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The problem is asking for the angle \( \sigma \) of the direction opposite to the point \( \left( 9, \frac{5\pi}{4} \right) \).

In polar coordinates, if you want to represent the same point but with a negative radius \( r < 0 \), the angle \( \sigma \) must be shifted by \( \pi \) radians (180 degrees) to point in the opposite direction on the unit circle. The given angle is \( \frac{5\pi}{4} \).

To find \( \sigma \), we add \( \pi \) to this angle:

\[
\sigma = \frac{5\pi}{4} + \pi = \frac{5\pi}{4} + \frac{4\pi}{4} = \frac{9\pi}{4}
\]

So, the angle \( \sigma \) is \( \frac{9\pi}{4} \).

Would you like more details on polar coordinates or how this calculation works?

Here are 5 follow-up questions you might explore:
1. What are the properties of polar coordinates compared to Cartesian coordinates?
2. How would you convert polar coordinates to Cartesian coordinates?
3. What happens when an angle exceeds \( 2\pi \) in polar coordinates?
4. How can we interpret negative radii geometrically?
5. How can angles be adjusted using different quadrants?

**Tip:** Always remember, when shifting an angle by \( \pi \), you are rotating the direction by 180 degrees on the polar plane.

Find the opposite angle for the point (9, 5π/4) in polar coordinates.

Learn how to determine the angle opposite to a given point in polar coordinates using angle addition. Explore the properties of polar coordinates, how to handle negative radii, and apply these concepts to solve the problem for point (9, 5π/4).

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