How many solutions exist to this trig relation? sin ⁡ ( 1 2 θ ) = 2 cos ⁡ ( 2 θ )

Question

How many solutions exist to this trig relation?  sin  ⁡      (          1      2        θ    )    =  2  cos  ⁡      (    2    θ    )

mentest91k99 · Accepted Answer

Hint: The left hand side has period   4  π the right hand side has period   π. So if   θ is a solution, so is   θ  +  4  k  π for any integer k, so your answer has to either be 0 solutions or an infinite number of solutions.Now you can just look at   sin  ⁡  (      1    2    θ  )  −  2  cos  ⁡  (  2  θ  ). If that is always negative or always positive, you have no solution. If you can find one positive and one negative, then somewhere it is 0 in between by continuity and you have an infinite number of solutions.

Frankie Burnett · Accepted Answer

Put   x  =  sin  ⁡  (      θ    2    )    ⟹    x  =  2  (  −  1  +  2      cos    2    ⁡  θ  )  =  −  2  +  4  (  1  −  2      sin    2    ⁡  (      θ    2    )      )    2    =  −  2  +  4  (  1  −  2      x    2        )    2    =  −  2  +  4  −  16      x    2    +  16      x    4      ⟹    16      x    4    −  16      x    2    −  x  +  2  =  0Wolfram Alpha says the roots are all irrational numbers.

How many solutions exist to this trig relation? sin(1/2 theta)=2 cos(2 theta)

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