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Solving trigonometric equations

Problem 13.410 · 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.
Reviewed
\[ \left. \sin{\left(\theta \right)} + \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{5 \pi}{4} }} = 0 \]
θ = 5*pi/4 works.✓ Proved
\[ \left. \sin{\left(\theta \right)} + \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{7 \pi}{4} }} = 0 \]
θ = 7*pi/4 works.✓ Proved

Answer \( \theta = \frac{5 \pi}{4},\ \frac{7 \pi}{4} \)

Lines: 2 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	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
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: pass — The solution correctly identifies the two angles in the specified interval where sine is negative and has magnitude sqrt(2)/2. The verification steps confirm the solutions are valid.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly identifies the two angles in the specified interval where sine is negative and has magnitude sqrt(2)/2. The verification steps confirm the solutions are valid.
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 systematically find all solutions or why no others exist.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (misleading) 2026-10-20 — The solution verifies specific answers but fails to explain how they were derived (e.g., using reference angles and quadrant analysis). This teaches students to guess and check rather than solve trigonometric equations systematically.
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.