Question

Point A (-3, 4) is rotated to become A' (-4, -3).



What degree of rotation is this? (Counterclockwise)

Answer 1

To determine the degree of rotation from point A (-3, 4) to point A' (-4, -3), we can use the properties of rotations in the coordinate plane.

1. **Find the original and rotated coordinates**:
- Point A: \((-3, 4)\)
- Point A': \((-4, -3)\)

2. **Identify the rotation transformation**:
The rotation of points in the plane can be characterized by their transformation under counterclockwise rotation.

3. **Use the standard rotation matrix**:
The standard rotation matrix for a counterclockwise rotation by an angle \(\theta\) is:
\[
R(\theta) = \begin{pmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{pmatrix}
\]

4. **Calculate the angle of rotation**:
To find the angle of rotation, we first determine the initial and final positions of the point:

For a point \((x, y)\), the angle \(\theta\) can be found using the tangent:
\[
\tan(\theta) = \frac{y}{x}
\]

For point A:
\[
\tan(\theta_A) = \frac{4}{-3}
\]
This gives us an angle in the second quadrant:
\[
\theta_A = \tan^{-1}(-\frac{4}{3}) + 180^\circ \approx 180^\circ - 53.13^\circ = 126.87^\circ
\]

For point A':
\[
\tan(\theta_{A'}) = \frac{-3}{-4} = \frac{3}{4}
\]
This is in the fourth quadrant:
\[
\theta_{A'} = \tan^{-1}(\frac{3}{4}) + 360^\circ \approx 36.87^\circ
\]

5. **Calculate the difference in angles**:
The rotation angle \(\phi\) (counterclockwise) can be calculated as:
\[
\phi = \theta_{A'} - \theta_A = 36.87^\circ - 126.87^\circ
\]
Since this calculation yields a negative angle, we can add 360 degrees to express it as a positive angle:
\[
\phi = 360^\circ + (36.87^\circ - 126.87^\circ) = 360^\circ - 90^\circ = 270^\circ
\]

Thus, the degree of counterclockwise rotation from point A to point A' is \(270^\circ\).