Converting between polar and rectangular coordinates
Problem 8.198 · easy
Convert the polar point \( \displaystyle \left(1, \frac{7 \pi}{6}\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{3}}{2}\\- \frac{1}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{3}}{2}\\- \frac{1}{2}\end{matrix}\right] \]x = r cos θ, y = r sin θ.✓ Proved
Answer \( \left(- \frac{\sqrt{3}}{2}, - \frac{1}{2}\right) \)
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 | floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (style) — The solution lacks the necessary setup sentences stating the conversion formulas and the specific values of r and theta used, jumping directly to the final matrix equality.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (style) 2026-10-10 — The solution lacks the necessary setup sentences stating the conversion formulas and the specific values of r and theta used, jumping directly to the final matrix equality.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution provides no derivation or explanation, merely stating the final answer as an identity. It fails to show the application of the conversion formulas x = r cos(θ) and y = r sin(θ) with the given values r=1 and θ=7π/6.
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 2026-10-10 with SymPy 1.14.0.