I have interesting trigonometric expression for professionals in mathematical science. So, here it is: sin ⁡ 3 π 14 − sin ⁡ π 14 − sin ⁡ 5 π 14 ;

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I have interesting trigonometric expression for professionals in mathematical science. So, here it is:  sin  ⁡                    3        π            14        −  sin  ⁡            π      14        −  sin  ⁡                    5        π            14        ;

Cahokiavv · Accepted Answer

For simplicity, let   x  =      π    14  , then we want to simplify:  sin  ⁡  3  x  −  sin  ⁡  x  −  sin  ⁡  5  xMultiply by   cos  ⁡  x to get:      sin    ⁡    3    x    cos    ⁡    x    −      sin    ⁡    x    cos    ⁡    x    −      sin    ⁡    5    x    cos    ⁡    x      (  ∗  )With   sin  ⁡  α  cos  ⁡  β  =            1      2            (    sin    ⁡    (    α    +    β    )    +    sin    ⁡    (    α    −    β    )    )  , you have:      sin    ⁡    3    x    cos    ⁡    x    =                  1        2                    (      sin      ⁡      4      x      +      sin      ⁡      2      x      )              and          sin    ⁡    5    x    cos    ⁡    x    =                  1        2                    (      sin      ⁡      6      x      +      sin      ⁡      4      x      )      and       sin    ⁡    x    cos    ⁡    x    =                  1        2              sin    ⁡    2    x    (  ∗  )              1      2            (          sin      ⁡      4      x        +          sin      ⁡      2      x        )    −                    1        2              sin    ⁡    2    x    −            1      2            (    sin    ⁡    6    x    +          sin      ⁡      4      x        )    =  −            1      2        sin  ⁡  6  xDivide again by   cos  ⁡  x  −            1      2                  sin      ⁡      6      x              cos      ⁡      x        =  −            1      2                  sin      ⁡                        6          π                14                    cos      ⁡              π        14              =  −            1      2                  cos      ⁡              (                  π          2                −                              6            π                    14                )                    cos      ⁡              π        14              =  −            1      2                  cos      ⁡              π        14                    cos      ⁡              π        14              =  −      1    2

Emmy Knox · Accepted Answer

Let   14  x  =  π  S  =  sin  ⁡  3  x  −  sin  ⁡  x  −  sin  ⁡  5  x  =  sin  ⁡  3  x  +  sin  ⁡  (  −  x  )  +  sin  ⁡  (  −  5  x  )Using How can we sum up sin and cos series when the angles are in arithmetic progression? ,  sin  ⁡  (  −  2  x  )  ⋅  S  =  cos  ⁡  5  x  −  cos  ⁡  7  x  −  sin  ⁡  2  x  ⋅  S  =  cos  ⁡  5  x  =  sin  ⁡  2  xAs   cos  ⁡  7  x  =  0  ,  cos  ⁡  5  x  =  sin  ⁡  2  x as   5  x  +  2  x  =            π      2

I have interesting trigonometric expression for professionals in mathematical science. So, here it i

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