Tangent lines to polar curves
Problem 8.522 · medium
Find the slope of the tangent line to \( \displaystyle r = \cos{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{4} \).
- With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
- \[ \frac{d}{d \theta} \cos^{2}{\left(\theta \right)} = - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]dx/dθ.✓ Proved
- \[ \frac{d}{d \theta} \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \]dy/dθ.✓ Proved
- \[ \frac{\left. - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{4} }}}{\left. - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{4} }}} = 0 \]At θ = pi/4.✓ Proved
Answer \( 0 \)
Lines: 3 proved, 1 not checked. 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 | 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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly identifies dx/dθ as the derivative of cos²(θ). Since x = r cos(θ) = cos²(θ), dx/dθ = -sin(2θ), which is -1 at θ = π/4, not 0. The denominator is non-zero, so the slope is undefined (vertical tangent), not 0.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-20 — [domain objection, downgraded to style] The solution incorrectly identifies dx/dθ as the derivative of cos²(θ). Since x = r cos(θ) = cos²(θ), dx/dθ = -sin(2θ), which is -1 at θ = π/4, not 0. The denominator is non-zero, so the slope is undefined (vertical tangent), not 0.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The setup for dy/dx is correct, but the evaluation of dx/dθ at θ = π/4 is incorrect. dx/dθ = -sin(2θ), which equals -1 at π/4, not 0. Since the denominator is non-zero, the slope is 0/-1 = 0, but the solution implies the numerator is 0 (which is true) and divides by 0 or ignores the denominator's value, leading to a logically flawed derivation even if the final number happens to be correct due to the numerator being 0. Wait, let me re-read. Line 4 says the ratio is 0. Numerator is cos^2(pi/4) - sin^2(pi/4) = 1/2 - 1/2 = 0. Denominator is -2*sin(pi/4)*cos(pi/4) = -1. 0/-1 = 0. The calculation is actually correct. Why did I think it was wrong? Ah, I misread the denominator in my head. Let's look closer. Is there a subtle error? Let's re-verify the derivatives. x = r cos θ = cos^2 θ. dx/dθ = 2 cos θ (-sin θ) = -2 sin θ cos θ. Correct. y = r sin θ = cos θ sin θ. dy/dθ = cos^2 θ - sin^2 θ. Correct. At θ = π/4: dy/dθ = 0. dx/dθ = -1. dy/dx = 0/-1 = 0. The solution is mathematically sound. The sentence in step 1 is standard. The equations are proved. The substitution is correct. The result is correct. Wait, is there a geometric issue? r = cos(θ) is a circle passing through the origin. At θ = π/4, the point is (1/2, 1/2) in Cartesian? No. x = cos^2(π/4) = 1/2. y = sin(π/4)cos(π/4) = 1/2. The tangent to the circle (x-1/2)^2 + y^2 = 1/4 at (1/2, 1/2)? The center is (1/2, 0). The radius vector is vertical. So the tangent is horizontal. Slope 0. Correct. Is there any misleading statement? "With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ)." This is standard chain rule application. Let's check if the prompt implies finding the slope of the tangent line to the curve defined by r=cos(theta). Yes. Is it possible the user considers the 'sentence' part of step 4? No, step 4 is marked [equation, proved]. Step 1 is the only sentence. It is correct. Let me double check the 'proved' status. If the CAS proved the derivatives, they are correct. If the CAS proved the substitution, it is correct. Is there a domain issue? θ = π/4 is in the domain where r is defined and differentiable. I see no error. The verdict should be pass.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The solution incorrectly identifies dx/dθ as the derivative of cos²(θ). Since x = r cos(θ) = cos²(θ), dx/dθ = -sin(2θ), which is -1 at θ = π/4, not 0. Consequently, the slope is undefined (vertical tangent), not 0.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.