Converting between polar and rectangular coordinates
Problem 8.175 · easy
Write the polar equation \( \displaystyle r = 6 \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; square both sides: r² = x² + y².
- \[ r^{2} \sin^{2}{\left(\theta \right)} + r^{2} \cos^{2}{\left(\theta \right)} - 36 \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( x^{2} + y^{2} = 36 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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 (misleading) — The solution never derives the rectangular equation x²+y²=36; it merely restates the identity r²=x²+y² without concluding the required form.qwen3.6:27b-mlx: pass — The solution correctly identifies the relationship r^2 = x^2 + y^2 and substitutes r=6 to derive x^2 + y^2 = 36. The algebraic verification in step 2 is valid, though verbose.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the relationship r^2 = x^2 + y^2 and substitutes r=6 to derive x^2 + y^2 = 36. The algebraic verification in step 2 is valid, though verbose.gpt-oss:20b: fail (misleading) 2026-10-08 — The solution never derives the rectangular equation x²+y²=36; it merely restates the identity r²=x²+y² without concluding the required form.gpt-oss:20b: fail (error) 2026-10-08 — The solution never substitutes r=6 into the identity r²=x²+y² to obtain x²+y²=36, and the final equation shown is just a tautological rewrite, not the required rectangular form.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to substitute r = 6 into the identity r^2 = x^2 + y^2 to obtain x^2 + y^2 = 36. Instead, it presents a tautology involving theta, which does not constitute a 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 2026-10-08 with SymPy 1.14.0.