Let's break down the problem of identifying the coordinates of a point in polar form based on the given conditions:

Polar Coordinates:

In polar coordinates, a point is represented as $(r, \theta)$, where:

• $r$ is the radial distance from the origin.
• $\theta$ is the angle with respect to the positive x-axis, measured counterclockwise.

Case 1: $r > 0$ and $0 < \theta < 2\pi$

• Here, the radial distance $r$ is positive, so the point lies on the direction of angle $\theta$ from the origin.
• Since $0 < \theta < 2\pi$, the angle is measured within a full circle (between 0 and $2\pi$, excluding 0 but including all values up to $2\pi$).

In this case, the point is straightforward: the coordinates of the point are $(r, \theta)$, where $r > 0$.

Case 2: $r < 0$ and $0 < \theta < 2\pi$

• In this case, $r$ is negative. A negative $r$ means that the point is in the direction opposite to the angle $\theta$. To correctly express this in polar coordinates, we would adjust $\theta$ by adding $\pi$ radians to it (since a negative $r$ flips the point by 180 degrees).

Thus, for negative $r$, the coordinates of the point can be re-expressed as $(-r, \theta + \pi)$, where $r > 0$ and $\theta + \pi$ remains within the range $0 < \theta < 2\pi$.

Summary:

1. For $r > 0$ and $0 < \theta < 2\pi$: The coordinates are $(r, \theta)$.
2. For $r < 0$ and $0 < \theta < 2\pi$: The coordinates can be re-expressed as $(-r, \theta + \pi)$.

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Would you like more detailed explanations or have further questions?

Here are 5 related questions:

1. How can you convert between polar and Cartesian coordinates?
2. What happens to polar coordinates when $\theta$ exceeds $2\pi$?
3. Can polar coordinates be used for 3D points? If so, how?
4. How would you graph a point given in polar coordinates?
5. How do the sine and cosine functions relate to polar coordinates?

Tip: When dealing with negative $r$ values, always think about how the direction is flipped by $\pi$ radians.