Solve for x 2sin(x)=-1

Problem

2*sin(x)=−1

Solution

1. Isolate the sine function by dividing both sides of the equation by 2

sin(x)=−1/2

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

α=π/6

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

x=π+π/6

x=2*π−π/6

4. Calculate the primary solutions within the interval [0,2*π)

x=(7*π)/6

x=(11*π)/6

5. Generalize the solution by adding multiples of the period 2*π where n is any integer.

x=(7*π)/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

x=(7*π)/6+2*n*π,(11*π)/6+2*n*π

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