Home›Calculus 2›Converting between polar and rectangular coordinates›Problem 8.170

Converting between polar and rectangular coordinates

Problem 8.170 · easy

Convert the polar point \( \displaystyle \left(1, \frac{4 \pi}{3}\right) \) to rectangular coordinates.

\[ \left[\begin{matrix}- \frac{1}{2}\\\frac{\left(-1\right) \sqrt{3}}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{1}{2}\\- \frac{\sqrt{3}}{2}\end{matrix}\right] \]
x = r cos θ, y = r sin θ.✓ Proved

Answer \( \left(- \frac{1}{2}, - \frac{\sqrt{3}}{2}\right) \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the conversion formulas x = r cos θ and y = r sin θ with r=1 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=1 and θ=4π/3, yielding the correct rectangular coordinates.
gpt-oss:20b: pass 2026-10-08
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly show the substitution of r=1 and theta=4pi/3 into the conversion formulas, jumping straight to the final matrix equality without demonstrating the calculation steps required for a complete solution.

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.