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

Jairo Hodges

Jairo Hodges

Answered question

2022-11-10

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

Answer & Explanation

mentest91k99

mentest91k99

Beginner2022-11-11Added 17 answers

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

Frankie Burnett

Beginner2022-11-12Added 2 answers

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 = 0
Wolfram Alpha says the roots are all irrational numbers.

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