Home›Calculus 2›Tangent lines to polar curves›Problem 8.517

Tangent lines to polar curves

Problem 8.517 · medium

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

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
\[ \frac{\left. \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}}{\left. - \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}} = 0 \]
At θ = pi/2.✓ Proved

Answer \( 0 \)

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) by ignoring that r is a function of theta. It fails to apply the product rule to differentiate x = r(theta)cos(theta) and y = r(theta)sin(theta), leading to incorrect derivatives for dx/dtheta and dy/dtheta.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution incorrectly assumes x = cos(theta) and y = sin(theta) by ignoring that r is a function of theta. It fails to apply the product rule to differentiate x = r(theta)cos(theta) and y = r(theta)sin(theta), leading to incorrect derivatives for dx/dtheta and dy/dtheta.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution incorrectly computes dx/dθ and dy/dθ as derivatives of cos(θ) and sin(θ) alone, ignoring that r is a function of θ (even if constant, the product rule must be applied or r explicitly substituted before differentiating). While the final numerical answer happens to be correct for this specific case, the method shown is fundamentally flawed and would fail for any r(θ) not equal to 1.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution incorrectly computes dx/dθ and dy/dθ by differentiating cos(θ) and sin(θ) directly, ignoring that r is a function of θ (even though r=1 here, the general formula requires the product rule or explicit substitution first). While the numerical result happens to be correct because r is constant, the method shown is fundamentally flawed for polar curves where r varies, and specifically fails to apply the chain rule/product rule structure dy/dx = (r' sin θ + r cos θ) / (r' cos θ - r sin θ). A student would learn to ignore r' entirely.
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.