Solving trigonometric equations
Problem 13.260 · 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.
- \[ \left. \sin{\left(2 \theta \right)} + 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{7 \pi}{12} }} = 0 \]θ = 7*pi/12 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{11 \pi}{12} }} = 0 \]θ = 11*pi/12 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{19 \pi}{12} }} = 0 \]θ = 19*pi/12 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{23 \pi}{12} }} = 0 \]θ = 23*pi/12 works.✓ Proved
Answer \( \theta = \frac{7 \pi}{12},\ \frac{11 \pi}{12},\ \frac{19 \pi}{12},\ \frac{23 \pi}{12} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution only verifies four specific values but fails to demonstrate that these are the only solutions in the given interval. It does not show the derivation of the general solution or the enumeration of all valid candidates within [0, 4π) for u, leaving open the possibility of missing solutions.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution only verifies four specific values but fails to demonstrate that these are the only solutions in the given interval. It does not show the derivation of the general solution or the enumeration of all valid candidates within [0, 4π) for u, leaving open the possibility of missing solutions.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution only verifies that the listed values are solutions but fails to demonstrate that they are the only solutions. It provides no derivation or completeness argument, leaving the student without a method to find the answers or confirm no others exist.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution lists only 4 solutions, but there are 8 solutions in the interval [0, 2π) because the period of sin(2θ) is π, resulting in 2 solutions per period over 2 periods. The missing solutions are π/12, 5π/12, 13π/12, and 17π/12.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 2026-10-18 with SymPy 1.14.0.