Cylindrical and spherical coordinates
Problem 9.452 · easy
Convert the cylindrical point \( \displaystyle (r, \theta, z) = \left(1, \frac{\pi}{6}, 5\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}\frac{\sqrt{3}}{2}\\1 \cdot \frac{1}{2}\\5\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{3}}{2}\\\frac{1}{2}\\5\end{matrix}\right] \]x = r cos θ, y = r sin θ, z = z.✓ Proved
Answer \( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}, 5\right) \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the conversion formulas and computes the values accurately.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the conversion formulas and computes the values accurately.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution incorrectly calculates y = r sin(π/6) as 1/2 instead of 1/2 * 1 = 1/2? Wait, sin(π/6) is 1/2. So y = 1 * 1/2 = 1/2. The stated answer is (sqrt(3)/2, 1/2, 5). The calculation is actually correct. Let me re-read carefully. r=1, theta=pi/6. x = 1*cos(pi/6) = sqrt(3)/2. y = 1*sin(pi/6) = 1/2. z=5. The result is correct. Why did I think it was wrong? Ah, I misread the 'unchecked' tag or something? No, the prompt says 'unchecked' for the equation but the algebra is trivial. The sentence says 'x = r cos θ, y = r sin θ, z = z'. This is the correct formula. The result matches. Is there a subtle error? No. It seems correct. Wait, looking at the JSON output requirement. If it is sound, say pass. It is sound.
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 2026-10-09 with SymPy 1.14.0.