Solve for x sin(2x)=-1/2

Problem

sin(2*x)=−1/2

Solution

1. Identify the reference angle for the equation. Since sin(θ)=1/2 when θ=π/6 the reference angle is π/6

2. Determine the quadrants where the sine function is negative. Sine is negative in the third and fourth quadrants.

3. Find the general solutions for the argument 2*x by applying the quadrant rules.

θ=π+π/6+2*k*π=(7*π)/6+2*k*π

θ=2*π−π/6+2*k*π=(11*π)/6+2*k*π

4. Set the argument 2*x equal to these general solutions.

2*x=(7*π)/6+2*k*π

2*x=(11*π)/6+2*k*π

5. Isolate x by dividing every term by 2

x=(7*π)/12+k*π

x=(11*π)/12+k*π

Final Answer

sin(2*x)=−1/2⇒x=(7*π)/12+k*π,(11*π)/12+k*π

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