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

Problem

sin(x)=−1/2

Solution

1. Identify the reference angle by finding the value of α such that sin(α)=1/2 in the first quadrant.

α=π/6

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

Quadrant III:x=π+α

Quadrant IV:x=2*π−α

3. Calculate the primary solutions within the interval [0,2*π) by substituting the reference angle.

x=π+π/6=(7*π)/6

x=2*π−π/6=(11*π)/6

4. Generalize the solution to include all possible values of x by adding integer multiples of the period 2*π where k is any integer.

x=(7*π)/6+2*k*π

x=(11*π)/6+2*k*π

Final Answer

sin(x)=−1/2⇒x=(7*π)/6+2*k*π,(11*π)/6+2*k*π

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