Converting between polar and rectangular coordinates
Problem 8.147 · easy
Write the polar equation \( \displaystyle r^{2} \sin{\left(2 \theta \right)} = 6 \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; sin 2θ = 2 sin θ cos θ.
- \[ r^{2} \sin{\left(\theta \right)} \cos{\left(\theta \right)} - 3 \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( x y = 3 \)
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’s second line is incorrect: it merely repeats an identity and never substitutes the polar equation r²sin(2θ)=6 or the expressions for sinθ, cosθ, and r². Consequently, the conclusion xy=3 is not justified by the given steps.qwen3.6:27b-mlx: fail (error) — The solution fails to actually perform the conversion to rectangular coordinates. It states the necessary identities but then presents a tautology in polar coordinates rather than substituting x and y to derive xy = 3.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The solution’s second line is incorrect: it merely repeats an identity and never substitutes the polar equation r²sin(2θ)=6 or the expressions for sinθ, cosθ, and r². Consequently, the conclusion xy=3 is not justified by the given steps.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to actually perform the conversion to rectangular coordinates. It states the necessary identities but then presents a tautology in polar coordinates rather than substituting x and y to derive xy = 3.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to perform the actual conversion. It states the identities but then presents a tautology (A - 3 = A - 3) that does not substitute the rectangular variables x and y, nor does it derive the final answer xy = 3.gpt-oss:20b: fail (error) 2026-10-06 — The second line does not correctly substitute the identities; it merely repeats the same expression and does not derive xy=3. The step is missing the use of sin2θ=2sinθcosθ and the conversion of sinθ cosθ to xy/r², so the conclusion is unsupported.
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 2026-10-06 with SymPy 1.14.0.