Converting between polar and rectangular coordinates
Problem 8.115 · easy
Write the polar equation \( \displaystyle r \sin{\left(\theta \right)} = 4 \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; r sin θ = y.
- \[ r \sin{\left(\theta \right)} - 4 \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( y = 4 \)
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 does not correctly substitute r sinθ with y; it merely repeats the left‑hand side, so the final rectangular equation is not derived.qwen3.6:27b-mlx: fail (error) — The solution fails to actually perform the conversion. It states the relationship r sin(theta) = y but then presents a tautology (r*sin(theta) - 4 = r*sin(theta) - 4) instead of substituting y for r sin(theta) to arrive at y = 4.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually perform the conversion. It states the relationship r sin(theta) = y but then presents a tautology (r*sin(theta) - 4 = r*sin(theta) - 4) instead of substituting y for r sin(theta) to arrive at y = 4.gpt-oss:20b: fail (error) 2026-10-05 — The solution does not correctly substitute r sinθ with y; it merely repeats the left‑hand side, so the final rectangular equation is not derived.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to actually perform the substitution to convert the polar equation to rectangular form. It states the conversion rule but then presents a tautology in polar coordinates rather than the required rectangular equation y = 4.gpt-oss:20b: fail (error) 2026-10-05 — The solution never derives the rectangular equation y=4; it merely repeats the original polar expression without substitution. The second line is incomplete and does not provide the required result.
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.