Concept CheckSuppose that ―90° < θ < 90° .Find the sign of each function value. cos(θ―180°)

Verified step by step guidance

1. Recognize that the problem involves finding the sign of the cosine function for an angle transformation.
2. Use the identity for cosine of a shifted angle: \( \cos(\theta - 180^\circ) = -\cos(\theta) \).
3. Since \( -90^\circ < \theta < 90^\circ \), \( \theta \) is in the first or fourth quadrant.
4. In the first quadrant, \( \cos(\theta) > 0 \), and in the fourth quadrant, \( \cos(\theta) > 0 \) as well.
5. Therefore, \( -\cos(\theta) < 0 \) for both quadrants, meaning \( \cos(\theta - 180^\circ) \) is negative.

Key Concepts

Understanding Angle Measures

In trigonometry, angles can be measured in degrees or radians. The question specifies that θ is between -90° and 90°, which indicates that θ is in the first or fourth quadrant. This range is crucial for determining the signs of trigonometric functions, as the sign of a function can vary depending on the quadrant in which the angle lies.

Cosine Function and Its Properties

The cosine function, denoted as cos(θ), represents the x-coordinate of a point on the unit circle corresponding to the angle θ. The cosine function is positive in the first quadrant and negative in the second quadrant. Understanding how the cosine function behaves with respect to angle transformations, such as cos(θ - 180°), is essential for determining its sign.

Angle Transformation

Angle transformation involves adjusting an angle by adding or subtracting a specific value. In this case, cos(θ - 180°) indicates a shift of the angle θ by 180°, which effectively reflects the angle across the origin on the unit circle. This transformation changes the sign of the cosine value, making it important to analyze how this affects the function's output based on the original angle θ.