Analyze special branches and hidden axis solutions. View tables, plots, exports, and conversion details instantly. Check coordinates fast using one clean responsive workspace today.
These sample rows show the circle branch and the special axis angle.
| Theta (deg) | Theta (rad) | Circle branch r | x | y | Note |
|---|---|---|---|---|---|
| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | Circle branch point. |
| 30 | 0.5236 | 3.0000 | 2.5981 | 1.5000 | Circle branch point. |
| 60 | 1.0472 | 5.1962 | 2.5981 | 4.5000 | Circle branch point. |
| 90 | 1.5708 | 6.0000 | 0.0000 | 6.0000 | Axis branch also allows any radius on x=0. |
| 120 | 2.0944 | 5.1962 | -2.5981 | 4.5000 | Circle branch point. |
| 180 | 3.1416 | 0.0000 | 0.0000 | 0.0000 | Circle branch point. |
Circle branch and y-axis branch shown together.
This calculator studies the relation rcos(theta)=3sin(2theta). It is a useful conversion exercise. It also teaches an important algebra lesson. Some steps can hide a branch. This page helps you avoid that mistake.
Start with standard polar identities. The term rcos(theta) equals x. Also, sin(2theta) equals 2sin(theta)cos(theta). Replace sine and cosine with y/r and x/r. Then simplify the expression carefully. You get x(x2+y2)=6xy. Factor x from the left side. The rectangular form becomes x(x2+y2-6y)=0.
The factored form shows two branches. The first branch is x=0. That is the entire y-axis. The second branch is x2+y2=6y. Completing the square gives x2+(y-3)2=9. That branch is a circle. Its center is (0,3). Its radius is 3.
Many learners divide by cos(theta) too early. That creates r=6sin(theta). This equation draws only the circle branch. It misses the y-axis branch. This calculator highlights that detail. It also checks points numerically. That makes the conversion more reliable.
You can enter theta and inspect the branch behavior. You can enter a radius and verify the original equation. You can also test x and y values directly. The results show polar values, rectangular values, and residual checks. A graph displays the circle and the axis branch together.
The page also includes exports. Use CSV for quick tabular saving. Use PDF for reports or study notes. The example table gives sample angles and coordinates. The formula section summarizes each step. The instructions section shows a clean workflow. This makes the page practical for revision, homework, and classroom use.
Special angles make the structure easy to see. At 30 degrees, the circle branch gives r=3. At 90 degrees, the original equation becomes 0=0. That means every point on the y-axis with that angle works. At 270 degrees, the same axis behavior appears again. These cases explain why the factored rectangular form is safer than an over-simplified polar step.
Use the graph to compare geometry and algebra together. Check a point, then inspect the residual. A near-zero residual confirms the relation. Large residuals show that the point does not belong to the set. This feedback is useful for study, teaching, and quick verification.
The full rectangular conversion is x(x2 + y2 - 6y) = 0. It represents two branches. One branch is the y-axis. The other branch is a circle centered at (0,3) with radius 3.
That step divides by cos(theta). Division by zero is not allowed. When cos(theta)=0, the original equation still holds for every radius on the y-axis. So r = 6sin(theta) keeps the circle but drops the axis branch.
It is a circle. The center is (0,3). The radius is 3. This circle is the non-axis branch of the converted set. It appears when the equation is simplified without the special x=0 branch.
It means the full y-axis belongs to the relation. In polar terms, this happens at angles where cos(theta)=0. At those angles, the left side becomes zero and the right side also becomes zero, regardless of radius.
Yes. Enter theta and a radius. The calculator converts the point to x and y, evaluates both sides of the original relation, and shows residuals. Small residuals mean the point fits the equation closely.
Yes. Enter x and y values in the form. The tool computes the rectangular residual from x(x2 + y2 - 6y). A near-zero result shows that the point lies on the converted set.
Because the conversion produces two branches. The y-axis comes from x=0. The circle comes from x2 + (y - 3)2 = 9. Showing both branches prevents incomplete interpretation of the original equation.
The CSV button saves the current results and example table in spreadsheet-friendly text. The PDF button creates a compact report with the main results. These options help with revision notes, homework files, and sharing.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.