          cos              5                                      π              3                                2                    Since,             sin              (        5                  π          6                )              =          sin              (        π        -                  π          6                )              ,    5          π      6       lies in the       I          I              n        d             Quadrant,       +    v    e   sign has to be taken  But,             cos      5              π      3        =          cos              (        2        π        -                  π          3                )              =          cos              (                  π          3                )              =          1      2        .        ∴          sin      5              π      6        =                                        1            -                          1              2                                2                      =                            1          4                      =          1      2        .  Therefore,             csc              (        5                  π          6                )              =          1              sin                  (          5                      π            6                    )                      =    2    .

Admiddevadxvi · Accepted Answer

We can assess csc ((5pi)/6) without using half angle formula.

\csc (\frac{5 π}{6}) = \frac{1}{\sin (\frac{5 π}{6})}

.
Find

\sin (\frac{5 π}{5}) .

Trig table, and unit circle -->

\sin (\frac{5 π}{6}) = \sin (- \frac{π}{6} + \frac{6 π}{6}) = \sin (- \frac{π}{6} + π) =

= \sin (\frac{π}{6}) = \frac{1}{2}

Hence,

\csc (\frac{5 π}{6}) = \frac{1}{\sin} = 2

How to find the exact value of csc ((5pi)/6) using the half angle formula?

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