Home›Calculus 2›Converting between polar and rectangular coordinates›Problem 8.171

Converting between polar and rectangular coordinates

Problem 8.171 · easy

Write the polar equation \( \displaystyle r = 5 \) in rectangular form.

Use x = r cos θ, y = r sin θ, r² = x² + y²; square both sides: r² = x² + y².
\[ r^{2} \sin^{2}{\left(\theta \right)} + r^{2} \cos^{2}{\left(\theta \right)} - 25 \]
Substituting back gives an equation equivalent to the polar one.✓ Proved

Answer \( x^{2} + y^{2} = 25 \)

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 (error) — The solution does not use the given polar equation r=5 to derive x²+y²=25. It merely states the identity r²=x²+y² and then presents a tautological equation that does not incorporate r=5, so the conclusion is unsupported.
qwen3.6:27b-mlx: fail (error) — The solution fails to substitute r = 5 into the identity r^2 = x^2 + y^2 to obtain x^2 + y^2 = 25. Instead, it presents a tautology involving theta that does not eliminate the polar coordinates.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to substitute r = 5 into the identity r^2 = x^2 + y^2 to obtain x^2 + y^2 = 25. Instead, it presents a tautology involving theta that does not eliminate the polar coordinates.
gpt-oss:20b: fail (error) 2026-10-08 — The solution does not use the given polar equation r=5 to derive x²+y²=25. It merely states the identity r²=x²+y² and then presents a tautological equation that does not incorporate r=5, so the conclusion is unsupported.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (misleading) 2026-10-08 — The solution fails to explicitly substitute r = 5 into the identity r^2 = x^2 + y^2 to derive x^2 + y^2 = 25. Instead, it presents a tautological identity that does not demonstrate the conversion process.

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.