sin ⁡ x + sin ⁡ 2 x + sin ⁡ 3 x = 1 + cos ⁡ x + cos ⁡ 2 x - solving a trigonometric equation

Question

sin  ⁡      x    +  sin  ⁡      2    x    +  sin  ⁡      3    x    =  1  +  cos  ⁡      x    +  cos  ⁡      2    x   - solving a trigonometric equation

Colin Moran · Accepted Answer

sin  ⁡  x  +  sin  ⁡  3  x  =  2  ⋅  sin  ⁡  2  x  ⋅  cos  ⁡  x  cos  ⁡  2  x  =  2      cos    2    ⁡  x  −  1Then  2  ⋅  sin  ⁡      2    x    ⋅  cos  ⁡  x  +  sin  ⁡      2    x    =  1  +  cos  ⁡      x    +  2      cos    2    ⁡  x  −  1  sin  ⁡      2    x    ⋅  (  2  cos  ⁡  x  +  1  )  =  cos  ⁡      x    ⋅  (  2  cos  ⁡  x  +  1  )  (  2  cos  ⁡  x  +  1  )  ⋅  (  sin  ⁡  2  x  −  cos  ⁡  x  )  =  0  (  2  cos  ⁡  x  +  1  )  ⋅  (  cos  ⁡  x  )  ⋅  (  2  sin  ⁡  x  −  1  )  =  0  cos  ⁡  x  =  −      1    2    ⇒  x  =  ±            2      π        3    +  2  k  π  cos  ⁡  x  =  0  ⇒  x  =      π    2    +  k  π  sin  ⁡  x  =      1    2    ⇒  x  =      {          π      6        ,                  5        π            6        }    +  2  k  π

sin ⁡ <mrow class="MJX-TeXAtom-ORD"> x </mrow> + sin &#x

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