Calculator
Example Data Table
| Polar Input | Identity Used | Rectangular Output | Notes |
|---|---|---|---|
| r = 5 | r^2 = x^2 + y^2 | x^2 + y^2 = 25 | Circle centered at origin. |
| r = 6 cos(theta) | r cos(theta) = x | x^2 + y^2 = 6x | Circle with shifted center. |
| r = 4 sin(theta) | r sin(theta) = y | x^2 + y^2 = 4y | Circle shifted upward. |
| theta = 45 degrees | tan(theta) = y / x | y = x | Line through origin. |
| r = 10 / (1 + 0.5 cos(theta)) | r cos(theta) = x | sqrt(x^2 + y^2) + 0.5x = 10 | Conic form. |
Formula Used
The calculator uses standard polar to rectangular identities:
x = r cos(theta)y = r sin(theta)r^2 = x^2 + y^2tan(theta) = y / xcos(theta) = x / sqrt(x^2 + y^2)sin(theta) = y / sqrt(x^2 + y^2)
For example, r = a cos(theta) is multiplied by r. This gives r^2 = a r cos(theta). Then substitutions give x^2 + y^2 = ax.
How to Use This Calculator
- Select point mode or equation mode.
- Choose the closest polar equation pattern.
- Enter the required constant, radius, angle, or eccentricity.
- Select degrees or radians when an angle is used.
- Choose the decimal precision for rounded values.
- Click Calculate to show the result below the header.
- Use CSV or PDF export for saved records.
Guide Article
Understanding Polar Conversion
Polar equations describe curves by distance and angle. Rectangular equations describe the same curve with x and y. This calculator helps bridge those views. It accepts common polar forms. It also converts polar points into rectangular coordinates. Each result includes the equation, identities, and clean steps.
Core Method
The main idea is substitution. The identity r squared equals x squared plus y squared is the base. The identity x equals r cosine theta is also important. The identity y equals r sine theta completes the link. With these three rules, many polar equations become familiar rectangular forms.
Common Curve Results
A circle often starts as r equals a. Its rectangular form is x squared plus y squared equals a squared. A cardioid or conic may keep a square root term first. Then the squared form is shown when it is algebraically useful. Lines using theta become slopes through the origin. Sine and cosine circle forms can be completed into center radius form.
Point and Equation Work
Use the point mode for exact coordinate work. Enter radius and angle. Choose degrees or radians. The tool computes x and y. It also shows the formulas used. This helps students check graphs, tables, and homework values.
Use the equation mode for curve conversion. Choose the closest polar pattern. Enter the required constant. Add eccentricity when converting conic forms. The result panel appears above the form after submission. You can review the steps before exporting.
Exporting and Checking
Exports are helpful for notes and reports. The CSV file stores the selected inputs and final results. The PDF file stores a readable summary. Both options keep the workflow simple.
This converter is best for standard forms. Very complex symbolic equations may need manual algebra. Still, the substitution guide can show the correct path. Start by replacing r squared. Then replace r cosine theta with x. Replace r sine theta with y. Simplify carefully. Check restrictions when squaring equations, because extraneous branches may appear.
The example table shows typical entries. Compare them with your own values. Small changes in angle or constant can change the curve strongly. Always verify units, signs, and pattern choice before using the final rectangular equation. Graphing after conversion can reveal hidden mistakes quickly.
FAQs
1. What does polar to rectangular conversion mean?
It changes equations written with radius and angle into equations written with x and y. Both forms can describe the same curve, but rectangular form is often easier for algebra and graph comparison.
2. Which identities are most important?
The key identities are x = r cos(theta), y = r sin(theta), and r^2 = x^2 + y^2. These three substitutions solve many standard polar equation conversions.
3. Can this calculator convert polar points?
Yes. Select point mode, enter r and theta, then choose degrees or radians. The calculator returns the rectangular ordered pair using x = r cos(theta) and y = r sin(theta).
4. Why does r = a become a circle?
When r = a, squaring both sides gives r^2 = a^2. Since r^2 equals x^2 + y^2, the rectangular form is x^2 + y^2 = a^2.
5. What happens with theta equals an angle?
A constant theta creates a line through the origin. The rectangular form normally uses y = tan(theta)x. If the angle is vertical, the line becomes x = 0.
6. Can custom polar equations be solved fully?
The custom option gives a substitution guide. It does not perform complete symbolic algebra for every expression. Use it to identify correct replacements before simplifying manually.
7. Why can squaring create issues?
Squaring may add extra branches that were not in the original equation. Always compare restrictions, signs, and domains after converting equations that contain square roots.
8. What do the export buttons save?
The CSV export saves inputs, results, and steps in spreadsheet format. The PDF export saves a readable result summary for notes, homework records, or project documentation.