Solving sin 2 ⁡ θ + cos ⁡ θ = 1

Question

Solving       sin    2    ⁡  θ  +  cos  ⁡  θ  =  1

glorietka4b · Accepted Answer

Given      sin    2    ⁡  θ  +  cos  ⁡  θ  =  1  ,and using the circular identity      sin    2    ⁡  θ  +      cos    2    ⁡  θ  =  1  ,,it follows that      cos    2    ⁡  θ  =  cos  ⁡  θ  ,,or  (  cos  ⁡  θ  −  1  )  cos  ⁡  θ  =  0..Hence  cos  ⁡  θ  ∈  {  0  ,  1  }  ,,which implies  θ  ∈  {                    (        2        k        +        1        )            2        π  ,  2  k  π  }  ,    k  ∈      Z    .You can check this: if   θ is an integer multiple of   2  π, then   sin  ⁡  θ  =  0 and   cos  ⁡  θ  =  1, so       sin    2    ⁡  θ  +  cos  ⁡  θ  =  1. If   θ is an odd multiple of   π      /    2, then   sin  ⁡  θ  =  ±  1, so       sin    2    ⁡  θ  =  1, and   cos  ⁡  θ  =  0, which also checks out.

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