Solve for x tan(x)<0

Problem

tan(x)<0

Solution

1. Identify the definition of the tangent function in terms of sine and cosine.

tan(x)=sin(x)/cos(x)

2. Determine the conditions for the tangent function to be negative. The quotient is negative when sin(x) and cos(x) have opposite signs.

sin(x)>0* and *cos(x)<0

sin(x)<0* and *cos(x)>0

3. Locate the quadrants on the unit circle where these conditions are met. The first case occurs in Quadrant II, and the second case occurs in Quadrant IV.

π/2<x<π

(3*π)/2<x<2*π

4. Generalize the solution for all real numbers by adding the period of the tangent function, which is π

π/2+k*π<x<π+k*π

Final Answer

tan(x)<0⇒x∈∪(π/2+k*π,π+k*π)

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