Converting between polar and rectangular coordinates
Problem 8.116 · easy
Write the polar equation \( \displaystyle r = 2 \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; square both sides: r² = x² + y².
- \[ r^{2} \sin^{2}{\left(\theta \right)} + r^{2} \cos^{2}{\left(\theta \right)} - 4 \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( x^{2} + y^{2} = 4 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | points on the polar curve at four angles satisfy the rectangular equation |
Reviewers
gpt-oss:20b: fail (error) — The solution never derives the final rectangular equation x^2+y^2=4; it stops at an unsimplified expression and does not show the substitution of r=2.qwen3.6:27b-mlx: fail (error) — The solution fails to actually convert the given equation r = 2 into rectangular form. It merely states the conversion formulas and then presents a tautology involving theta, never substituting r=2 into r^2 = x^2 + y^2 to derive x^2 + y^2 = 4.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually convert the given equation r = 2 into rectangular form. It merely states the conversion formulas and then presents a tautology involving theta, never substituting r=2 into r^2 = x^2 + y^2 to derive x^2 + y^2 = 4.gpt-oss:20b: fail (error) 2026-10-05 — The solution never derives the final rectangular equation x^2+y^2=4; it stops at an unsimplified expression and does not show the substitution of r=2.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually convert the equation. It states the conversion formulas but then presents a tautology involving theta instead of substituting r^2 = 4 into x^2 + y^2 = r^2 to get x^2 + y^2 = 4.gpt-oss:20b: fail (error) 2026-10-05 — The solution never substitutes r=2 into the relation r²=x²+y², so it does not derive x²+y²=4. The second line is just an identity and does not provide the required rectangular equation.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/polar_conversion, checked 2026-10-05 with SymPy 1.14.0.