What are the solutions to sine left parenthesis 2 theta right parenthesis equals negative one half in the i

Question

Solve the equation on the interval 0 less than or equals theta less than 2 pi .0≤θ<2π.

sine left parenthesis 2 theta right parenthesis equals negative one halfsin(2θ)=−12

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Part 1

What are the solutions to

sine left parenthesis 2 theta right parenthesis equals negative one halfsin(2θ)=−12

in the interval

0 less than or equals theta less than 2 pi0≤θ<2π​?

  Select the correct choice and fill in any answer boxes in your choice below.

A.

The solution set is

Answer 1

sin(2θ) = -1/2

Solve for θ on interval from [0,2π):

2θ = arcsin(-1/2), note sine is negative in 3rd and 4th quadrants

2θ = 7π/6 , 11π/6, 19π/6 , 23π/6, 31π/6, ... Note add 2π (or 12π/6) to the 1st two results

θ = 7π/12, 11π/12, 19π/12, 23π/12, 31π/12

The last is rejected because outside the target interval.

Visit this graph for confirmation:

desmos.com/calculator/qgavczcgug