Home›Calculus 2›Converting between polar and rectangular coordinates›Problem 8.196

Converting between polar and rectangular coordinates

Problem 8.196 · easy

Convert the polar point \( \displaystyle \left(5, \frac{5 \pi}{4}\right) \) to rectangular coordinates.

\[ \left[\begin{matrix}- \frac{5 \sqrt{2}}{2}\\- \frac{5 \sqrt{2}}{2}\end{matrix}\right] \]
x = r cos θ, y = r sin θ.✓ Proved

Answer \( \left(- \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}\right) \)

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: fail (error) — The solution merely states an equality of identical matrices and claims it follows from x=r cosθ, y=r sinθ, but it does not actually compute cos(5π/4) or sin(5π/4). The setup is incomplete and does not demonstrate the conversion process.
qwen3.6:27b-mlx: fail (error) — The solution provides only a tautological identity check of the final answer rather than showing the derivation steps (substituting r=5 and theta=5pi/4 into x = r cos theta and y = r sin theta). It fails to demonstrate the conversion process.

Every verdict on record (4)

gpt-oss:20b: fail (error) 2026-10-10 — The solution merely states an equality of identical matrices and claims it follows from x=r cosθ, y=r sinθ, but it does not actually compute cos(5π/4) or sin(5π/4). The setup is incomplete and does not demonstrate the conversion process.
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution provides only a tautological identity check of the final answer rather than showing the derivation steps (substituting r=5 and theta=5pi/4 into x = r cos theta and y = r sin theta). It fails to demonstrate the conversion process.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution provides no derivation or explanation of how the polar coordinates were converted to rectangular coordinates. It merely states the final answer in a matrix format without showing the application of x = r cos(θ) and y = r sin(θ) with the given values r=5 and θ=5π/4.

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.