Tangent lines to polar curves
Problem 8.279 · medium
Find the slope of the tangent line to \( \displaystyle r = 3 \cos{\left(2 \theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{2} \).
- With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
- \[ \frac{d}{d \theta} 3 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 3 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 6 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \]dx/dθ.✓ Proved
- \[ \frac{d}{d \theta} 3 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 6 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 3 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \]dy/dθ.✓ Proved
- \[ \frac{\left. - 6 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 3 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}}{\left. - 3 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 6 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}} = 0 \]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: fail (error) — The derivative of y with respect to θ is computed incorrectly: it should be -3 sinθ sin2θ + 6 cosθ cos2θ, not -6 sinθ sin2θ + 3 cosθ cos2θ. This mis‑calculation violates the algebraic step, though the final numerical result happens to be correct.qwen3.6:27b-mlx: pass — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the algebraic evaluation is verified as correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the algebraic evaluation is verified as correct.gpt-oss:20b: fail (error) 2026-10-17 — The derivative of y with respect to θ is computed incorrectly: it should be -3 sinθ sin2θ + 6 cosθ cos2θ, not -6 sinθ sin2θ + 3 cosθ cos2θ. This mis‑calculation violates the algebraic step, though the final numerical result happens to be correct.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the algebraic evaluation is verified.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the formula for the slope of a tangent line in polar coordinates and the algebraic evaluation is correct.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.