To determine the angle of rotation that maps the point [tex]\((0, -2)\)[/tex] to [tex]\((2, 0)\)[/tex], we can follow these steps:
1. Identify the coordinates:
- The initial point is [tex]\((0, -2)\)[/tex].
- The final point is [tex]\((2, 0)\)[/tex].
2. Determine the initial angle:
- The initial angle [tex]\(\theta_1\)[/tex] is the angle between the positive x-axis and the line connecting the origin to the point [tex]\((0, -2)\)[/tex].
- Since [tex]\((0, -2)\)[/tex] lies directly below the origin on the y-axis, the angle [tex]\(\theta_1\)[/tex] is -90 degrees (or alternatively [tex]\(-\frac{\pi}{2}\)[/tex] radians).
3. Determine the final angle:
- The final angle [tex]\(\theta_2\)[/tex] is the angle between the positive x-axis and the line connecting the origin to the point [tex]\((2, 0)\)[/tex].
- Since [tex]\((2, 0)\)[/tex] lies directly on the positive x-axis from the origin, the angle [tex]\(\theta_2\)[/tex] is 0 degrees (or 0 radians).
4. Calculate the angle of rotation:
- The angle of rotation is the difference between the final angle [tex]\(\theta_2\)[/tex] and the initial angle [tex]\(\theta_1\)[/tex].
- [tex]\(\theta_{\text{rotation}} = \theta_2 - \theta_1\)[/tex].
5. Convert angles to radians (if necessary):
- Initial angle: [tex]\(\theta_1 = -\frac{\pi}{2}\)[/tex] radians.
- Final angle: [tex]\(\theta_2 = 0\)[/tex] radians.
6. Compute the rotation in radians:
- [tex]\(\theta_{\text{rotation}} = 0 - (-\frac{\pi}{2}) = \frac{\pi}{2}\)[/tex] radians.
7. Convert the rotation angle from radians to degrees:
- To convert radians to degrees, we use the conversion factor [tex]\(180^\circ / \pi\)[/tex].
- [tex]\(\frac{\pi}{2} \times \frac{180^\circ}{\pi} = 90^\circ\)[/tex].
Therefore, the angle measure of the rotation that maps the point [tex]\((0, -2)\)[/tex] to [tex]\((2, 0)\)[/tex] is [tex]\(90\)[/tex] degrees.
