Question
(-7,2) rotated 270 degrees clockwise around the origin
Answers
GPT-4o mini
To rotate the point \((-7, 2)\) 270 degrees clockwise around the origin, you can follow these steps:
1. **Convert the clockwise rotation to a counterclockwise rotation**: A 270-degree clockwise rotation is equivalent to a 90-degree counterclockwise rotation.
2. **Use the rotation formulas**: The rotation of a point \((x, y)\) counterclockwise by an angle \(\theta\) around the origin can be calculated using:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
For a 90-degree counterclockwise rotation, \(\theta = 90^\circ\), so:
\[
\cos(90^\circ) = 0 \quad \text{and} \quad \sin(90^\circ) = 1
\]
3. **Plug in the values**:
\[
x' = -7 \cdot 0 - 2 \cdot 1 = -2
\]
\[
y' = -7 \cdot 1 + 2 \cdot 0 = -7
\]
So, after rotating the point \((-7, 2)\) 270 degrees clockwise around the origin, the new coordinates are \((-2, -7)\).
1. **Convert the clockwise rotation to a counterclockwise rotation**: A 270-degree clockwise rotation is equivalent to a 90-degree counterclockwise rotation.
2. **Use the rotation formulas**: The rotation of a point \((x, y)\) counterclockwise by an angle \(\theta\) around the origin can be calculated using:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
For a 90-degree counterclockwise rotation, \(\theta = 90^\circ\), so:
\[
\cos(90^\circ) = 0 \quad \text{and} \quad \sin(90^\circ) = 1
\]
3. **Plug in the values**:
\[
x' = -7 \cdot 0 - 2 \cdot 1 = -2
\]
\[
y' = -7 \cdot 1 + 2 \cdot 0 = -7
\]
So, after rotating the point \((-7, 2)\) 270 degrees clockwise around the origin, the new coordinates are \((-2, -7)\).