To rotate a point \( Q(x, y) \) about the origin by an angle \( \theta \), you can use the rotation formulas:
\[ Q' = (x', y') = (x \cdot \cos(\theta) - y \cdot \sin(\theta), x \cdot \sin(\theta) + y \cdot \cos(\theta)) \]
In this case, the point \( Q(-5, -9) \) is rotated by \( -270^\circ \).
First, let's convert the angle to radians for easier calculations. However, we can also directly use the values of sine and cosine corresponding to this angle, as follows:
- For \( -270^\circ \):
- \( \cos(-270^\circ) = 0 \)
- \( \sin(-270^\circ) = -1 \)
Now, applying the formulas with \( x = -5 \) and \( y = -9 \):
\[ x' = -5 \cdot 0 - (-9) \cdot (-1) = 0 - 9 = -9 \] \[ y' = -5 \cdot (-1) + (-9) \cdot 0 = 5 + 0 = 5 \]
Therefore, the new coordinates after the rotation are:
\[ Q' = (-9, 5) \]
Thus, the location of point \( Q' \) after rotating \( Q(-5, -9) \) by -270 degrees about the origin is:
\[ \boxed{(-9, 5)} \]
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