To determine the new coordinates of the point \((-4, 5)\) after rotating it counterclockwise about the origin by 270 degrees, we can use the rotation formula.
When rotating a point \((x, y)\) counterclockwise by an angle \(\theta\), the new coordinates \((x', y')\) are given by:
\[ x' = x \cos(\theta) - y \sin(\theta) \] \[ y' = x \sin(\theta) + y \cos(\theta) \]
For a rotation of 270 degrees, we first need the values of \(\cos(270^\circ)\) and \(\sin(270^\circ)\):
- \(\cos(270^\circ) = 0\)
- \(\sin(270^\circ) = -1\)
Plugging these values into the rotation formulas, we get:
\[ x' = x \cdot 0 - y \cdot (-1) = y \] \[ y' = x \cdot (-1) + y \cdot 0 = -x \]
Now, substituting in the coordinates of our point \((-4, 5)\):
- \(x' = 5\) (which is \(y\))
- \(y' = -(-4) = 4\) (which is \(-x\))
Thus, after rotating the point \((-4, 5)\) counterclockwise about the origin by 270 degrees, the new point is:
\[ (5, 4) \]
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