Tangent lines to parametric curves
Problem 8.65 · hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 2 \).
- dy/dx = (dy/dt)/(dx/dt).Reviewed
- \[ \frac{d}{d t} 2 t = 2 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]dy/dt.✓ Proved
- \[ \frac{\cos{\left(2 \right)}}{2} \]The slope at t = 2.✓ Proved
- \[ \left[\begin{matrix}4\\\sin{\left(2 \right)}\end{matrix}\right] \]The point.✓ Proved
- \[ \frac{\left(x - 4\right) \cos{\left(2 \right)}}{2} + \sin{\left(2 \right)} = \frac{x \cos{\left(2 \right)}}{2} - 2 \cos{\left(2 \right)} + \sin{\left(2 \right)} \]Point-slope form.✓ Proved
Answer \( y = \frac{x \cos{\left(2 \right)}}{2} - 2 \cos{\left(2 \right)} + \sin{\left(2 \right)} \)
✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 |
| 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: pass — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t=2, and uses the point-slope form to derive the final equation. The algebraic simplification is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t=2, and uses the point-slope form to derive the final equation. The algebraic simplification is correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t=2, and uses the point-slope form to derive the correct linear equation.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.