Solving trigonometric equations
Problem 13.251 · medium
Solve \( \displaystyle \sin{\left(2 \theta \right)} = \frac{\sqrt{3}}{2} \) 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.
- \[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]θ = pi/6 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{\pi}{3} }} = 0 \]θ = pi/3 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{7 \pi}{6} }} = 0 \]θ = 7*pi/6 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{4 \pi}{3} }} = 0 \]θ = 4*pi/3 works.✓ Proved
Answer \( \theta = \frac{\pi}{6},\ \frac{\pi}{3},\ \frac{7 \pi}{6},\ \frac{4 \pi}{3} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution only verifies that the four listed values are solutions, but it fails to prove that these are the *only* solutions in the given interval. A complete solution must derive the general solution for u and filter for the specific range, rather than just checking candidates.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution only verifies that the four listed values are solutions, but it fails to prove that these are the *only* solutions in the given interval. A complete solution must derive the general solution for u and filter for the specific range, rather than just checking candidates.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution verifies that the stated answers are correct 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 satisfy the equation.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution is incomplete because it only verifies specific values rather than deriving the full set of solutions. It fails to explain how the four solutions were found or why there are no others, leaving the student without the method to solve the equation generally.gpt-oss:20b: fail (error) 2026-10-15 — The solution only tests a few candidate angles and omits many valid solutions. The correct set of solutions for θ in [0,2π) is π/6, π/3, 2π/3, 5π/6, 7π/6, 4π/3, 5π/3, 11π/6. The method of checking individual substitutions does not guarantee completeness and leads to an incorrect answer.
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 2026-10-18 with SymPy 1.14.0.