Solving trigonometric equations
Problem 13.254 · 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.
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{2 \pi}{3} }} = 0 \]θ = 2*pi/3 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{5 \pi}{6} }} = 0 \]θ = 5*pi/6 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{5 \pi}{3} }} = 0 \]θ = 5*pi/3 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{11 \pi}{6} }} = 0 \]θ = 11*pi/6 works.✓ Proved
Answer \( \theta = \frac{2 \pi}{3},\ \frac{5 \pi}{6},\ \frac{5 \pi}{3},\ \frac{11 \pi}{6} \)
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 lists four values but provides no derivation or justification for why these are the only solutions. It relies entirely on checking specific values without establishing the general solution set for sin(u) = -sqrt(3)/2 over the interval [0, 4pi), leaving the completeness of the answer unproven.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution lists four values but provides no derivation or justification for why these are the only solutions. It relies entirely on checking specific values without establishing the general solution set for sin(u) = -sqrt(3)/2 over the interval [0, 4pi), leaving the completeness of the answer unproven.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution lists only 4 solutions, but there are 8 solutions in the interval [0, 2pi) for sin(2theta) = -sqrt(3)/2. The substitution u = 2theta maps [0, 2pi) to [0, 4pi), which contains 4 solutions for sin(u) = -sqrt(3)/2 in [0, 2pi) and 4 in [2pi, 4pi), totaling 8 values for theta. The provided answer is incomplete.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution lists only 4 values, but the equation sin(2θ) = -√3/2 has 8 solutions in the interval [0, 2π). The setup correctly identifies the range for u as [0, 4π), but the final answer is incomplete.gpt-oss:20b: fail (error) 2026-10-15 — The solution lists only four values, but the correct solutions are θ=7π/12, 11π/12, 19π/12, 23π/12. The given answer omits two solutions and includes incorrect ones. The sentence about setting u=2θ is fine, but the subsequent steps do not correctly solve sin(2θ)=-√3/2.
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.