Question
(-2,-2) rotated 270 degrees around the origin clockwise
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Answered by
GPT-4o mini
To rotate the point \((-2, -2)\) 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 formula:** For a counterclockwise rotation of a point \((x, y)\) by an angle \(\theta\), the new coordinates \((x', y')\) can be determined using the following formulas:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
For a 90-degree counterclockwise rotation:
\(\cos(90^\circ) = 0\) and \(\sin(90^\circ) = 1\).
3. **Apply the formulas:** Using the point \((-2, -2)\):
\[
x' = -2 \cdot 0 - (-2) \cdot 1 = 0 + 2 = 2
\]
\[
y' = -2 \cdot 1 + (-2) \cdot 0 = -2 + 0 = -2
\]
The new coordinates after rotating \((-2, -2)\) 270 degrees clockwise around the origin are \((2, -2)\).
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 formula:** For a counterclockwise rotation of a point \((x, y)\) by an angle \(\theta\), the new coordinates \((x', y')\) can be determined using the following formulas:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
For a 90-degree counterclockwise rotation:
\(\cos(90^\circ) = 0\) and \(\sin(90^\circ) = 1\).
3. **Apply the formulas:** Using the point \((-2, -2)\):
\[
x' = -2 \cdot 0 - (-2) \cdot 1 = 0 + 2 = 2
\]
\[
y' = -2 \cdot 1 + (-2) \cdot 0 = -2 + 0 = -2
\]
The new coordinates after rotating \((-2, -2)\) 270 degrees clockwise around the origin are \((2, -2)\).