Graph tan(theta)>0

Problem

tan(θ)>0

Solution

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

tan(θ)=sin(θ)/cos(θ)

2. Determine the conditions for the quotient to be positive. The expression tan(θ)>0 is true when sin(θ) and cos(θ) have the same sign.

3. Analyze the quadrants of the unit circle. Both functions are positive in Quadrant I (0<θ<π/2, and both functions are negative in Quadrant III (π<θ<(3*π)/2.

4. Generalize the solution for all real numbers. Since the tangent function has a period of π the inequality holds for intervals repeating every π radians.

n*π<θ<n*π+π/2

5. Describe the graph. On a coordinate plane where the horizontal axis is θ and the vertical axis is y=tan(θ) the graph consists of the portions of the tangent curves that lie above the θaxis. On a unit circle, the solution corresponds to the interior of the first and third quadrants, excluding the axes.

Final Answer

tan(θ)>0⇒θ∈(⋃_n∈ℤ^)(n*π,n*π+π/2)

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