Cylindrical and spherical coordinates
Problem 9.374 · easy
Convert the cylindrical point \( \displaystyle (r, \theta, z) = \left(4, \frac{\pi}{6}, 5\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}2 \sqrt{3}\\2\\5\end{matrix}\right] \]x = r cos θ, y = r sin θ, z = z.✓ Proved
Answer \( \left(2 \sqrt{3}, 2, 5\right) \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 cylindrical coordinates. The resulting rectangular coordinates are correct.
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 cylindrical coordinates. The resulting rectangular coordinates are correct.qwen3.6:27b-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 cylindrical coordinates, yielding the correct rectangular coordinates.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.