Trigonometry equation cos ⁡ ( x ) − cos ⁡ ( 2 x ) + cos ⁡ ( 3 x ) = 0

Question

Trigonometry equation   cos  ⁡  (  x  )  −  cos  ⁡  (  2  x  )  +  cos  ⁡  (  3  x  )  =  0

Zayden Andrade · Accepted Answer

Notice,  cos  ⁡  x  −  cos  ⁡  2  x  +  cos  ⁡  3  x  =  0  (  cos  ⁡  x  +  cos  ⁡  3  x  )  −  cos  ⁡  2  x  =  0  2  cos  ⁡      (                  x        +        3        x            2        )    cos  ⁡      (                  x        −        3        x            2        )    −  cos  ⁡  2  x  =  0  2  cos  ⁡  2  x  cos  ⁡  x  −  cos  ⁡  2  x  =  0  cos  ⁡  2  x  (  2  cos  ⁡  x  −  1  )  =  0Now, solving for x as follows  cos  ⁡  2  x  =  0    ⟹    2  x  =  (  2  k  +  1  )      π    2    x  =  (  2  k  +  1  )      π    4        x    =          90      ∘        k    +          45      ∘      or  2  cos  ⁡  x  −  1  =  0  cos  ⁡  x  =      1    2    =  cos  ⁡      π    3    x  =  2  k  π  ±      π    3        x    =          360      ∘        k    ±          60      ∘      Where, k is any integer

Emmy Dillon · Accepted Answer

First we need to simplify   cos  ⁡  (  x  )  −  cos  ⁡  (  2  x  )  +  cos  ⁡  (  3  x  )  =  0  1  −  2  cos  ⁡  (  x  )  −  2      cos    2    ⁡  (  x  )  +  4      cos    3    ⁡  (  x  )  =  0  (  2  cos  ⁡  (  x  )  −  1  )  (  2      cos    2    ⁡  (  x  )  −  1  )  =  0Split to two equations:  2  cos  ⁡  (  x  )  −  1  =  0And it&#039;s easy to solve:  x  =      π    3    +  2  π      C    1   ;  x  =            5      π        3    +  2  π      C    2   ;  x  =      π    4    +  2  π      C    3  ;  x  =            7      π        4    +  2  π      C    4        C    1    ,      C    2    ,      C    3    ,      C    4    ∈      Z

Trigonometry equation cos ⁡ ( x ) − cos &

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