Solving trigonometric equations
Problem 13.82 · medium
Solve \( \displaystyle \sin{\left(2 \theta \right)} = 0 \) 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.
- \[ \left. \sin{\left(2 \theta \right)} \right|_{\substack{ \theta=0 }} = 0 \]θ = 0 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }} = 0 \]θ = pi/2 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} \right|_{\substack{ \theta=\pi }} = 0 \]θ = pi works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{3 \pi}{2} }} = 0 \]θ = 3*pi/2 works.✓ Proved
Answer \( \theta = 0,\ \frac{\pi}{2},\ \pi,\ \frac{3 \pi}{2} \)
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 verifies four candidate values but fails to prove that these are the *only* solutions. It does not derive the general solution or demonstrate completeness within the given interval.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution verifies four candidate values but fails to prove that these are the *only* solutions. It does not derive the general solution or demonstrate completeness within the given interval.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution only verifies four specific values but fails to prove that these are the only solutions in the given interval. It does not derive the general solution or explicitly check for other roots within the range [0, 4π) for u.gpt-oss:20b: fail (error) 2026-10-14 — The solution lists specific values but never justifies that they are the only solutions. It does not solve sin(u)=0 for u∈[0,4π) and therefore fails to establish completeness of the solution set.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution verifies specific values but fails to demonstrate that these are the *only* solutions in the given interval. It lacks the general derivation (e.g., 2θ = nπ) required to solve the equation completely.gpt-oss:20b: fail (error) 2026-10-14 — The solution only checks a few candidate values but does not justify that these are the only solutions. It omits the general solution sin u = 0 ⇒ u = kπ and the mapping back to θ. Thus the reasoning is incomplete and could mislead a student into thinking the test cases alone prove completeness.
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 2026-10-17 with SymPy 1.14.0.