Question

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, you can use the rotation transformation formulas.

The general formula for rotating a point \((x, y)\) counterclockwise by an angle \(\theta\) is given by:

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

For a \(270\) degrees rotation, we first convert this angle to radians:

\[
\theta = 270^\circ = \frac{3\pi}{2} \text{ radians}
\]

Now, we find the cosine and sine of \(270\) degrees:

\[
\cos(270^\circ) = 0
\]
\[
\sin(270^\circ) = -1
\]

Substituting these values into the rotation formulas:

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

Thus, the new coordinates after rotating \((x, y)\) by \(270\) degrees counterclockwise are:

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

So the new point after the rotation is \((y, -x)\).