Cylindrical and spherical coordinates
Problem 9.379 · easy
Convert the cylindrical point \( \displaystyle (r, \theta, z) = \left(5, \pi, 3\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}-5\\0\\3\end{matrix}\right] \]x = r cos θ, y = r sin θ, z = z.✓ Proved
Answer \( \left(-5, 0, 3\right) \)
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 | converted back to rectangular in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the conversion formulas x = r cos θ, y = r sin θ, z = z to the given point, resulting in the correct rectangular coordinates.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the conversion formulas x = r cos θ, y = r sin θ, z = z to the given point, resulting in the correct rectangular coordinates.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution line is marked 'unchecked', but the problem statement implies algebraic verification is handled by a CAS. More importantly, the solution provides no explanatory sentences to justify the conversion formulas or the evaluation of trigonometric functions, failing the requirement to check that 'each conclusion follows from the lines before it' via logical steps. However, strictly following the prompt's instruction to review SENTENCES: there are no sentences. The single line is an equation. If we must judge the 'solution' as a whole, it is technically correct but lacks the required textual justification. But wait, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is an equation. It is marked 'unchecked'. The prompt says 'do not re-check the algebra of a line marked proved'. This line is NOT marked proved. It is marked 'unchecked'. This is ambiguous. Let's look at the content. The equation is true. The comment explains the formulas. Is there a sentence? No. Is the setup correct? Yes. Is the answer correct? Yes. Why would it fail? Perhaps because it's just an equation with a comment, and the prompt asks to review sentences. If there are no sentences, is it a pass? Or is the lack of sentences a style issue? Or is the 'unchecked' tag an error in the solution format? Let's assume the 'unchecked' tag means the CAS didn't verify it, so I should check it. The algebra is correct. x = 5 cos(pi) = -5. y = 5 sin(pi) = 0. z = 3. The result is correct. The comment is correct. There are no sentences to critique for logical flow. I will mark it as pass because the math is correct and there are no false statements.gpt-oss:20b: pass 2026-10-07
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/cylindrical_spherical_coords, checked 2026-10-07 with SymPy 1.14.0.