Home›Calculus 2›Tangent lines to parametric curves›Problem 8.70

Tangent lines to parametric curves

Problem 8.70 · hard

Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).

dy/dx = (dy/dt)/(dx/dt).
Reviewed
\[ \frac{d}{d t} t^{2} = 2 t \]
dx/dt.✓ Proved
\[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]
dy/dt.✓ Proved
\[ \frac{\left(-1\right) \cos{\left(1 \right)}}{2} = - \frac{\cos{\left(1 \right)}}{2} \]
The slope at t = -1.Reviewed
\[ \left[\begin{matrix}1\\- \sin{\left(1 \right)}\end{matrix}\right] \]
The point.Reviewed
\[ \frac{\left(1 - x\right) \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} = - \frac{x \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} + \frac{\cos{\left(1 \right)}}{2} \]
Point-slope form.✓ Proved

Answer \( y = - \frac{x \cos{\left(1 \right)}}{2} - \sin{\left(1 \right)} + \frac{\cos{\left(1 \right)}}{2} \)

✓ Nihil obstat Lines: 3 proved, 3 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t = -1, and derives the correct tangent line equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the chain rule for parametric equations, evaluates the derivatives and coordinates at t = -1, and derives the correct tangent line equation.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly uses the point (1, -sin(1)) instead of the correct point (1, sin(1)) at t = -1, since sin(-1) = -sin(1). This leads to an incorrect tangent line equation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution uses the point (1, -sin(1)) in the point-slope formula, but the correct point at t = -1 is (1, sin(-1)) = (1, -sin(1)). Wait, sin(-1) i
gpt-oss:20b: pass 2026-09-28

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.