In Exercises 39–40, let θ be an angle in standard position. Name the quadrant in which θ lies. tan θ > 0 and cos θ < 0

Key Concepts

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants based on the signs of the x (cosine) and y (sine) coordinates. Quadrant I has both coordinates positive, Quadrant II has a negative x and positive y, Quadrant III has both negative, and Quadrant IV has a positive x and negative y. Understanding these quadrants is essential for determining the signs of trigonometric functions in relation to the angle θ.

Trigonometric Functions and Their Signs

The tangent function (tan θ) is the ratio of sine to cosine (sin θ/cos θ). Therefore, tan θ is positive when both sine and cosine are either both positive or both negative. Conversely, cosine (cos θ) is negative in Quadrants II and III. Recognizing the relationships between these functions helps in identifying the quadrant where the angle θ lies.

Inequalities in Trigonometry

Inequalities involving trigonometric functions provide critical information about the angle's position. In this case, the condition tan θ > 0 indicates that both sine and cosine must have the same sign, while cos θ < 0 specifies that cosine is negative. Analyzing these inequalities together allows us to deduce the specific quadrant for the angle θ.