Tangent lines to polar curves
Problem 8.400 · medium
Find the slope of the tangent line to \( \displaystyle r = 4 - 4 \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
- With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).Reviewed
- \[ \frac{d}{d \theta} \left(4 - 4 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} = \left(4 \sin{\left(\theta \right)} - 4\right) \sin{\left(\theta \right)} - 4 \cos^{2}{\left(\theta \right)} \]dx/dθ.✓ Proved
- \[ \frac{d}{d \theta} \left(4 - 4 \sin{\left(\theta \right)}\right) \sin{\left(\theta \right)} = \left(4 - 4 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]dy/dθ.✓ Proved
- \[ \frac{\left. \left(4 - 4 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. \left(4 \sin{\left(\theta \right)} - 4\right) \sin{\left(\theta \right)} - 4 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = -1 \]At θ = 2*pi/3.✓ Proved
Answer \( -1 \)
Lines: 3 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the algebraic steps are verified.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the algebraic steps are verified.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the algebraic steps are verified.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the subsequent calculations are correct.gpt-oss:20b: fail (error) 2026-10-15 — The computed slope at θ=2π/3 is +1, not –1. The algebraic evaluation of dy/dx yields 1, so the stated answer is incorrect.
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 2026-10-18 with SymPy 1.14.0.