How to find polar coordinates from rectangular coordinates?
Understand the Problem
The question is asking how to convert rectangular coordinates (x, y) into polar coordinates (r, θ). The process involves using the formulas r = √(x² + y²) for the radius and θ = arctan(y/x) for the angle.
Answer
The polar coordinates are $(r, \theta)$, where $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan\left(\frac{y}{x}\right) + k \cdot \pi$ for quadrant adjustment.
Answer for screen readers
The polar coordinates are $(r, \theta)$, where: $$ r = \sqrt{x^2 + y^2} $$ $$ \theta = \arctan\left(\frac{y}{x}\right) + k \cdot \pi $$ depending on the quadrant adjustment.
Steps to Solve
- Find the Radius (r)
To convert from rectangular coordinates $(x, y)$ to polar coordinates, we first need to find the radius $r$.
Using the formula: $$ r = \sqrt{x^2 + y^2} $$
Simply replace $x$ and $y$ with their respective values.
- Find the Angle (θ)
Next, we calculate the angle $\theta$ using the arctangent function.
The formula is: $$ \theta = \arctan\left(\frac{y}{x}\right) $$
Make sure to consider the correct quadrant based on the signs of $x$ and $y$.
- Adjusting the Angle (θ)
Depending on the quadrant in which the point $(x, y)$ lies, you may need to adjust the angle:
- If $x > 0$ and $y \geq 0$: $\theta$ is in Quadrant I, no adjustment needed.
- If $x < 0$: Add $π$ radians to the angle.
- If $y < 0$ and $x > 0$: $\theta$ is in Quadrant IV, no adjustment needed.
- If $y < 0$ and $x < 0$: Add $π$ to the angle.
- If $x = 0$ and $y > 0$: $\theta = \frac{π}{2}$.
- If $x = 0$ and $y < 0$: $\theta = \frac{3π}{2}$.
- Presenting the Results
Finally, write the polar coordinates as $(r, \theta)$.
The polar coordinates are $(r, \theta)$, where: $$ r = \sqrt{x^2 + y^2} $$ $$ \theta = \arctan\left(\frac{y}{x}\right) + k \cdot \pi $$ depending on the quadrant adjustment.
More Information
Polar coordinates are used in many fields such as physics and engineering to simplify calculations involving circular and rotational motion. Understanding how to switch between coordinate systems is essential for applications in trigonometry and geometry.
Tips
- Forgetting to adjust the angle $\theta$ based on the quadrant of the point $(x, y)$.
- Using the wrong sign for $x$ and $y$, which can lead to incorrect values for $r$ and $\theta$.
- Not being aware of the range of the arctangent function, which might result in angles outside the desired range.
