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

Key Concepts

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants based on the signs of the x and y coordinates. Quadrant I has both x and y positive, Quadrant II has x negative and y positive, Quadrant III has both negative, and Quadrant IV has x positive and y negative. Understanding these quadrants is essential for determining the signs of trigonometric functions based on the angle's position.

Trigonometric Functions and Their Signs

Each trigonometric function has specific signs in different quadrants. For instance, tangent (tan) is positive in Quadrants I and III, while secant (sec), being the reciprocal of cosine, is positive in Quadrants I and IV. Knowing these relationships helps in identifying the quadrant where the angle lies based on the given conditions of the functions.

Understanding Tangent and Secant

Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle, while secant is the reciprocal of cosine, representing the ratio of the hypotenuse to the adjacent side. The conditions tan θ > 0 and sec θ > 0 indicate that both functions are positive, which restricts θ to Quadrant I, where both sine and cosine are positive.