Converting between polar and rectangular coordinates
Problem 8.114 · easy
Convert the rectangular point \( \displaystyle \left(0, 2\right) \) to polar coordinates with \( \displaystyle r > 0 \) and \( \displaystyle 0 \le \theta < 2\pi \).
- \[ 2 \]r = √(x² + y²).✓ Proved
- The point is in an axis direction; θ = pi/2.Reviewed
- \[ \left[\begin{matrix}0\\2\end{matrix}\right] \]Converting back gives the point.✓ Proved
Answer \( \left(2, \frac{\pi}{2}\right) \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ 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 | r and θ from hypot and atan2 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the radius and angle for the point (0, 2) and verifies the conversion. The reasoning is sound and concise.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the radius and angle for the point (0, 2) and verifies the conversion. The reasoning is sound and concise.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the radius and angle for the point (0, 2) and verifies the conversion.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.