Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.356

Horizontal and vertical tangents of parametric curves

Problem 8.356 · medium

Find the points where \( \displaystyle x = \cos{\left(t \right)},\ y = \sin{\left(2 t \right)} \) for \( \displaystyle 0 \le t < 2\pi \) has a horizontal or a vertical tangent line.

\[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]
dx/dt.✓ Proved
\[ \frac{d}{d t} \sin{\left(2 t \right)} = 2 \cos{\left(2 t \right)} \]
dy/dt.✓ Proved
Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
\[ \left. 2 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{\pi}{4} }} = 0 \]
t = pi/4: horizontal, at (sqrt(2)/2, 1).✓ Proved
\[ \left. 2 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{3 \pi}{4} }} = 0 \]
t = 3*pi/4: horizontal, at (-sqrt(2)/2, -1).✓ Proved
\[ \left. 2 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{5 \pi}{4} }} = 0 \]
t = 5*pi/4: horizontal, at (-sqrt(2)/2, 1).✓ Proved
\[ \left. 2 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{7 \pi}{4} }} = 0 \]
t = 7*pi/4: horizontal, at (sqrt(2)/2, -1).✓ Proved
\[ \left. - \sin{\left(t \right)} \right|_{\substack{ t=0 }} = 0 \]
t = 0: vertical, at (1, 0).✓ Proved
\[ \left. - \sin{\left(t \right)} \right|_{\substack{ t=\pi }} = 0 \]
t = pi: vertical, at (-1, 0).✓ Proved

Answer \( \text{horizontal at } (sqrt(2)/2, 1), (-sqrt(2)/2, -1), (-sqrt(2)/2, 1), (sqrt(2)/2, -1);\ \text{vertical at } (1, 0), (-1, 0) \)

Lines: 8 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	a sentence; read, not computed
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
7	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
8	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
9	✓ 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	roots of dx/dt and dy/dt found by Newton's method from a dense grid

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to check the condition dx/dt != 0 for horizontal tangents. At t = 3*pi/4 and 7*pi/4, both dx/dt and dy/dt are zero, creating singular points (cusps) rather than smooth horizontal tangents. These points should be excluded or analyzed separately, but the solution lists them as standard horizontal tangents.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to check the condition dx/dt != 0 for horizontal tangents. At t = 3*pi/4 and 7*pi/4, both dx/dt and dy/dt are zero, creating singular points (cusps) rather than smooth horizontal tangents. These points should be excluded or analyzed separately, but the solution lists them as standard horizontal tangents.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the conditions for horizontal and vertical tangents and verifies that the non-zero derivative condition holds for all candidate points. The coordinates and parameter values are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to check the condition dx/dt ≠ 0 for horizontal tangents and dy/dt ≠ 0 for vertical tangents. At t = 0 and t = π, both dx/dt and dy/dt are zero, resulting in singular points (cusps) rather than vertical tangents. The solution incorrectly classifies these as vertical tangents.
gpt-oss:20b: pass 2026-10-14

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_hv, checked 2026-10-18 with SymPy 1.14.0.