To find the angle of rotation when the center of rotation is at the origin, and the transformation is given by [tex]\( A(2,3) \longrightarrow A'(-3,2) \)[/tex], follow these steps:
1. Convert the coordinates of each point into polar coordinates: This involves finding the angle each point makes with the positive x-axis.
2. Calculate the angle [tex]\(\theta_A\)[/tex] for [tex]\(A(2, 3)\)[/tex]:
[tex]\[
\theta_A = \arctan\left(\frac{3}{2}\right)
\][/tex]
From the given data, [tex]\(\theta_A \approx 0.9828\)[/tex] radians.
3. Calculate the angle [tex]\(\theta_{A'}\)[/tex] for [tex]\(A'(-3, 2)\)[/tex]:
[tex]\[
\theta_{A'} = \arctan\left(\frac{2}{-3}\right)
\][/tex]
Bearing in mind that this angle lies in the second quadrant, we adjust as necessary:
From the given data, [tex]\(\theta_{A'} \approx 2.5536\)[/tex] radians.
4. Find the difference between the angles: The rotation angle is the difference between [tex]\(\theta_{A'}\)[/tex] and [tex]\(\theta_A\)[/tex]:
[tex]\[
\Delta\theta = \theta_{A'} - \theta_A
\][/tex]
Substituting the values, we get:
[tex]\[
\Delta\theta \approx 2.5536 - 0.9828 = 1.5708 \text{ radians}
\][/tex]
5. Convert the rotation angle from radians to degrees: Since angles are often more easily understood in degrees,
[tex]\[
\Delta\theta_{\text{degrees}} = \Delta\theta \cdot \frac{180}{\pi}
\][/tex]
Using the given data, [tex]\(\Delta\theta_{\text{degrees}} \approx 90.0^\circ \)[/tex].
So, the angle of rotation for the transformation [tex]\( A(2, 3) \longrightarrow A'(-3, 2) \)[/tex] when the center of rotation is at the origin is [tex]\( \boxed{90^\circ} \)[/tex].
