Concept Check Suppose that 90° < θ < 180° .   Find the sign of each function value. tan θ/2

Verified step by step guidance

1. Identify the quadrant where \( \theta \) lies. Since \( 90^\circ < \theta < 180^\circ \), \( \theta \) is in the second quadrant.
2. Determine the range of \( \frac{\theta}{2} \). Since \( \theta \) is between \( 90^\circ \) and \( 180^\circ \), dividing by 2 gives \( 45^\circ < \frac{\theta}{2} < 90^\circ \). This means \( \frac{\theta}{2} \) lies in the first quadrant.
3. Recall the signs of tangent in each quadrant. Tangent is positive in the first and third quadrants, and negative in the second and fourth quadrants.
4. Since \( \frac{\theta}{2} \) is in the first quadrant, \( \tan \frac{\theta}{2} \) is positive.
5. Therefore, the sign of \( \tan \frac{\theta}{2} \) is positive for \( 90^\circ < \theta < 180^\circ \).

Key Concepts

Angle Quadrants and Their Ranges

The coordinate plane is divided into four quadrants, each with specific angle ranges and sign conventions for trigonometric functions. For 90° < θ < 180°, θ lies in the second quadrant, where sine is positive and cosine is negative. Understanding the quadrant helps determine the sign of trigonometric values.

Half-Angle Relationships

The half-angle θ/2 is half the measure of the original angle θ. When θ is between 90° and 180°, θ/2 lies between 45° and 90°, placing it in the first quadrant. This affects the sign of trigonometric functions of θ/2, as signs depend on the quadrant of the angle.

Sign of the Tangent Function

The tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants. Since θ/2 lies between 45° and 90° (first quadrant), tan(θ/2) is positive. Recognizing the quadrant of θ/2 is essential to determine the sign of tan(θ/2).