Calculator
Use the default entries for r = 1/(1 - sinθ). Change the options for related conic forms.
Formula Used
Polar relation: x = r cosθ, y = r sinθ, and r² = x² + y².
Requested conversion: r = 1/(1 - sinθ).
Algebra: r(1 - sinθ) = 1. Then r - r sinθ = 1. Since r sinθ = y, we get r - y = 1.
Final form: r = y + 1. Then x² + y² = (y + 1)². So x² = 2y + 1.
How to Use This Calculator
- Keep the default values for the requested equation.
- Change numerator, sign, or trig function for related forms.
- Enter a sample theta to test one polar point.
- Select a decimal precision for displayed values.
- Press the conversion button to view the result.
- Use the CSV or print option for saving work.
Example Data Table
| Polar Form | Key Substitution | Rectangular Result | Conic Type |
|---|---|---|---|
r = 1/(1 - sinθ) |
r sinθ = y |
x² = 2y + 1 |
Parabola |
r = 1/(1 + sinθ) |
r sinθ = y |
x² = -2y + 1 |
Parabola |
r = 1/(1 - cosθ) |
r cosθ = x |
y² = 2x + 1 |
Parabola |
Detailed Guide For Polar To Rectangular Conversion
Understanding This Polar Conversion
The equation r = 1/(1 - sin theta) is a polar conic. It describes a parabola in rectangular form. The denominator contains sine, so the y coordinate appears naturally. Since y equals r sin theta, the expression becomes easier to change.
Why The Equation Becomes A Parabola
Start with r = 1/(1 - sin theta). Multiply both sides by the denominator. This gives r(1 - sin theta) = 1. Distribute r across both terms. The new statement is r - r sin theta = 1. Replace r sin theta with y. The result is r - y = 1. Move y to the other side. Now r = y + 1. Square both sides because r contains a square root. Since r squared equals x squared plus y squared, the equation becomes x squared plus y squared equals y plus one squared. Expand the right side. Then cancel y squared from both sides. The final form is x squared equals 2y plus 1.
What The Rectangular Form Shows
The rectangular equation x squared = 2y + 1 is easier to graph. It can also be written as x squared = 2(y + 1/2). This shows a vertical parabola. Its vertex is at (0, -1/2). The value 4p equals 2. Therefore p equals 1/2. The focus is at (0, 0). The directrix is y = -1. These values match the polar conic structure. The pole becomes the focus.
Why This Calculator Helps
Manual conversion can feel confusing. Small sign errors can change the conic type. This tool displays each algebra step. It also supports related polar forms. You can change the numerator, constant term, sign, coefficient, and trigonometric function. The calculator then builds the matching rectangular equation. It also checks one angle point. That point helps confirm the result.
Common Learning Notes
Always place parentheses around the denominator. The expression means one divided by the full term. It does not mean one divided by one, then minus sine. Use theta in degrees when entering sample angles here. The tool converts degrees to radians internally. Sine equations usually produce vertical conics. Cosine equations usually produce horizontal conics. When the squared coefficient cancels, a parabola appears. When it does not cancel, another conic may appear.
Practical Uses
This conversion is useful in precalculus, calculus, analytic geometry, and graphing tasks. It helps compare polar graphs with rectangular graphs. It also supports checking homework solutions. Teachers can use it to show each step. Students can use it to practice structure. The result connects polar focus information with rectangular shape details. That connection builds stronger graph understanding.
Best Practice Tips
Check each replacement before simplifying. Replace r squared with x squared plus y squared. Replace r sin theta with y. Replace r cos theta with x. Keep signs clear at every stage. A neat line of work prevents most mistakes during exams.
FAQs
What is the rectangular form of r = 1/(1 - sin theta)?
The rectangular form is x² = 2y + 1. It comes from multiplying by the denominator, replacing r sinθ with y, then squaring with r² = x² + y².
Why does sine connect to y?
In polar and rectangular conversion, y = r sinθ. That means any expression containing r sinθ can be replaced by y. This makes sine based polar equations easier to simplify.
Why is r squared replaced with x squared plus y squared?
The distance from the origin is r. By the distance formula, r² = x² + y². This identity is the main bridge from polar form to rectangular form.
Is this equation a parabola?
Yes. The final equation x² = 2y + 1 is a vertical parabola. Its vertex is (0, -1/2), and it opens upward.
What is the focus of the converted parabola?
For x² = 2(y + 1/2), 4p = 2. So p = 1/2. The vertex is (0, -1/2), so the focus is (0, 0).
What is the directrix?
The directrix is y = -1. This follows from the vertex (0, -1/2) and p = 1/2. Move one half unit downward from the vertex.
Can I use this for cosine equations?
Yes. Select cosine in the trig function field. Cosine conversions use r cosθ = x. They often create horizontal conics instead of vertical ones.
What happens if I change the denominator sign?
The sign changes the direction of the conic. For example, 1/(1 + sinθ) gives x² = -2y + 1. That parabola opens downward.
Why does the calculator square both sides?
After substitution, the equation usually contains r. Squaring allows replacement with x² + y². This removes the polar distance term and creates a rectangular equation.
What does the residual mean?
The residual checks a sample point in the final equation. A value near zero means the converted rectangular equation matches the polar point. Minor decimals can appear from rounding.
Can the denominator become zero?
Yes. Some angles can make the denominator zero. At those angles, the polar value is undefined. The calculator reports this when a sample angle causes division by zero.