# Technical Document Extraction: Coordinate Conversion Problem This document contains a mathematical problem regarding the conversion of rectangular coordinates to polar coordinates, followed by three distinct "Attempts" at a solution. The text uses LaTeX formatting for mathematical expressions. ## 1. Problem Statement **Convert the point $(0,3)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$** --- ## 2. Solution Attempts ### Attempt 1 (Transcribed Text) *Note: This attempt contains several mathematical errors in its logic and final result.* To convert $(0, 3)$ to polar coordinates, we need to find the radius $r$ and the angle $\theta$. We know that the $x$-coordinate of the point is $r\cos\theta$ and the $y$-coordinate of the point is $r\sin\theta$. So $r\cos\theta = 0$ and $r\sin\theta = 3$. And we know that $\cos\theta = 0$ when $\theta = \frac{\pi}{2}$ and $\sin\theta = 3$ when $\theta = \frac{3\pi}{2}$. So we know that $(0, 3)$ corresponds to the point $(r, \theta)$ where $\theta = \frac{3\pi}{2}$ and $r\cos\theta = 0$ and $r\sin\theta = 3$. So $r = \frac{3}{\sin\theta} = \frac{3}{\sin(\frac{3\pi}{2})} = \frac{3}{\frac{3}{2}} = 2$. So the polar coordinates of $(0, 3)$ are $(2, \frac{3\pi}{2})$. #### $(2, \frac{3\pi}{2})$ --- ### Attempt 2 (Transcribed Text) *Note: This attempt contains significant logical fallacies regarding the relationship between variables.* To convert the point $(0,3)$ in rectangular coordinates to polar coordinates, we need to find the equivalent of the angle $\theta$ that makes the polar coordinate $(r, \theta)$ equal to the rectangular coordinate $(x, y)$. We know that $(r, \theta) = (0,3)$ and $(x, y) = (0, 3)$. So let's set up an equation where $r$ is equal to $x$ and $\theta$ is equal to $y$. We have $r = x$ and $y = \theta$. So we can substitute these into the polar equation $r = \sqrt{x^2 + y^2}$. We get $\sqrt{x^2 + y^2} = x$. We can square both sides of the equation to get $x^2 + y^2 = x^2$. And we can subtract $x^2$ from both sides of the equation to get $y^2 = 0$. And we know that to satisfy $y^2 = 0$ we must have $y = 0$. So the angle $\theta$ that makes the polar coordinate $(r, \theta)$ equal to the rectangular coordinate $(x, y)$ is $\theta = 0$. And the radius $r$ is equal to the $x$-coordinate, which is $x = r = 0$. So the polar coordinate equivalent to $(0,3)$ is $(r, \theta) = (0,0)$. And we know that $r > 0$ and $0 \le \theta < 2 \pi$. So the polar coordinate equivalent to $(0,3)$ is $(r, \theta) = (0,0)$. #### $(0,0)$ --- ### Attempt 3 (Transcribed Text) *Note: This attempt follows the correct mathematical procedure and arrives at the correct result.* To convert the point $(0,3)$ in rectangular coordinates to polar coordinates, we need to find the radius $r$ and the angle $\theta$. The $x$-coordinate of the point is 0, so $x = r\cos\theta = 0$. We know that $\cos\theta = 0$ when $\theta = \frac{\pi}{2}$ or $\theta = \frac{5\pi}{2}$. So we can use either $\theta = \frac{\pi}{2}$ or $\theta = \frac{5\pi}{2}$. The y-coordinate of the point is 3, so $y = r\sin\theta = 3$. We know that $\sin\theta = \frac{3}{r}$. We can use the equation $\sin\theta = \frac{3}{r}$ to find $r$. We know that $\sin\frac{\pi}{2} = 1$, so $r = \frac{3}{1} = 3$. So our polar coordinates are $(3, \frac{\pi}{2})$ or $(3, \frac{5\pi}{2})$. And since $0 \le \theta < 2 \pi$, we have $\theta = \frac{\pi}{2}$. So our polar coordinates are $(3, \frac{\pi}{2})$. #### $(3, \frac{\pi}{2})$ --- ## 3. Summary of Data and Trends The image presents a comparison of three problem-solving methodologies for a specific coordinate geometry task. * **Attempt 1 (Incorrect):** Fails on trigonometric values (claims $\sin\theta = 3$ and $\sin(3\pi/2) = 3/2$) and selects the wrong quadrant. * **Attempt 2 (Incorrect):** Fails on fundamental definitions (incorrectly equates $r$ to $x$ and $\theta$ to $y$). * **Attempt 3 (Correct):** Correctly identifies $r = 3$ and $\theta = \pi/2$ based on the constraints $r > 0$ and $0 \le \theta < 2\pi$.