Question

how would you need to rotate (-4,-9) to get it to (-9,4)

Answer 1

To find the rotation needed to transform the point (-4, -9) to the point (-9, 4), we can follow these steps:

1. **Determine the Origin**: We assume that the rotation is about the origin (0,0).

2. **Find the angle of rotation**: We can use the formula for rotating a point around the origin. The rotation of a point \((x, y)\) by an angle \(\theta\) is given by:

\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]

where \((x', y')\) is the new point after rotation.

3. **Set the equations**: Set up the equations based on the original and new coordinates:

\[
-9 = -4 \cos(\theta) + 9 \sin(\theta) \quad (1)
\]
\[
4 = -4 \sin(\theta) - 9 \cos(\theta) \quad (2)
\]

4. **Solve the equations**: We can solve these equations to find \(\theta\).

Let's start with equation (1):

Rearranging it:
\[
-9 = -4 \cos(\theta) + 9 \sin(\theta)
\]

Or:
\[
4 \cos(\theta) + 9 \sin(\theta) = 9 \quad (3)
\]

Now for equation (2):
\[
4 = -4 \sin(\theta) - 9 \cos(\theta)
\]

Or:
\[
4 \sin(\theta) + 9 \cos(\theta) = -4 \quad (4)
\]

5. **Use a graphical or numerical method or by substituting**: Solving these equations simultaneously can be complicated, but numerically or graphically, we can determine that the angle of rotation required to transform (-4, -9) to (-9, 4) is approximately \( \theta = 270^\circ\) or \( \theta = -90^\circ\) counter-clockwise.

Thus, the point (-4, -9) can be transformed to (-9, 4) by rotating it approximately \( \theta = 270^\circ\) counter-clockwise around the origin.