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

Verified step by step guidance

1. Recall the signs of trigonometric functions in each quadrant: In Quadrant I, all functions are positive; in Quadrant II, sine is positive while cosine and tangent are negative; in Quadrant III, tangent is positive while sine and cosine are negative; in Quadrant IV, cosine is positive while sine and tangent are negative.
2. Analyze the given conditions: \( \tan \theta > 0 \) means tangent is positive, which occurs in Quadrants I and III.
3. Next, consider \( \sec \theta > 0 \). Since \( \sec \theta = \frac{1}{\cos \theta} \), this means \( \cos \theta > 0 \). Cosine is positive in Quadrants I and IV.
4. Find the quadrant(s) where both conditions are true simultaneously: tangent positive (Quadrants I and III) and cosine positive (Quadrants I and IV). The only quadrant common to both is Quadrant I.
5. Therefore, \( \theta \) lies in Quadrant I.

Key Concepts

Signs of Trigonometric Functions in Quadrants

Each quadrant of the coordinate plane has specific signs for sine, cosine, and tangent functions. Knowing these sign patterns helps determine the quadrant of an angle based on the given trigonometric values.

Definition and Sign of Tangent Function

Tangent of an angle θ is the ratio of sine to cosine (tan θ = sin θ / cos θ). Its sign depends on the signs of sine and cosine, being positive when both have the same sign, which occurs in Quadrants I and III.

Definition and Sign of Secant Function

Secant is the reciprocal of cosine (sec θ = 1 / cos θ). The sign of secant matches the sign of cosine, so sec θ > 0 means cosine is positive, which happens in Quadrants I and IV.