Calculator Input
Formula Used
Main substitutions:
x = r cos(θ)
y = r sin(θ)
x^2 + y^2 = r^2
The calculator replaces each rectangular variable with its polar equivalent. Then it simplifies common equations when a known form is detected.
Common Transformations
x = abecomesr = a / cos(θ).y = abecomesr = a / sin(θ).x^2 + y^2 = a^2becomesr = a.x^2 + y^2 = axbecomesr = a cos(θ).x^2 + y^2 = aybecomesr = a sin(θ).
Example Data Table
| Rectangular Equation | Polar Form | Rule |
|---|---|---|
x^2 + y^2 = 25 |
r^2 = 25; r = 5 |
Circle centered at pole |
x^2 + y^2 = 6x |
r = 6 cos(θ) |
Circle with x term |
x^2 + y^2 = 8y |
r = 8 sin(θ) |
Circle with y term |
x = 4 |
r = 4 / cos(θ) |
Vertical line |
y = 3 |
r = 3 / sin(θ) |
Horizontal line |
How to Use This Calculator
- Enter a rectangular equation with x and y variables.
- Use ^2 for squared terms, such as x^2.
- Select the angle notation you prefer.
- Choose decimal precision for numeric square roots.
- Press the convert button.
- Review the result above the form.
- Download the work as CSV or PDF when needed.
Understanding Rectangular to Polar Equation Conversion
What the Conversion Means
Rectangular equations use x and y coordinates. Polar equations use radius and angle. The radius is written as r. The angle is usually written as θ. Both systems describe the same plane. They only use different measurements. A rectangular equation can often become cleaner after conversion.
Why Polar Form Helps
Polar form is useful for circles, spirals, rays, and many symmetric curves. It also helps when a curve depends on distance from the origin. A circle centered near the origin may need fewer symbols in polar form. A line may also become a compact fraction with sine and cosine terms.
Core Idea
The conversion starts with three identities. Replace x with r cos(θ). Replace y with r sin(θ). Replace x squared plus y squared with r squared. These identities come from a right triangle built from the point, the origin, and the x axis.
Common Patterns
Some equations have direct results. The equation x equals a becomes r equals a divided by cos(θ). The equation y equals a becomes r equals a divided by sin(θ). A circle like x squared plus y squared equals a squared becomes r equals a. A circle like x squared plus y squared equals ax becomes r equals a cos(θ).
Using the Result
After conversion, check the domain. Polar equations can represent points more than once. Negative r values may also describe points in the opposite direction. For classwork, show the substitution step before writing the final result. This calculator provides both the direct substituted form and a simplified form when possible.
Accuracy Notes
Algebra rules still matter. Parentheses can change the meaning of an equation. Use clear input. Avoid missing equal signs. For complex equations, the direct substituted form may be the safest result. Then you can simplify it further by hand or with a symbolic algebra tool.
FAQs
1. What is a rectangular equation?
A rectangular equation uses x and y coordinates. It describes points on a plane using horizontal and vertical distances.
2. What is a polar equation?
A polar equation uses r and θ. The value r measures distance from the origin. The angle θ measures direction.
3. Which formulas are used?
The calculator uses x = r cos(θ), y = r sin(θ), and x^2 + y^2 = r^2.
4. Can it convert circle equations?
Yes. It handles common circle forms, including x^2 + y^2 = a, x^2 + y^2 = ax, and x^2 + y^2 = ay.
5. Can it convert line equations?
Yes. It supports vertical lines, horizontal lines, slope intercept lines, and many general linear equations.
6. Why does my answer look unsimplified?
Some equations do not match a simple stored pattern. In that case, the calculator shows the direct substitution form.
7. Should I use degrees or radians?
The symbolic conversion is the same. Numerical graphing usually uses radians unless your graphing tool states otherwise.
8. Can I export my result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for printable reports and homework records.