Solving trigonometric equations
Problem 13.481 · medium
Solve \( \displaystyle 2 \cos^{2}{\left(\theta \right)} + 3 \cos{\left(\theta \right)} + 1 = 0 \) for \( \displaystyle 0 \le \theta < 2\pi \).
- \[ 2 \cos^{2}{\left(\theta \right)} + 3 \cos{\left(\theta \right)} + 1 = \left(\cos{\left(\theta \right)} + 1\right) \left(2 \cos{\left(\theta \right)} + 1\right) \]Factor as a quadratic in cos θ.✓ Proved
- Set each factor to zero; then find every angle in [0, 2π) with those values.
- \[ \left. 2 \cos^{2}{\left(\theta \right)} + 3 \cos{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\frac{2 \pi}{3} }} = 0 \]θ = 2*pi/3 works.✓ Proved
- \[ \left. 2 \cos^{2}{\left(\theta \right)} + 3 \cos{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\pi }} = 0 \]θ = pi works.✓ Proved
- \[ \left. 2 \cos^{2}{\left(\theta \right)} + 3 \cos{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\frac{4 \pi}{3} }} = 0 \]θ = 4*pi/3 works.✓ Proved
Answer \( \theta = \frac{2 \pi}{3},\ \pi,\ \frac{4 \pi}{3} \)
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | Newton's method from 1440 starting points finds exactly 3 roots in [0, 2π), the same ones |
Reviewers
gpt-oss:20b: fail (error) — The solution only checks three specific angles but does not justify that these are the only solutions. It fails to solve the equations cosθ = -1 and cosθ = -1/2 and to show that the corresponding angles are the only ones in [0,2π).qwen3.6:27b-mlx: fail (misleading) — The solution verifies that the stated answers are roots but fails to demonstrate that they are the *only* solutions in the given interval. It skips the crucial step of solving for cos(θ) and finding all corresponding angles, which could lead a student to believe that checking specific values is sufficient for solving an equation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-11-09 — The solution verifies that the stated answers are roots but fails to demonstrate that they are the *only* solutions in the given interval. It skips the crucial step of solving for cos(θ) and finding all corresponding angles, which could lead a student to believe that checking specific values is sufficient for solving an equation.gpt-oss:20b: fail (error) 2026-11-09 — The solution only checks three specific angles but does not justify that these are the only solutions. It fails to solve the equations cosθ = -1 and cosθ = -1/2 and to show that the corresponding angles are the only ones in [0,2π).qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly factors the quadratic in cos(theta), identifies the roots, and verifies that the resulting angles satisfy the original equation within the specified domain.gpt-oss:20b: fail (misleading) 2026-11-08 — The solution skips the crucial step of solving the equations cosθ+1=0 and 2cosθ+1=0 for all θ in [0,2π). It merely verifies the listed angles without showing that these are the only solutions. This omission could mislead a student into thinking that testing a few angles suffices.qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly factors the quadratic, identifies the valid cosine values, and verifies the resulting angles within the specified domain.gpt-oss:20b: fail (misleading) 2026-11-08 — The solution skips the step of solving the equations cos θ = –1 and cos θ = –½, and merely checks the listed angles. A student would learn that simply substituting candidate values is sufficient, which is not a correct method for finding all solutions.
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 2026-11-09 with SymPy 1.14.0.