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Solving trigonometric equations

Problem 13.404 · easy

Solve \( \displaystyle \sin{\left(\theta \right)} = \frac{\sqrt{2}}{2} \) for \( \displaystyle 0 \le \theta < 2\pi \).

Isolate sin θ and find the angles in [0, 2π) with that value.
\[ \left. \sin{\left(\theta \right)} - \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{\pi}{4} }} = 0 \]
θ = pi/4 works.✓ Proved
\[ \left. \sin{\left(\theta \right)} - \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{3 \pi}{4} }} = 0 \]
θ = 3*pi/4 works.✓ Proved

Answer \( \theta = \frac{\pi}{4},\ \frac{3 \pi}{4} \)

Lines: 2 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	Not checked	—	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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	Newton's method from 1440 starting points finds exactly 2 roots in [0, 2π), the same ones

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (misleading) — The solution verifies specific candidates but fails to explain how they were derived or why they are the only solutions in the given interval. It omits the crucial step of identifying the reference angle and using the symmetry of the sine function (positive in quadrants I and II) to find all solutions.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-21 — The solution verifies specific candidates but fails to explain how they were derived or why they are the only solutions in the given interval. It omits the crucial step of identifying the reference angle and using the symmetry of the sine function (positive in quadrants I and II) to find all solutions.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (misleading) 2026-10-20 — The solution only verifies that the proposed answers are correct but fails to demonstrate that they are the *only* solutions in the given interval. A student would not learn how to derive the solutions or ensure completeness.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly identifies the two solutions in the specified interval and verifies them. The reasoning is sound and complete for this level of problem.
gpt-oss:20b: pass 2026-10-20

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/trig_equations, checked 2026-10-21 with SymPy 1.14.0.