Try to find the number of solutions of the equation sin ⁡ θ + cos ⁡ θ = sin ⁡ 2 θin the interval [ − π , π ]

Question

Try to find the number of solutions of the equation  sin  ⁡  θ  +  cos  ⁡  θ  =  sin  ⁡  2  θin the interval   [  −  π  ,  π  ]

Tasinazzokbc · Accepted Answer

By only considering   sin  ⁡  2  θ  =                    1        −                  5                    2      , one gets four roots since   sin  ⁡  2  θ has a period of π and the interval   [  −  π  ,  π  ] is twice that length.But one also has to consider the equality of   sin  ⁡  θ  +  cos  ⁡  θ  =                    1        −                  5                    2      . Since  sin  ⁡  θ  +  cos  ⁡  θ  =      2    sin  ⁡      (          π      4        +    θ    )  has a period of   2  π, actual number of roots is only 2.

Hayley Mcclain · Accepted Answer

If you use the multiple angle formula   θ  =  2      tan          −      1        ⁡  (  x  )  sin  ⁡  (  θ  )  +  cos  ⁡  (  θ  )  −  sin  ⁡  (  2  θ  )  =  0    ⟹        x    4    −  6      x    3    +  2  x  −  1  =  0which has only two real solutions      x    1    =            3      +              5              2    +            1      2              (      11      +      5              5            )            x    2    =            3      +              5              2    −            1      2              (      11      +      5              5            )      Then       θ    1    ∼  2.80847 and       θ    2    ∼  −  1.23768 that you observed plotting.Be always careful when you square.

Try to find the number of solutions of the equation sin theta+cos theta=sin 2 theta in the interval [−pi,pi].

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