To rotate a point \((x, y)\) by an angle \(\theta\) counterclockwise about the origin, we can use the following formulas:
\[ x' = x \cos(\theta) - y \sin(\theta) \] \[ y' = x \sin(\theta) + y \cos(\theta) \]
For a 270-degree counterclockwise rotation, we need to convert 270 degrees to radians:
\[ 270^\circ = \frac{3\pi}{2} \text{ radians} \]
Now, we can calculate the cosine and sine of \(270^\circ\):
\[ \cos(270^\circ) = 0 \] \[ \sin(270^\circ) = -1 \]
Now, we can substitute the point \((5, 8)\) into the formulas:
\[ x' = 5 \cdot 0 - 8 \cdot (-1) = 0 + 8 = 8 \] \[ y' = 5 \cdot (-1) + 8 \cdot 0 = -5 + 0 = -5 \]
Therefore, the new point after rotating the point \((5, 8)\) 270 degrees counterclockwise about the origin is:
\[ (8, -5) \]
1 answer