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Solving trigonometric equations

Problem 13.487 · easy

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

Isolate sin θ and find the angles in [0, 2π) with that value.
\[ \left. 2 \sin{\left(\theta \right)} - \sqrt{2} \right|_{\substack{ \theta=\frac{\pi}{4} }} = 0 \]
θ = pi/4 works.✓ Proved
\[ \left. 2 \sin{\left(\theta \right)} - \sqrt{2} \right|_{\substack{ \theta=\frac{3 \pi}{4} }} = 0 \]
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 (error) — The solution only verifies that the stated answers are solutions, but fails to demonstrate that they are the *only* solutions in the given interval. A complete solution requires deriving the general solution or explicitly checking the quadrants to ensure no roots are missed.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-09 — The solution only verifies that the stated answers are solutions, but fails to demonstrate that they are the *only* solutions in the given interval. A complete solution requires deriving the general solution or explicitly checking the quadrants to ensure no roots are missed.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution only verifies that the given answers are correct but fails to demonstrate that they are the only solutions in the specified interval. It does not show the derivation (e.g., sin(θ) = 1/√2 implies θ = π/4 or 3π/4), leaving open the possibility of other solutions.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly isolates sin(θ) = √2/2 and identifies the two solutions in the specified interval [0, 2π). The verification steps confirm the answers.
gpt-oss:20b: pass 2026-11-08

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 2026-11-09 with SymPy 1.14.0.