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Solving trigonometric equations

Problem 13.486 · medium

Solve \( \displaystyle \sin{\left(2 \theta \right)} = \frac{1}{2} \) for \( \displaystyle 0 \le \theta < 2\pi \).

Let u = 2θ. Since 0 ≤ θ < 2π, u runs over [0, 4π): two full turns, so twice as many solutions.
\[ \left. \sin{\left(2 \theta \right)} - 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{\pi}{12} }} = 0 \]
θ = pi/12 works.✓ Proved
\[ \left. \sin{\left(2 \theta \right)} - 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{5 \pi}{12} }} = 0 \]
θ = 5*pi/12 works.✓ Proved
\[ \left. \sin{\left(2 \theta \right)} - 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{13 \pi}{12} }} = 0 \]
θ = 13*pi/12 works.✓ Proved
\[ \left. \sin{\left(2 \theta \right)} - 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{17 \pi}{12} }} = 0 \]
θ = 17*pi/12 works.✓ Proved

Answer \( \theta = \frac{\pi}{12},\ \frac{5 \pi}{12},\ \frac{13 \pi}{12},\ \frac{17 \pi}{12} \)

Lines: 4 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
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ 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 4 roots in [0, 2π), the same ones

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution verifies that the listed values are solutions but fails to demonstrate that they are the *only* solutions. It does not derive the general solution or explain why no other values in the interval [0, 2π) satisfy the equation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-09 — The solution verifies that the listed values are solutions but fails to demonstrate that they are the *only* solutions. It does not derive the general solution or explain why no other values in the interval [0, 2π) satisfy the equation.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution verifies specific values but fails to demonstrate that these are the only solutions in the given interval. It does not solve the equation generally or justify completeness.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution provides a list of answers and verifies them, but fails to demonstrate the derivation or prove that the list is exhaustive. It does not show how the solutions were found from the equation sin(u) = 1/2.
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.