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!