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Converting between polar and rectangular coordinates

Problem 8.150 · easy

Convert the rectangular point \( \displaystyle \left(- \frac{5}{2}, \frac{5 \sqrt{3}}{2}\right) \) to polar coordinates with \( \displaystyle r > 0 \) and \( \displaystyle 0 \le \theta < 2\pi \).

\[ 5 \]
r = √(x² + y²).✓ Proved
The point is in quadrant 2; θ = 2*pi/3.
Reviewed
\[ \left[\begin{matrix}- \frac{5}{2}\\\frac{5 \sqrt{3}}{2}\end{matrix}\right] \]
Converting back gives the point.✓ Proved

Answer \( \left(5, \frac{2 \pi}{3}\right) \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the quadrant and the corresponding angle, and the radius calculation is correct. The steps are logically sound and lead to the correct answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the quadrant and the corresponding angle, and the radius calculation is correct. The steps are logically sound and lead to the correct answer.
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the quadrant and the reference angle, leading to the correct polar coordinates. The steps are logically sound and the final answer matches the stated answer.
gpt-oss:20b: pass 2026-10-06

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 2026-10-06 with SymPy 1.14.0.