How to use the half angle formula to solve tan ( θ 2 ) = sin θ ?

Question

How to use the half angle formula to solve             tan              (                  θ          2                )              =          sin      θ      ?

plumingagcqr · Accepted Answer

Use             sin      θ        =                  2                  tan                      (                          θ              2                        )                                      1        +                              tan            2                                (                          θ              2                        )                              Here,            tan              (                  θ          2                )              =          sin      θ        =                  2                  tan                      (                          θ              2                        )                                      1        +                              tan            2                                (                          θ              2                        )                              So,             tan              (                  θ          2                )                    (              (        1        +                              tan            2                                (                          θ              2                        )                          -        2        )            =      0      And so,             tan              (                  θ          2                )              =    0   gives             θ      2        =    k    π    ,    k    =    0    ,    1    ,    2    ,    3    ,    .    .  the other factor = 0 gives tan^(theta/2)=1 to tan(theta/2)=+-1 to theta/2=kpi+-pi/4. Combining both for values of       θ  ,.       θ    =    2    k    π                and              θ    =          (      2      k      ±              1        2            )        π    ,    k    =    0    ,    1    ,    2    ,    3    ,    ...

Charlie Hatfield · Accepted Answer

Implement the trig identity.sin a = 2sin (a/2).cos (a/2)Change the formula:                              sin                      (                          t              2                        )                                                cos                      (                          t              2                        )                                =    2          sin              (                  t          2                )              .          cos              (                  t          2                )            Simplify both sides by sin (t/2(, we get:            1                        cos                      (                          t              2                        )                                =    2          cos              (                  t          2                )                  2                  cos        2                    (                  t          2                )              =    1                      cos        2                    (                  t          2                )              =          1      2                  cos              (                  t          2                )              =    ±          1              2              =    ±                  2            2      Trig table of special arcs, and trig unit circle give 4 solution arcs:a.             cos      x        =                  2            2       --&amp;gt;       x    =    ±          π      4      Two solution arcs:      x    =          π      4       and       x    =                  7        π            4       (arc co-terminal to arc                     -        3        π            4      )b.             cos      x        =    -                  2            2       --&amp;gt;       x    =    ±                  3        π            4      Two solution arcs:      x    =                  3        π            4      , and       x    =                  5        π            4       (arc co-terminal to arc                     -        3        π            4      )Answers for             (      0      ,      2      π      )                  π      4        ;                  3        π            4        ,                  5        π            4        ,                  7        π            4

How to use the half angle formula to solve Tan(theta/2) = Sin theta?

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