Concept Check Suppose that ―90° < θ < 90° .   Find the sign of each function value. sec θ/2

Verified step by step guidance

1. Recall that the domain for \( \theta \) is \( -90^\circ < \theta < 90^\circ \), which means \( \theta \) lies in Quadrant I or IV.
2. Since \( \theta \) is between \( -90^\circ \) and \( 90^\circ \), dividing by 2 gives \( \frac{\theta}{2} \) in the interval \( -45^\circ < \frac{\theta}{2} < 45^\circ \).
3. Determine the quadrant of \( \frac{\theta}{2} \) based on this interval. Because \( \frac{\theta}{2} \) lies between \( -45^\circ \) and \( 45^\circ \), it is in Quadrant I or IV.
4. Recall that the secant function is defined as \( \sec x = \frac{1}{\cos x} \). So, the sign of \( \sec \frac{\theta}{2} \) depends on the sign of \( \cos \frac{\theta}{2} \).
5. Since cosine is positive in Quadrants I and IV, conclude that \( \cos \frac{\theta}{2} > 0 \), and therefore \( \sec \frac{\theta}{2} > 0 \).

Key Concepts

Domain and Range of the Angle θ

Understanding that θ lies between -90° and 90° helps determine the quadrant in which θ/2 lies. Since θ/2 will be between -45° and 45°, this restricts the angle to the first or fourth quadrant, which affects the sign of trigonometric functions.

Definition and Sign of the Secant Function

Secant is the reciprocal of cosine, sec(α) = 1/cos(α). The sign of sec(α) depends on the sign of cos(α). If cos(α) is positive, sec(α) is positive; if cos(α) is negative, sec(α) is negative.

Sign of Cosine in Quadrants I and IV

Cosine is positive in both the first and fourth quadrants. Since θ/2 lies between -45° and 45°, it is in these quadrants, ensuring cos(θ/2) is positive, and thus sec(θ/2) is also positive.