Converting between polar and rectangular coordinates
Problem 8.178 · easy
Write the polar equation \( \displaystyle r \cos{\left(\theta \right)} = 4 \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; r cos θ = x.
- \[ r \cos{\left(\theta \right)} - 4 \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( x = 4 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ 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 | points on the polar curve at four angles satisfy the rectangular equation |
Reviewers
gpt-oss:20b: fail (error) — The solution never derives x=4; it merely restates the original equation. The second line does not transform the polar equation into rectangular form, so the conclusion is missing.qwen3.6:27b-mlx: fail (error) — The solution fails to actually perform the conversion. It states the relationship r cos(theta) = x but then presents a tautology (r*cos(theta) - 4 = r*cos(theta) - 4) instead of substituting x for r cos(theta) to arrive at x = 4.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to actually perform the conversion. It states the relationship r cos(theta) = x but then presents a tautology (r*cos(theta) - 4 = r*cos(theta) - 4) instead of substituting x for r cos(theta) to arrive at x = 4.gpt-oss:20b: fail (error) 2026-10-08 — The solution never derives x=4; it merely restates the original equation. The second line does not transform the polar equation into rectangular form, so the conclusion is missing.gpt-oss:20b: fail (error) 2026-10-08 — The solution does not actually transform the polar equation into rectangular form; it merely repeats the left‑hand side and does not isolate x. The conclusion that x=4 is not justified by the given steps.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly substitute x for r cos(theta) to arrive at the final rectangular equation x = 4. Instead, it presents a tautology (0=0) as the result, which does not answer the prompt.
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.