Solving trigonometric equations
Problem 13.77 · medium
Solve \( \displaystyle 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} = 0 \) for \( \displaystyle 0 \le \theta < 2\pi \).
- \[ 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} = \left(2 \cos{\left(\theta \right)} + 1\right) \cos{\left(\theta \right)} \]Factor as a quadratic in cos θ.✓ Proved
- Set each factor to zero; then find every angle in [0, 2π) with those values.Reviewed
- \[ \left. 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }} = 0 \]θ = pi/2 works.✓ Proved
- \[ \left. 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }} = 0 \]θ = 2*pi/3 works.✓ Proved
- \[ \left. 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{4 \pi}{3} }} = 0 \]θ = 4*pi/3 works.✓ Proved
- \[ \left. 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{3 \pi}{2} }} = 0 \]θ = 3*pi/2 works.✓ Proved
Answer \( \theta = \frac{\pi}{2},\ \frac{2 \pi}{3},\ \frac{4 \pi}{3},\ \frac{3 \pi}{2} \)
Lines: 5 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly factors the equation, identifies the necessary cosine values, and verifies all four solutions within the specified interval [0, 2π).
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly factors the equation, identifies the necessary cosine values, and verifies all four solutions within the specified interval [0, 2π).gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution verifies that the listed values are roots but fails to demonstrate that they are the *only* solutions in the interval [0, 2π). It omits the crucial step of solving cos(θ) = 0 and cos(θ) = -1/2 to derive the angles, leaving a gap in logical completeness.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution lists four solutions but fails to show the derivation for the two solutions arising from 2*cos(theta) + 1 = 0. It only verifies the answers rather than solving the equation, and it omits the logical step of finding theta where cos(theta) = -1/2.gpt-oss:20b: pass 2026-10-14
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.