Home›Calculus 2›Tangent lines to polar curves›Problem 8.564

Tangent lines to polar curves

Problem 8.564 · medium

Find the slope of the tangent line to \( \displaystyle r = 1 \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).

With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
\[ \frac{d}{d \theta} \cos{\left(\theta \right)} = - \sin{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \sin{\left(\theta \right)} = \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = \frac{\sqrt{3}}{3} \]
At θ = 2*pi/3.✓ Proved

Answer \( \frac{\sqrt{3}}{3} \)

Lines: 3 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. How we verify

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
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	the secant through two points on the curve 2·10⁻¹⁰ apart

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly assumes x = cos(theta) and y = sin(theta), ignoring that r = 1 is a constant. While the numerical result happens to be correct because dr/dtheta = 0, the setup fails to model the general case for polar curves (x = r(theta)cos(theta)), teaching the student to ignore the product rule when r is not constant.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-05 — The solution incorrectly assumes x = cos(theta) and y = sin(theta), ignoring that r = 1 is a constant. While the numerical result happens to be correct because dr/dtheta = 0, the setup fails to model the general case for polar curves (x = r(theta)cos(theta)), teaching the student to ignore the product rule when r is not constant.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution incorrectly assumes x = cos(theta) and y = sin(theta), ignoring that r = 1 is a constant. It fails to apply the product rule to x = r cos(theta) and y = r sin(theta), leading to incorrect derivatives dx/dtheta and dy/dtheta.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution incorrectly assumes r is constant (r=1) when computing dx/dθ and dy/dθ, omitting the terms involving dr/dθ. Although dr/dθ is zero for this specific problem, the general formula dy/dx = (dy/dθ)/(dx/dθ) requires x and y to be expressed in terms of θ first (x = r(θ)cosθ, y = r(θ)sinθ), and the derivatives must reflect the product rule. Stating the derivatives of cos(θ) and sin(θ) directly as dx/dθ and dy/dθ is mathematically incorrect for a general polar curve and teaches a false method.
gpt-oss:20b: pass 2026-11-04

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_tangent, checked 2026-11-05 with SymPy 1.14.0.