Tangent lines to parametric curves
Problem 8.11 · hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} 2 t = 2 \]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.
- \[ \left[\begin{matrix}-2\\- \sin{\left(1 \right)}\end{matrix}\right] \]The point.
- \[ \frac{\left(x + 2\right) \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} = \frac{x \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} + \cos{\left(1 \right)} \]Point-slope form.✓ Proved
Answer \( y = \frac{x \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} + \cos{\left(1 \right)} \)
Lines: 3 proved, 3 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 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 5 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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 incorrectly evaluates the slope at t = 1 instead of t = -1. Since cos(t) is an even function, cos(-1) = cos(1), so the numerical value of the slope is correct, but the reasoning implies evaluating at the wrong parameter value. Additionally, the point coordinates are correct (x=-2, y=-sin(1)), but the label 'The point' is vague. The primary error is the implicit evaluation at t=1 for the derivative step, which is technically incorrect procedure even if the result coincidentally matches due to symmetry.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly evaluates the slope at t = 1 instead of t = -1. Since cos(t) is an even function, cos(-1) = cos(1), so the numerical value of the slope is correct, but the reasoning implies evaluating at the wrong parameter value. Additionally, the point coordinates are correct (x=-2, y=-sin(1)), but the label 'The point' is vague. The primary error is the implicit evaluation at t=1 for the derivative step, which is technically incorrect procedure even if the result coincidentally matches due to symmetry.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer returned a non-objectgpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly evaluates the slope at t = -1 as cos(1)/2 instead of cos(-1)/2. Although cos(1) equals cos(-1), the step implies evaluating at t=1, which is conceptually wrong and dangerous for non-even functions. Additionally, the final answer format 'Eq(y, ...)' is non-standard.gpt-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 2026-10-13 with SymPy 1.14.0.