Eliminate θ from the equation x cos ⁡ ( 3 θ ) + y sin ⁡ ( 3 θ ) = a cos ⁡ ( θ ) x sin ⁡ ( 3 θ ) + y cos ⁡ ( 3 θ ) = a cos ⁡ ( θ + π 6 )I tried squaring a adding but got nowhere. Also got x and y as linear equations in form of θ but cant see what to do next.

Question

Eliminate   θ from the equation  x  cos  ⁡  (  3  θ  )  +  y  sin  ⁡  (  3  θ  )  =  a  cos  ⁡  (  θ  )    x  sin  ⁡  (  3  θ  )  +  y  cos  ⁡  (  3  θ  )  =  a  cos  ⁡  (  θ  +      π    6    )I tried squaring a adding but got nowhere. Also got   x and   y as linear equations in form of   θ but cant see what to do next.

Bettoldi1l · Accepted Answer

Not elegant at all!Squaring &amp;amp; adding                              x          2                +                  y          2                +        2        x        y        sin        ⁡        6        θ                    a        2              =      cos    2    ⁡  θ  +      cos    2    ⁡      (    θ    +                  π        6              )    =  1  +  cos  ⁡            π      6        cos  ⁡      (    2    θ    +                  π        6              )                (  1  )using Prove that   cos  ⁡  (  A  +  B  )  cos  ⁡  (  A  −  B  )  =            cos      2        A  −            sin      2        BAdding &amp;amp; squaring                    (        x        +        y                  )          2                (        1        +        sin        ⁡        6        θ        )                    a        2              =            [      cos      ⁡      θ      +      cos      ⁡              (        θ        +                              π            6                          )            ]        2    =      (    1    +    cos    ⁡                  π        6              )        (    1    +    cos    ⁡          (      2      θ      +                        π          6                    )        )                (  2  )using Prosthaphaeresis Formula   cos  ⁡  C  +  cos  ⁡  D and Double angle formula  2  θ  +            π      6        =  t  ,  6  θ  =  3  t  −            π      2      From (1),                              x          2                +                  y          2                −        2        x        y        cos        ⁡        3        t                    a        2              =  1  +  cos  ⁡            π      6        cos  ⁡  t              (  3  )From (2),                    (        x        +        y                  )          2                (        1        −        cos        ⁡        3        t        )                    a        2              =      (    1    +    cos    ⁡                  π        6              )        (    1    +    cos    ⁡    t    )                (  4  )Solve (3),(4) for   cos  ⁡  t  ,  cos  ⁡  3  tand use   cos  ⁡  3  t  =  4      cos    3    ⁡  t  −  3  cos  ⁡  t to eliminate   t

Eliminate θ from the equation x cos ⁡ ( 3

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