If cos ⁡ ( x ) cos ⁡ ( 2 x ) cos ⁡ ( 3 x ) = 4 7 and S = 1 cos 2 ⁡ x + 1 cos 2 ⁡ 2 x + 1 cos 2 ⁡ 3 x when S ∈ R + then S=?

Question

If   cos  ⁡  (  x  )  cos  ⁡  (  2  x  )  cos  ⁡  (  3  x  )  =      4    7   and   S  =            1                        cos          2                ⁡                  x                      +            1                        cos          2                ⁡                  2          x                      +            1                        cos          2                ⁡                  3          x                     when   S  ∈            R              +       then S=?

Jovan Wong · Accepted Answer

Let   a  =  cos  ⁡  (  x  ),   b  =  cos  ⁡  (  2  x  ),   c  =  cos  ⁡  (  3  x  ), then:  a  b  c  =      4    7    S  =            1              a        2              +            1              b        2              +            1              c        2            Because   cos  ⁡  (  2  x  )  =  2      cos    2    ⁡  (  x  )  −  1,   ∴  b  =  2      a    2    −  1Because   cos  ⁡  (  3  x  )  =  cos  ⁡  (  x  )  cos  ⁡  (  2  x  )  −  sin  ⁡  (  x  )  (  2  sin  ⁡  (  x  )  cos  ⁡  (  x  )  )  =  a  b  −  2  a      sin    2    ⁡  (  x  )As   2  a      sin    2    ⁡  (  x  )  =  2  a  (  1  −      cos    2    ⁡  (  x  )  )  =  2  a  (  1  −      a    2    )  ∴  cos  ⁡  (  3  x  )  =  a  b  (  2      a    2    −  1  )  −  2  a  (  1  −      a    2    )  =  4      a    3    −  3  a  ∴  c  =  4      a    3    −  3  aSo we get the following equation:  a  (  2      a    2    −  1  )  (  4      a    3    −  3  a  )  =      4    7    56      a    6    −  70      a    4    +  21      a    2    −  4  =  0As I have know special way of solving such polynomials I turned to wolfram alpha and got an approximate root of a=−0.96392 and a=0.96392. But a=cos(x), so when a=−0.96292 x=164.562(3.dp) and when a=0.96392 x=15.4378(4.dp). Putting those values into the equation of S, we get:  S  =            1                        cos          2                ⁡        (        164.562        )              +            1                        cos          2                ⁡        (        2        ∗        164.562        )              +            1                        cos          2                ⁡        (        3        ∗        164.562        )              ∴  S  =  10.673Or  S  =            1                        cos          2                ⁡        (        15.4378        )              +            1                        cos          2                ⁡        (        2        ∗        15.4378        )              +            1                        cos          2                ⁡        (        3        ∗        15.4378        )              ∴  S  =  4.54217Answer:  S  =  4.54217  ,  S  =  10.673

If cos ⁡ ( x ) cos ⁡ ( 2 x )

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