Home›Calculus 2›Tangent lines to polar curves›Problem 8.568

Tangent lines to polar curves

Problem 8.568 · medium

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

With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 4 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 4 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 8 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 4 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 8 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 4 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - 8 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 4 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}}{\left. - 4 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 8 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}} = \frac{7 \sqrt{3}}{3} \]
At θ = pi/3.✓ Proved

Answer \( \frac{7 \sqrt{3}}{3} \)

✓ 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 applies the formula for the slope of a tangent line in polar coordinates and the algebraic evaluation is verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-05 — 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 2026-11-05
qwen3.6:27b-mlx: pass 2026-11-04
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: pass 2026-11-04 — 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 2026-11-04

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.