now with this understanding, solve: Find polar coordinates to represent (2,23)(2,23​)​ with r≥0*r*≥0​ and 0≤θ<2π0≤*θ*<2*π*​​.
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Find polar coordinates to represent (2,23)(2,23​)​ with r≤0*r*≤0​ and 0≤θ<2π0≤*θ*<2*π*​.

Let's find the polar coordinates for the point \( (2, 23) \) under both conditions: \( r \geq 0 \) and \( r \leq 0 \).

### Part 1: Polar coordinates with \( r \geq 0 \)

Polar coordinates are represented as \( (r, \theta) \), where:
- \( r \) is the radial distance from the origin to the point,
- \( \theta \) is the angle from the positive \( x \)-axis to the line connecting the origin to the point.

Given the Cartesian coordinates \( (x, y) = (2, 23) \), the formulas to convert to polar coordinates are:
\[
r = \sqrt{x^2 + y^2}
\]
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]

1. **Compute \( r \)**:
\[
r = \sqrt{2^2 + 23^2} = \sqrt{4 + 529} = \sqrt{533} \approx 23.086
\]

2. **Compute \( \theta \)**:
\[
\theta = \tan^{-1}\left(\frac{23}{2}\right) \approx \tan^{-1}(11.5) \approx 1.740 \, \text{radians}
\]
Since the point is in the first quadrant, \( \theta \) remains \( 1.740 \) radians.

Thus, the polar coordinates are approximately:
\[
(r, \theta) \approx (23.086, 1.740)
\]

### Part 2: Polar coordinates with \( r \leq 0 \)

For \( r \leq 0 \), the radial distance will be negative. In this case, we first compute the positive radial distance (which is still \( 23.086 \)) but then use a negative value for \( r \). To adjust for the negative \( r \), we must add \( \pi \) to \( \theta \) (since this corresponds to pointing in the opposite direction).

\[
r \approx -23.086
\]
\[
\theta = 1.740 + \pi \approx 1.740 + 3.142 = 4.882 \, \text{radians}
\]

Thus, the polar coordinates with \( r \leq 0 \) are approximately:
\[
(r, \theta) \approx (-23.086, 4.882)
\]

### Final Results:
1. **With \( r \geq 0 \)**: \( (23.086, 1.740) \)
2. **With \( r \leq 0 \)**: \( (-23.086, 4.882) \)

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Would you like more details or clarifications?

Here are 5 related questions for practice:
1. What are the polar coordinates of \( (5, 12) \)?
2. How do you convert polar coordinates back to Cartesian form?
3. What is the effect of changing the sign of \( r \) in polar coordinates?
4. How do you handle \( \theta \) when the point is in different quadrants?
5. How do polar and Cartesian coordinates relate to trigonometric functions?

**Tip**: When converting Cartesian to polar coordinates, always ensure \( \theta \) is within the correct range \( [0, 2\pi) \), especially when dealing with different quadrants!

Find polar coordinates to represent (2, 23) with r ≥ 0 and 0 ≤ θ < 2π. Also, find polar coordinates to represent (2, 23) with r ≤ 0 and 0 ≤ θ < 2π.

Learn how to find the polar coordinates for the point (2, 23) with both positive and negative radial distances. This detailed solution covers key trigonometric formulas, such as r = sqrt(x^2 + y^2) and θ = tan^(-1)(y/x). Ideal for high school students learning about polar and Cartesian coordinate systems.

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