Converting between polar and rectangular coordinates
Problem 8.144 · easy
Convert the polar point \( \displaystyle \left(2, \frac{\pi}{3}\right) \) 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=π/3 to arrive at the correct rectangular coordinates.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the conversion formulas x = r cos θ and y = r sin θ with r=2 and θ=π/3 to arrive at the correct rectangular coordinates.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution claims to use the formulas x = r cos θ and y = r sin θ, but the resulting coordinates (1, sqrt(3)) correspond to r=2, θ=π/3 only if the input was (2, π/3). Wait, 2*cos(π/3) = 1 and 2*sin(π/3) = sqrt(3). The calculation is actually correct. However, the line is marked 'unchecked' but contains a tautology 'Matrix(...) = Matrix(...)'. The text says 'x = r cos θ, y = r sin θ' but does not show the substitution step. More importantly, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. This line is labeled '[equation, unchecked]'. If it's unchecked, I must check the algebra. 2*cos(pi/3) = 1. 2*sin(pi/3) = sqrt(3). The math is correct. But the format is weird. Let's look closer. The problem asks to convert (2, pi/3). The answer is (1, sqrt(3)). The solution line shows the result equals the result. It doesn't show the derivation. Is this an error? The instructions say 'Check that... each conclusion follows from the lines before it'. There are no lines before it. It just states the answer. This is a 'style' or 'misleading' issue because it doesn't show the work, but the result is correct. However, usually 'unchecked' implies I need to verify the math. The math is correct. But the line is just a tautology. It doesn't actually perform the conversion in the text, it just asserts the final state. A student might not learn how to get there. But is it 'wrong'? No. Is it 'misleading'? Maybe. Let's look at the severity definitions. 'error' is a step or setup is wrong. 'misleading' is a student would learn something false. 'style' is correct but could be clearer. This is definitely 'style' or 'misleading'. But wait, if I have to find what is wrong... The line is labeled 'unchecked'. If I check it, the equation is true. The text 'x = r cos θ, y = r sin θ' is the method. The equation is the result. It's a bit of a jump, but not false. However, often these adversarial reviews catch subtle errors. Did I miss one? r=2, theta=pi/3. x = 2 * 1/2 = 1. y = 2 * sqrt(3)/2 = sqrt(3). The answer is correct. The setup is correct. The only issue is the lack of intermediate steps. Is that an 'error'? Probably not. Is it 'misleading'? It might imply that the conversion is trivial or just a lookup. I will mark it as 'style' because the solution is technically correct but lacks the explanatory step of substitution, which is poor pedagogical style for a 'solution'.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.