Home›Calculus 2›Tangent lines to polar curves›Problem 8.399

Tangent lines to polar curves

Problem 8.399 · medium

Find the slope of the tangent line to \( \displaystyle r = 4 \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).

With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - 4 \sin^{2}{\left(\theta \right)} + 4 \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 4 \sin^{2}{\left(\theta \right)} = 8 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. 8 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - 4 \sin^{2}{\left(\theta \right)} + 4 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = \sqrt{3} \]
At θ = 2*pi/3.✓ Proved

Answer \( \sqrt{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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: pass
qwen3.6:27b-mlx: pass — The solution correctly sets up the parametric derivatives for polar coordinates and evaluates them at the specified angle. The algebraic steps are verified as correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly sets up the parametric derivatives for polar coordinates and evaluates them at the specified angle. The algebraic steps are verified as correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly sets up the conversion from polar to Cartesian coordinates and applies the chain rule for dy/dx. The derivatives and substitution are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the conversion from polar to Cartesian coordinates and applies the chain rule for dy/dx. The derivatives and substitution are correct.
gpt-oss:20b: pass 2026-10-15

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.