To determine the signs of the coordinates [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for a point [tex]\( P(x, y) \)[/tex] lying on the terminal side of the angle [tex]\(\theta = -60^\circ\)[/tex] in standard position, let's follow these steps:
1. Understand the Angle's Position:
- An angle of [tex]\(-60^\circ\)[/tex] means that you move 60 degrees clockwise from the positive [tex]\( x \)[/tex]-axis.
- Since angles are typically measured counterclockwise from the positive [tex]\( x \)[/tex]-axis, a negative angle indicates a clockwise rotation.
2. Convert Negative Angle to Positive Equivalent:
- To better understand the angle's location, we can convert the negative angle into its positive equivalent by adding [tex]\( 360^\circ \)[/tex]:
[tex]\[
\theta = 360^\circ + (-60^\circ) = 300^\circ
\][/tex]
- Thus, we are now considering [tex]\( \theta = 300^\circ \)[/tex].
3. Determine the Quadrant:
- The standard coordinate plane is divided into four quadrants:
- Quadrant I: [tex]\(0^\circ < \theta < 90^\circ\)[/tex]
- Quadrant II: [tex]\(90^\circ < \theta < 180^\circ\)[/tex]
- Quadrant III: [tex]\(180^\circ < \theta < 270^\circ\)[/tex]
- Quadrant IV: [tex]\(270^\circ < \theta < 360^\circ\)[/tex]
- Since [tex]\( 300^\circ \)[/tex] lies between [tex]\( 270^\circ \)[/tex] and [tex]\( 360^\circ \)[/tex], it is in the fourth quadrant.
4. Sign of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the Fourth Quadrant:
- In the fourth quadrant:
- The [tex]\( x \)[/tex]-coordinate is positive because you are to the right of the [tex]\( y \)[/tex]-axis.
- The [tex]\( y \)[/tex]-coordinate is negative because you are below the [tex]\( x \)[/tex]-axis.
5. Conclusion:
- Therefore, for the point [tex]\( P(x, y) \)[/tex] on the terminal side of the angle [tex]\( \theta = -60^\circ \)[/tex] (or equivalently [tex]\( \theta = 300^\circ \)[/tex]), [tex]\( x \)[/tex] is positive and [tex]\( y \)[/tex] is negative.
The correct answer is:
[tex]\[
\boxed{x \text{ is positive, and } y \text{ is negative.}}
\][/tex]
