Solving trigonometric equations
Problem 13.328 · medium
Solve \( \displaystyle \sin{\left(2 \theta \right)} = - \frac{\sqrt{2}}{2} \) 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.
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{5 \pi}{8} }} = 0 \]θ = 5*pi/8 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{7 \pi}{8} }} = 0 \]θ = 7*pi/8 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{13 \pi}{8} }} = 0 \]θ = 13*pi/8 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{15 \pi}{8} }} = 0 \]θ = 15*pi/8 works.✓ Proved
Answer \( \theta = \frac{5 \pi}{8},\ \frac{7 \pi}{8},\ \frac{13 \pi}{8},\ \frac{15 \pi}{8} \)
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 listed values are solutions but fails to demonstrate that these are the *only* solutions in the given interval. A complete solution requires deriving the general solution for u and filtering for the range [0, 4π).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution only verifies that the listed values are solutions but fails to demonstrate that these are the *only* solutions in the given interval. A complete solution requires deriving the general solution for u and filtering for the range [0, 4π).gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution lists four values for theta, but fails to derive them or explain why these specific values are the only solutions. It merely verifies that they are solutions, which does not constitute a complete solution to 'Solve for theta'. A student would not learn how to find the solutions, only how to check them.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) = -√2/2 has 8 solutions in the interval [0, 2π). The values 3π/8, π/8, 11π/8, and 9π/8 are missing.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.