Solve for x 2sin(x)+1=0

Problem

2*sin(x)+1=0

Solution

1. Isolate the sine function by subtracting 1 from both sides of the equation.

2*sin(x)=−1

2. Divide both sides by 2 to solve for sin(x)

sin(x)=−1/2

3. Identify the reference angle. The value sin(θ)=1/2 corresponds to a reference angle of π/6 (or 30.

4. Determine the quadrants where the sine function is negative. Sine is negative in Quadrant III and Quadrant IV.

5. Calculate the specific angles within the interval [0,2*π)

x=π+π/6=(7*π)/6

x=2*π−π/6=(11*π)/6

6. Generalize the solution to include all possible rotations by adding multiples of 2*π where n is an 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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