How to find polar coordinates of a given point?
Understand the Problem
The question asks for a method or instructions on how to convert Cartesian coordinates (x, y) of a point to polar coordinates (r, θ). This involves finding the radial distance 'r' from the origin to the point and the angle 'θ' that the line connecting the origin to the point makes with the positive x-axis.
Answer
$r = \sqrt{x^2 + y^2}$, $\theta = \arctan(\frac{y}{x})$ (adjust $\theta$ based on the quadrant of $(x, y)$).
Answer for screen readers
To convert Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$:
-
Calculate $r$ using the formula: $r = \sqrt{x^2 + y^2}$
-
Calculate $\theta$ using the formula:
- If $x > 0$: $\theta = \arctan\left(\frac{y}{x}\right)$
- If $x < 0$ and $y \geq 0$: $\theta = \arctan\left(\frac{y}{x}\right) + \pi$
- If $x < 0$ and $y < 0$: $\theta = \arctan\left(\frac{y}{x}\right) + \pi$
- If $x = 0$ and $y > 0$: $\theta = \frac{\pi}{2}$
- If $x = 0$ and $y < 0$: $\theta = \frac{3\pi}{2}$
- If $x = 0$ and $y = 0$: $\theta = \text{undefined (or 0)}$
Steps to Solve
- Calculate the radial distance $r$
The radial distance $r$ is the distance from the origin $(0,0)$ to the point $(x, y)$. We can calculate it using the Pythagorean theorem:
$$r = \sqrt{x^2 + y^2}$$
- Calculate the angle $\theta$
The angle $\theta$ is the angle between the positive x-axis and the line connecting the origin to the point $(x, y)$. We can find $\theta$ using the arctangent function:
$$\theta = \arctan\left(\frac{y}{x}\right)$$
- Adjust $\theta$ based on the quadrant
The arctangent function has a range of $(-\frac{\pi}{2}, \frac{\pi}{2})$, which means it only gives angles in the first and fourth quadrants. To get the correct angle for all quadrants, we need to adjust $\theta$:
- If $x > 0$, $\theta = \arctan\left(\frac{y}{x}\right)$ (First and Fourth Quadrants)
- If $x < 0$ and $y \geq 0$, $\theta = \arctan\left(\frac{y}{x}\right) + \pi$ (Second Quadrant)
- If $x < 0$ and $y < 0$, $\theta = \arctan\left(\frac{y}{x}\right) + \pi$ (Third Quadrant)
- If $x = 0$ and $y > 0$, $\theta = \frac{\pi}{2}$
- If $x = 0$ and $y < 0$, $\theta = \frac{3\pi}{2}$
- If $x = 0$ and $y = 0$, $\theta$ is undefined (or can be taken as 0).
To convert Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$:
-
Calculate $r$ using the formula: $r = \sqrt{x^2 + y^2}$
-
Calculate $\theta$ using the formula:
- If $x > 0$: $\theta = \arctan\left(\frac{y}{x}\right)$
- If $x < 0$ and $y \geq 0$: $\theta = \arctan\left(\frac{y}{x}\right) + \pi$
- If $x < 0$ and $y < 0$: $\theta = \arctan\left(\frac{y}{x}\right) + \pi$
- If $x = 0$ and $y > 0$: $\theta = \frac{\pi}{2}$
- If $x = 0$ and $y < 0$: $\theta = \frac{3\pi}{2}$
- If $x = 0$ and $y = 0$: $\theta = \text{undefined (or 0)}$
More Information
Polar coordinates represent a point in terms of its distance from the origin ($r$) and the angle it makes with the positive x-axis ($\theta$). This representation is particularly useful in situations involving circular symmetry.
Tips
A common mistake is forgetting to adjust the angle $\theta$ based on the quadrant in which the point $(x, y)$ lies. The arctangent function only returns values in the range $(-\frac{\pi}{2}, \frac{\pi}{2})$, so you must add $\pi$ to the result if $x$ is negative to get the correct angle in the second or third quadrant. Also, remember the special cases when $x = 0$.
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