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 quotient to be positive. The expression is positive when both sin(x) and cos(x) have the same sign.

sin(x)>0* and *cos(x)>0

sin(x)<0* and *cos(x)<0

3. Analyze the unit circle quadrants. Both functions are positive in Quadrant I, and both functions are negative in Quadrant III.

Quadrant I: *0<x<π/2

Quadrant III: *π<x<(3*π)/2

4. Generalize the solution using the period of the tangent function, which is π

n*π<x<n*π+π/2

Final Answer

tan(x)>0⇒x∈(⋃_n∈ℤ^)(n*π,n*π+π/2)

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