Question

90 and 270 degree rotations practice
Given a point x,y what is the new point when the point is rotated counterclockwise about the origin 270 degrees

Answer 1

To rotate a point \((x, y)\) counterclockwise about the origin by 270 degrees, we can use the rotation formulas.

A rotation of a point \((x, y)\) counterclockwise by an angle \(\theta\) can be described by the following equations:

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

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

For a 270-degree rotation, we have:

- \(\theta = 270^\circ\)
- The cosine and sine of 270 degrees are:
- \(\cos(270^\circ) = 0\)
- \(\sin(270^\circ) = -1\)

Substituting these values into the rotation equations gives us:

\[
x' = x \cdot 0 - y \cdot (-1) = y
\]

\[
y' = x \cdot (-1) + y \cdot 0 = -x
\]

Thus, when you rotate a point \((x, y)\) counterclockwise by 270 degrees, the new coordinates \((x', y')\) will be:

\[
(x', y') = (y, -x)
\]

So the new point after a 270-degree counterclockwise rotation of the point \((x, y)\) is \((y, -x)\).