Converting between polar and rectangular coordinates
Problem 8.117 · easy
Write the polar equation \( \displaystyle r = 8 \cos{\left(\theta \right)} \) in rectangular form.
- Use x = r cos θ, y = r sin θ, r² = x² + y²; multiply by r: r² = 2a·r cos θ.
- \[ r^{2} \sin^{2}{\left(\theta \right)} + r^{2} \cos^{2}{\left(\theta \right)} - 8 r \cos{\left(\theta \right)} \]Substituting back gives an equation equivalent to the polar one.✓ Proved
Answer \( x^{2} + y^{2} = 8 x \)
Lines: 1 proved, 1 not checked. 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 | 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 (style) — [domain objection, downgraded to style] The first sentence incorrectly states the polar equation as r² = 2a·r cosθ and introduces an undefined constant a. It should be r² = 8r cosθ. The subsequent equation is also not derived correctly from the given polar form.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly substitute the rectangular identities (x = r cos θ, r² = x² + y²) into the equation r² = 8r cos θ to derive the final answer x² + y² = 8x. It only verifies that the polar equation is consistent with itself, omitting the actual conversion step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly substitute the rectangular identities (x = r cos θ, r² = x² + y²) into the equation r² = 8r cos θ to derive the final answer x² + y² = 8x. It only verifies that the polar equation is consistent with itself, omitting the actual conversion step.gpt-oss:20b: fail (style) 2026-10-05 — [domain objection, downgraded to style] The first sentence incorrectly states the polar equation as r² = 2a·r cosθ and introduces an undefined constant a. It should be r² = 8r cosθ. The subsequent equation is also not derived correctly from the given polar form.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to perform the actual conversion to rectangular coordinates. It substitutes the polar equation into an identity but never replaces r and theta with x and y to derive the final answer x^2 + y^2 = 8x.gpt-oss:20b: fail (error) 2026-10-05 — The first sentence incorrectly introduces a parameter 2a and claims r² = 2a·r cosθ, which does not follow from the given polar equation r = 8 cosθ. The correct step is r² = 8r cosθ, leading to x² + y² = 8x. The solution omits this and contains a misapplied formula.
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 2026-10-05 with SymPy 1.14.0.