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Tangent lines

Problem 2.783 · medium Video

Find an equation of the tangent line to \( \displaystyle y = - x + 2 \sin{\left(x \right)} \) at \( \displaystyle x = \pi \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
Reviewed
\[ \left. - x + 2 \sin{\left(x \right)} \right|_{\substack{ x=\pi }} = - \pi \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- x + 2 \sin{\left(x \right)}\right) = 2 \cos{\left(x \right)} - 1 \]
Differentiate.✓ Proved
\[ \left. 2 \cos{\left(x \right)} - 1 \right|_{\substack{ x=\pi }} = -3 \]
The slope at the point.✓ Proved
\[ - 3 x + 2 \pi \]
Point-slope form, then simplify.

Answer \( y = - 3 x + 2 \pi \)

The graph of y = -x + 2sin(x) near x = pi. A secant line through the point (pi, -pi) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at -3, giving y = -3x + 2pi.

Lines: 3 proved, 1 reviewed, 1 not checked. 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	claude-sonnet-5-5	claude-sonnet-5-5: This is the standard statement that a tangent line at x = a passes through (a, f(a)) with slope f'(a). It is a sentence, not an algebraic equivalence, and it is true.
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	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the line meets the curve at x = a, and its slope matches a central difference quotient of f there

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the point of tangency is -pi, but f(pi) = -pi + 2*sin(pi) = -pi. Wait, the equation in line 2 says 'Subs(...), -pi'. This is cor

Senior review claude-sonnet-5-5, 2026-10-12: pass — f(pi) = -pi and f'(pi) = 2cos(pi) - 1 = -3. Point-slope gives y = -3(x - pi) - pi = -3x + 2pi, which matches the stated answer. Line 5 folds point-slope and simplification into one line, but that is a minor presentation issue and not an error.

qwen3.6:27b-mlx: dismiss — The response was unparseable and its own reasoning confirmed f(pi) = -pi + 2 sin(pi) = -pi, so line 2 is correct and no defect was identified.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — f(pi) = -pi and f'(pi) = 2cos(pi) - 1 = -3. Point-slope gives y = -3(x - pi) - pi = -3x + 2pi, which matches the stated answer. Line 5 folds point-slope and simplification into one line, but that is a minor presentation issue and not an error.
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the point of tangency is -pi, but f(pi) = -pi + 2*sin(pi) = -pi. Wait, the equation in line 2 says 'Subs(...), -pi'. This is cor
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the point of tangency and the slope, and the final equation is algebraically correct.
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/tangent_line, checked 2026-10-13 with SymPy 1.14.0.