Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.354

Horizontal and vertical tangents of parametric curves

Problem 8.354 · medium

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

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

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

Lines: 8 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — 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 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to check for points where both dx/dt and dy/dt are zero (singular points). At t = pi/2 and t = 3pi/2, both derivatives vanish, creating cusps. The solution incorrectly omits these points from the vertical tangent analysis without justification, and the stated answer is incomplete.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the critical values of t within the given interval, and verifies that the other derivative is non-zero at these points. The resulting coordinates match the stated answer.
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.