Let's explore the concept of polar coordinates to determine whether the value of [tex]\(\theta\)[/tex] can be negative.
Polar coordinates represent a point in the plane using a distance [tex]\(r\)[/tex] from the origin and an angle [tex]\(\theta\)[/tex] from a reference direction, typically the positive [tex]\(x\)[/tex]-axis. The coordinate pair [tex]\((r, \theta)\)[/tex] provides the means to specify any point in the plane.
### Understanding the Angle [tex]\(\theta\)[/tex]:
1. Definition of [tex]\(\theta\)[/tex]:
- [tex]\(\theta\)[/tex] is the angle formed between the line connecting the point to the origin and the positive [tex]\(x\)[/tex]-axis.
- It can be measured in degrees or radians.
2. Range of [tex]\(\theta\)[/tex]:
- The angle [tex]\(\theta\)[/tex] is not restricted to positive values only.
- A positive angle typically represents a counterclockwise rotation from the positive [tex]\(x\)[/tex]-axis.
- A negative angle represents a clockwise rotation from the positive [tex]\(x\)[/tex]-axis.
3. Example:
- If [tex]\(\theta\)[/tex] is [tex]\(45^\circ\)[/tex] (or [tex]\(\pi/4\)[/tex] radians), it represents a counterclockwise rotation of 45 degrees.
- If [tex]\(\theta\)[/tex] is [tex]\(-45^\circ\)[/tex] (or [tex]\(-\pi/4\)[/tex] radians), it represents a clockwise rotation of 45 degrees.
### Conclusion:
Given that [tex]\(\theta\)[/tex] can represent both counterclockwise (positive) and clockwise (negative) rotations, it is clear that [tex]\(\theta\)[/tex] can indeed have negative values.
### Answer:
A. True
