dentify the coordinates of the point in polar form based upon the given conditions.
Use pi for π.

r>0r>0 and 0<θ<2π

&
r<0r<0 and 0<θ<2π

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.

Identify the coordinates of the point in polar form based upon the given conditions. Use pi for π. r > 0 and 0 < θ < 2π & r < 0 and 0 < θ < 2π

Discover how to identify polar coordinates when r is positive or negative and the angle θ is between 0 and 2π. Learn the transformation rule for negative radial distances and understand the relationship between radius and angle in polar form. Suitable for Grades 10-12.

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