Converting between polar and rectangular coordinates
Problem 8.193 · easy
Write the polar equation \( \displaystyle r^{2} \sin{\left(2 \theta \right)} = 12 \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; sin 2θ = 2 sin θ cos θ.
- \[ r^{2} \sin{\left(\theta \right)} \cos{\left(\theta \right)} - 6 \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( x y = 6 \)
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 second sentence claims an equivalent equation but the algebra is incorrect. It should substitute sinθ=y/r and cosθ=x/r, leading to xy=6, not the expression shown.qwen3.6:27b-mlx: fail (error) — The solution fails to actually perform the conversion. It lists identities but does not substitute them into the original equation to derive xy = 6. The provided 'equation' is a tautology that does not demonstrate the transformation from polar to rectangular coordinates.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — The second sentence claims an equivalent equation but the algebra is incorrect. It should substitute sinθ=y/r and cosθ=x/r, leading to xy=6, not the expression shown.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to actually perform the conversion. It lists identities but does not substitute them into the original equation to derive xy = 6. The provided 'equation' is a tautology that does not demonstrate the transformation from polar to rectangular coordinates.gpt-oss:20b: fail (misleading) 2026-10-10 — The solution omits the factor 2 from sin 2θ = 2 sinθ cosθ, so the intermediate equation is incorrect, though the final result happens to be right.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to actually perform the conversion. It lists identities but does not substitute them into the original equation to derive xy = 6. The provided equation is a tautology (0=0) that does not represent the conversion steps.
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 2026-10-10 with SymPy 1.14.0.