Using expansion of sin ⁡ 8 θ to find equation with given roots

Question

Using expansion of   sin  ⁡  8  θ to find equation with given roots

Xzavier Shelton · Accepted Answer

Using   sin  ⁡  2  A  =  2  sin  ⁡  A  cos  ⁡  A repeatedly  sin  ⁡  8  x  =  8  sin  ⁡  x  cos  ⁡  x  cos  ⁡  2  x  cos  ⁡  4  xNow if   sin  ⁡  8  x  =  0  ,,either   sin  ⁡  x  =  0    ⟹    x  =  m  πor   cos  ⁡  x  =  0    ⟹    x  =  (  2  m  +  1  )            π      2          (  mod    2  π  )or   cos  ⁡  2  x  =  0    ⟹    x  =  (  2  m  +  1  )            π      4          (  mod    2  π  )or   cos  ⁡  4  x  =  0    ⟹    x  =  (  2  m  +  1  )            π      8        ,  x  ≡  ±            π      8        ,  ±                    3        π            8          (  mod    2  π  )Now   cos  ⁡  4  x  =  2      cos    2    ⁡  2  x  −  1  =  2  (  1  −  2      sin    2    ⁡  x      )    2    −  1  =  1  −  8      sin    2    ⁡  x  +  8      sin    4    ⁡  xSo, the equation whose roots are   ±  sin  ⁡            π      8        ,  ±  sin  ⁡                    3        π            8       will be  1  −  8      s    2    +  8      s    4    =  0

Using expansion of sin ⁡

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