Solve for x sin(x)<0

Problem

sin(x)<0

Solution

1. Identify the quadrants where the sine function is negative. On the unit circle, sin(x) represents the ycoordinate.

2. Determine the intervals within one full rotation [0,2*π) The ycoordinate is negative in the third and fourth quadrants.

3. Define the boundaries for these quadrants. The third quadrant starts at π and the fourth quadrant ends at 2*π

4. Generalize the solution to account for the periodicity of the sine function. Since sin(x) has a period of 2*π add 2*k*π to the boundaries, where k is any integer.

Final Answer

π+2*k*π<x<2*π+2*k*π,k∈ℤ

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