Tangent lines to parametric curves
Problem 8.56 · hard
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} \left(t^{3} - t\right) = 3 t^{2} - 1 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]dy/dt.✓ Proved
- \[ \frac{\cos{\left(1 \right)}}{2} \]The slope at t = 1.✓ Proved
- \[ \left[\begin{matrix}0\\\sin{\left(1 \right)}\end{matrix}\right] \]The point.✓ Proved
- \[ \frac{x \cos{\left(1 \right)}}{2} + \sin{\left(1 \right)} \]Point-slope form.✓ Proved
Answer \( y = \frac{x \cos{\left(1 \right)}}{2} + \sin{\left(1 \right)} \)
Lines: 5 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.
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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ 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 | a secant through nearby points of the curve has the same slope; the line passes through the point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution uses the point (0, sin(1)) in the point-slope formula, but the x-coordinate at t=1 is 1^3 - 1 = 0, so the point is actually (0, sin(1)). Wait, 1^3 - 1 is 0. So the point is correct. Let me re-read carefully. x = t^3 - t. At t=1, x = 1 - 1 = 0. y = sin(1). So the point is (0, sin(1)). The slope is cos(1)/2. The equation is y - sin(1) = (cos(1)/2)(x - 0), which simplifies to y = x*cos(1)/2 + sin(1). The solution seems correct. Why did I think it was wrong? Let me check the matrix line. Matrix([ [0], [sin(1)] ]). This represents the point (0, sin(1)). This is correct. The final equation is correct. The steps are correct. Is there a subtle error? "Eq(y, x*cos(1)/2 + sin(1))" is the stated answer. The derivation leads to this. It seems sound. Let me look for style or misleading issues. The sentence "dy/dx = (dy/dt)/(dx/dt)" is standard. The derivatives are correct. The evaluation at t=1 is correct. The point calculation is correct. The point-slope form application is correct. I will mark it as pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution uses the point (0, sin(1)) in the point-slope formula, but the x-coordinate at t=1 is 1^3 - 1 = 0, so the point is actually (0, sin(1)). Wait, 1^3 - 1 is 0. So the point is correct. Let me re-read carefully. x = t^3 - t. At t=1, x = 1 - 1 = 0. y = sin(1). So the point is (0, sin(1)). The slope is cos(1)/2. The equation is y - sin(1) = (cos(1)/2)(x - 0), which simplifies to y = x*cos(1)/2 + sin(1). The solution seems correct. Why did I think it was wrong? Let me check the matrix line. Matrix([ [0], [sin(1)] ]). This represents the point (0, sin(1)). This is correct. The final equation is correct. The steps are correct. Is there a subtle error? "Eq(y, x*cos(1)/2 + sin(1))" is the stated answer. The derivation leads to this. It seems sound. Let me look for style or misleading issues. The sentence "dy/dx = (dy/dt)/(dx/dt)" is standard. The derivatives are correct. The evaluation at t=1 is correct. The point calculation is correct. The point-slope form application is correct. I will mark it as pass.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution uses the point (0, sin(1)) for the tangent line equation, but the x-coordinate at t=1 is 1^3 - 1 = 0, which is correct. However, the pogpt-oss:20b: pass 2026-09-26
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/parametric_tangent, checked 2026-09-26 with SymPy 1.14.0.