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Cylindrical and spherical coordinates

Problem 9.298 · easy

Convert the cylindrical point \( \displaystyle (r, \theta, z) = \left(5, \pi, 5\right) \) to rectangular coordinates.

\[ \left[\begin{matrix}-5\\0\\5\end{matrix}\right] \]
x = r cos θ, y = r sin θ, z = z.✓ Proved

Answer \( \left(-5, 0, 5\right) \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the conversion formulas x = r cos θ, y = r sin θ, z = z to the given point (5, π, 5), yielding (-5, 0, 5). The algebraic verification is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the conversion formulas x = r cos θ, y = r sin θ, z = z to the given point (5, π, 5), yielding (-5, 0, 5). The algebraic verification is correct.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution provides only a final numerical check without showing the conversion steps or explicitly stating the formulas used. While the result is correct, the lack of derivation makes it impossible to verify the method, and labeling the line 'unchecked' contradicts the prompt's instruction that equations are checked by a CAS.
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/cylindrical_spherical_coords, checked 2026-10-05 with SymPy 1.14.0.