Converting between polar and rectangular coordinates
Problem 8.111 · easy
Write the polar equation \( \displaystyle r^{2} \sin{\left(2 \theta \right)} = 4 \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; sin 2θ = 2 sin θ cos θ.
- \[ r^{2} \sin{\left(\theta \right)} \cos{\left(\theta \right)} - 2 \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( x y = 2 \)
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 skips the crucial step of substituting sinθ= y/r and cosθ= x/r, which turns r² sinθ cosθ into xy. It also omits the factor 2 from sin2θ=2sinθcosθ, so the derived equation r² sinθ cosθ = 2 is not shown. Thus the conclusion xy=2 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 sets up the identities but then presents a tautology involving polar terms rather than substituting x and y to derive xy = 2.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually perform the conversion to rectangular coordinates. It sets up the identities but then presents a tautology involving polar terms rather than substituting x and y to derive xy = 2.gpt-oss:20b: fail (error) 2026-10-05 — The solution skips the crucial step of substituting sinθ= y/r and cosθ= x/r, which turns r² sinθ cosθ into xy. It also omits the factor 2 from sin2θ=2sinθcosθ, so the derived equation r² sinθ cosθ = 2 is not shown. Thus the conclusion xy=2 is not justified by the given steps.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually perform the conversion. It states the necessary identities but then presents a tautology involving polar terms rather than substituting them into rectangular coordinates to derive xy = 2.gpt-oss:20b: fail (misleading) 2026-10-05 — The solution skips the crucial step of using sin 2θ = 2 sinθ cosθ, leading to 2 r² sinθ cosθ = 4. The subsequent equation shown is merely a tautology and does not correctly substitute the polar equation into rectangular coordinates, so a student would learn an incorrect derivation.
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.