To rotate a point \((x, y)\) counterclockwise about the origin by an angle \(\theta\), you can use the following transformation formulas:
\[ x' = x \cos(\theta) - y \sin(\theta) \] \[ y' = x \sin(\theta) + y \cos(\theta) \]
For a rotation of 270 degrees, we first convert the angle to radians. The angle of 270 degrees is equivalent to \(\frac{3\pi}{2}\) radians.
We know that:
- \(\cos\left(\frac{3\pi}{2}\right) = 0\)
- \(\sin\left(\frac{3\pi}{2}\right) = -1\)
Now substituting these values into the rotation formulas:
-
For the new x-coordinate \(x'\): \[ x' = x \cos\left(\frac{3\pi}{2}\right) - y \sin\left(\frac{3\pi}{2}\right) = x \cdot 0 - y \cdot (-1) = y \]
-
For the new y-coordinate \(y'\): \[ y' = x \sin\left(\frac{3\pi}{2}\right) + y \cos\left(\frac{3\pi}{2}\right) = x \cdot (-1) + y \cdot 0 = -x \]
Now, substituting in the point \((-4, 5)\):
- \(x = -4\)
- \(y = 5\)
Calculating \(x'\) and \(y'\):
- \(x' = y = 5\)
- \(y' = -x = -(-4) = 4\)
Thus, the new point after rotating the point \((-4, 5)\) counterclockwise by 270 degrees is:
\[ \boxed{(5, 4)} \]
1 answer