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Converting between polar and rectangular coordinates

Problem 8.120 · easy

Convert the polar point \( \displaystyle \left(2, 0\right) \) to rectangular coordinates.

\[ \left[\begin{matrix}2\\0\end{matrix}\right] \]
x = r cos θ, y = r sin θ.✓ Proved

Answer \( \left(2, 0\right) \)

Every line of this solution was proved by the computer algebra system SymPy. 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	✓ 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	floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the conversion formulas x = r cos(θ) and y = r sin(θ) as sentences, instead burying them in a comment. More critically, it presents a trivial matrix identity as the proof without showing the substitution of r=2 and θ=0 into the formulas.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the conversion formulas x = r cos(θ) and y = r sin(θ) as sentences, instead burying them in a comment. More critically, it presents a trivial matrix identity as the proof without showing the substitution of r=2 and θ=0 into the formulas.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the conversion formulas (x = r cos θ, y = r sin θ) and the substitution of values before presenting the result. It presents an identity matrix equality as the derivation, which is logically insufficient and confusing.
gpt-oss:20b: pass 2026-10-05

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.