Solve a given trigonometric equationSolve the following equation: sin ⁡ x + sin ⁡ ( x + 7 π 24 ) = 2 − 2 2 + 6 + 2 4 So far, I found out that 2 − 2 2 = sin ⁡ π 8 and 6 + 2 4 = sin ⁡ 7 π 12 .Thank you!

Question

Solve a given trigonometric equationSolve the following equation:  sin  ⁡  x  +  sin  ⁡      (    x    +                  7        π            24        )    =            2      −              2              2    +                    6            +              2              4  So far, I found out that             2      −              2              2    =  sin  ⁡      π    8   and                     6            +              2              4    =  sin  ⁡            7      π        12  .Thank you!

lesinetzgl5 · Accepted Answer

Hint Expand the second sin, to write yourequation as  A  cos  ⁡  (  x  )  +  B  sin  ⁡  (  x  )  =  1or  cos  ⁡  (  x  +  α  )  =      1                  A        2            +              B        2            second approachthe equation is  sin  ⁡  (  x  )  +  sin  ⁡  (  x  +            7      π        24    )  =  sin  ⁡  (      π    8    )  +  sin  ⁡  (            7      π        12    )  =  sin  ⁡  (      π    8    )  +  sin  ⁡      (    π    −    (          π      8        +                  7        π            24        )    )    =  sin  ⁡  (      π    8    )  +  sin  ⁡  (      π    8    +            7      π        24    )

Solve a given trigonometric equation sin x + sin ( x + (7 pi)/(24)) = ( sqrt(2 - sqrt 2))/(2) + (sqrt 6 + sqrt 2)/(4)

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