Converting between polar and rectangular coordinates
Problem 8.177 · easy
Convert the polar point \( \displaystyle \left(2, \frac{4 \pi}{3}\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}-1\\- \sqrt{3}\end{matrix}\right] \]x = r cos θ, y = r sin θ.✓ Proved
Answer \( \left(-1, - \sqrt{3}\right) \)
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: pass — The solution correctly applies the conversion formulas x = r cos θ and y = r sin θ with r=2 and θ=4π/3, yielding the correct rectangular coordinates.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the conversion formulas x = r cos θ and y = r sin θ with r=2 and θ=4π/3, yielding the correct rectangular coordinates.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution provides no derivation or explanation, merely stating the final result 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=2 and θ=4π/3.
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 2026-10-08 with SymPy 1.14.0.