Solve for x sin(3x)=-1/2

Problem

sin(3*x)=−1/2

Solution

1. Identify the reference angle for the equation by finding the value where sin(θ)=1/2

(θ_ref)=π/6

2. Determine the quadrants where the sine function is negative, which are Quadrant III and Quadrant IV.

(θ_1)=π+π/6=(7*π)/6

(θ_2)=2*π−π/6=(11*π)/6

3. Set up the general solution for the argument 3*x by adding multiples of 2*π where n is any integer.

3*x=(7*π)/6+2*n*π

3*x=(11*π)/6+2*n*π

4. Isolate x by dividing both sides of each equation by 3

x=(7*π)/18+(2*n*π)/3

x=(11*π)/18+(2*n*π)/3

Final Answer

sin(3*x)=−1/2⇒x=(7*π)/18+(2*n*π)/3,(11*π)/18+(2*n*π)/3

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