Solving trigonometric equations
Problem 13.326 · medium
Solve \( \displaystyle 2 \sin^{2}{\left(\theta \right)} - \sin{\left(\theta \right)} = 0 \) for \( \displaystyle 0 \le \theta < 2\pi \).
- \[ 2 \sin^{2}{\left(\theta \right)} - \sin{\left(\theta \right)} = \left(2 \sin{\left(\theta \right)} - 1\right) \sin{\left(\theta \right)} \]Factor as a quadratic in sin θ.✓ Proved
- Set each factor to zero; then find every angle in [0, 2π) with those values.
- \[ \left. 2 \sin^{2}{\left(\theta \right)} - \sin{\left(\theta \right)} \right|_{\substack{ \theta=0 }} = 0 \]θ = 0 works.✓ Proved
- \[ \left. 2 \sin^{2}{\left(\theta \right)} - \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]θ = pi/6 works.✓ Proved
- \[ \left. 2 \sin^{2}{\left(\theta \right)} - \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{5 \pi}{6} }} = 0 \]θ = 5*pi/6 works.✓ Proved
- \[ \left. 2 \sin^{2}{\left(\theta \right)} - \sin{\left(\theta \right)} \right|_{\substack{ \theta=\pi }} = 0 \]θ = pi works.✓ Proved
Answer \( \theta = 0,\ \frac{\pi}{6},\ \frac{5 \pi}{6},\ \pi \)
Lines: 5 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 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 |
| 6 | ✓ 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 verifies that the listed values are solutions but fails to demonstrate that they are the *only* solutions. It omits the crucial step of solving sin(θ) = 0 and sin(θ) = 1/2 to derive the candidate set, leaving the completeness of the answer unjustified.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution verifies that the listed values are solutions but fails to demonstrate that they are the *only* solutions. It omits the crucial step of solving sin(θ) = 0 and sin(θ) = 1/2 to derive the candidate set, leaving the completeness of the answer unjustified.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution verifies that the listed values are roots but fails to demonstrate that these are the *only* solutions in the interval [0, 2π). Specifically, it omits the derivation of sin(θ) = 0 and sin(θ) = 1/2, which is necessary to establish completeness.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to derive the solutions from the factored equation. It lists the correct answers and verifies them, but does not show the logical step of solving sin(theta) = 0 and sin(theta) = 1/2 to obtain those specific angles.gpt-oss:20b: pass 2026-10-15
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 2026-10-19 with SymPy 1.14.0.