The problem involves converting polar coordinates and identifying points that correspond to the same location but with different values of $r$ and $\theta$. Here is a breakdown of the information given:

• The point given is $\left(5, \frac{\pi}{4}\right)$.
• To find the point directly opposite this point, we add $\pi$ to the angle, resulting in the point $\left(5, \frac{\pi}{4} + \pi\right) = \left(5, \frac{5\pi}{4}\right)$.
• The problem then introduces the concept that using $-r$ (i.e., a negative radius) can also represent the same point if you adjust the angle accordingly.

Solution to the Blank:

The statement mentions that a pair of polar coordinates $(-r, \theta)$ is equivalent to $(r, \theta + \pi)$. Therefore, if we want to represent the point $\left(5, \frac{5\pi}{4}\right)$ with a negative radius, we take $r = -5$ and the angle remains $\frac{\pi}{4}$.

Thus, the polar coordinates for the same point using a negative radius are: $(-5, \frac{\pi}{4})$

Would you like more details or have any questions about this explanation?

Related Questions:

1. How do you convert between polar and Cartesian coordinates?
2. What is the significance of adding $\pi$ to an angle in polar coordinates?
3. How do negative values of $r$ affect the representation of a point in polar coordinates?
4. What are some examples of polar coordinates that represent the same point using different values of $r$ and $\theta$?
5. How does the conversion formula between polar and Cartesian coordinates change with negative $r$ values?

Tip:

In polar coordinates, adding $2\pi$ to any angle results in the same point, as it represents a full rotation around the origin. This property is useful when converting between different representations.