If A , B , C ∈ R and cos ⁡ ( A − B ) + cos ⁡ ( B − C ) + cos ⁡ ( C − A ) = − 3 2 , , Then ∑ cos 3 ⁡ ( θ + A ) ∏ cos ⁡ ( θ + A ) , , Where θ ∈ R

Question

If   A  ,  B  ,  C  ∈      R   and       cos    ⁡    (    A    −    B    )    +    cos    ⁡    (    B    −    C    )    +    cos    ⁡    (    C    −    A    )    =    −          3      2            ,  , Then                    ∑                  cos          3                ⁡        (        θ        +        A        )                    ∏        cos        ⁡        (        θ        +        A        )                  ,   , Where   θ  ∈      R

Brooks Butler · Accepted Answer

Consider the sum  cos  ⁡  (  θ  +  A  )  +  cos  ⁡  (  θ  +  B  )  +  cos  ⁡  (  θ  +  C  )  cos  ⁡  (  θ  +  A  )  +  cos  ⁡  (  θ  +  B  )  +  cos  ⁡  (  θ  +  C  )=0Using the identity       a    3    +      b    3    +      c    3    =  3  a  b  c if (a+b+c)=0, we get  ∑      cos    3    ⁡  (  θ  +  A  )  =  3  ∏  cos  ⁡  (  θ  +  A  )

If A , B , C ∈ <mrow class="MJX-TeXAtom-ORD"> <mi mathva

Answered question

Answer & Explanation

New Questions in Trigonometry