Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. cos θ > 0 , sec θ > 0

Verified step by step guidance

1. Recall that \( \cos \theta \) is positive in the first and fourth quadrants.
2. Recall that \( \sec \theta = \frac{1}{\cos \theta} \), so \( \sec \theta \) is positive wherever \( \cos \theta \) is positive.
3. Since both \( \cos \theta > 0 \) and \( \sec \theta > 0 \), the angle \( \theta \) must be in a quadrant where \( \cos \theta \) is positive.
4. Identify the quadrants where both conditions are satisfied: first and fourth quadrants.
5. Conclude that the possible quadrants for \( \theta \) are the first and fourth quadrants.

Key Concepts

Trigonometric Functions and Their Signs

In trigonometry, the signs of the trigonometric functions (sine, cosine, tangent, etc.) vary depending on the quadrant in which the angle θ lies. The four quadrants are defined as follows: Quadrant I (all positive), Quadrant II (sine positive), Quadrant III (tangent positive), and Quadrant IV (cosine positive). Understanding these signs is crucial for determining the possible quadrants based on given conditions.

Cosine and Secant Functions

The cosine function, cos θ, represents the x-coordinate of a point on the unit circle, while the secant function, sec θ, is the reciprocal of cosine (sec θ = 1/cos θ). Therefore, if cos θ > 0, it implies that sec θ must also be positive. This relationship helps in identifying the quadrants where these conditions hold true.

Quadrant Identification

To identify the quadrant of an angle based on the signs of trigonometric functions, one must analyze the conditions provided. For cos θ > 0, the angle must be in Quadrant I or IV, where cosine is positive. Since sec θ is also positive, this further restricts the angle to Quadrant I, where both cosine and secant are positive.