Solving trigonometric equations
Problem 13.253 · medium
Solve \( \displaystyle \sin{\left(2 \theta \right)} = -1 \) 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.
- \[ \left. \sin{\left(2 \theta \right)} + 1 \right|_{\substack{ \theta=\frac{3 \pi}{4} }} = 0 \]θ = 3*pi/4 works.✓ Proved
- \[ \left. \sin{\left(2 \theta \right)} + 1 \right|_{\substack{ \theta=\frac{7 \pi}{4} }} = 0 \]θ = 7*pi/4 works.✓ Proved
Answer \( \theta = \frac{3 \pi}{4},\ \frac{7 \pi}{4} \)
Lines: 2 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | Newton's method from 1440 starting points finds exactly 2 roots in [0, 2π), the same ones |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution claims there are 'twice as many solutions' but only lists two. The equation sin(2θ) = -1 has four solutions in the given interval: 3π/4, 7π/4, 11π/4, and 15π/4. The provided answer is incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution claims there are 'twice as many solutions' but only lists two. The equation sin(2θ) = -1 has four solutions in the given interval: 3π/4, 7π/4, 11π/4, and 15π/4. The provided answer is incomplete.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution checks two values but fails to derive them or verify that they are the only solutions in the given interval. It asserts 'twice as many solutions' without identifying which ones, leaving the derivation incomplete and the uniqueness unproven.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution claims there are 'twice as many solutions' in the interval [0, 4π) for u, but fails to actually solve sin(u) = -1 to find the four values of u (3π/2, 7π/2, 11π/2, 15π/2). It then arbitrarily lists two values for θ without deriving them from the four values of u, missing the other two valid solutions (θ = 7π/4 is listed, but 3π/4 is also listed; wait, let's check. sin(2*3pi/4) = sin(3pi/2) = -1. sin(2*7pi/4) = sin(7pi/2) = -1. The other solutions are u=11pi/2 => theta=11pi/4 (out of range? no, 11pi/4 < 2pi? 11/4 = 2.75 > 2. So 11pi/4 is out. u=15pi/2 => theta=15pi/4 > 2pi. Wait. u in [0, 4pi). sin(u)=-1 at u=3pi/2, 7pi/2. 11pi/2 is 5.5pi > 4pi. So there are only TWO solutions for u in [0, 4pi). Therefore there are only TWO solutions for theta. The stated answer is actually correct. However, the reasoning 'two full turns, so twice as many solutions' is vague and the step from u to theta is skipped entirely. The solution does not show how the specific theta values were derived from the u values. It just checks them. This is a 'proof by verification' which is not a valid method for 'Solve' problems unless all solutions are found and verified. Did it find all? u=3pi/2 -> theta=3pi/4. u=7pi/2 -> theta=7pi/4. Are there others? sin(u)=-1. Period 2pi. In [0, 4pi), solutions are 3pi/2 and 7pi/2. That's it. So the answer is correct. But the solution text says 'u runs over [0, 4pi): two full turns, so twice as many solutions.' This implies there are 4 solutions for u? No, 'twice as many' as what? As in [0, 2pi)? In [0, 2pi), sin(u)=-1 has 1 solution. So in [0, 4pi), it has 2 solutions. The text is confusing but not necessarily wrong if interpreted as 'twice the count of the base period'. However, the critical failure is that it does not derive the answers. It just states them and checks them. A student reading this would not learn how to solve the equation, only how to check answers. This is misleading/incomplete methodology.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.