Solving trigonometric equations
Problem 13.87 · easy
Solve \( \displaystyle \cos{\left(\theta \right)} = - \frac{\sqrt{3}}{2} \) for \( \displaystyle 0 \le \theta < 2\pi \).
- Isolate cos θ and find the angles in [0, 2π) with that value.
- \[ \left. \cos{\left(\theta \right)} + \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{5 \pi}{6} }} = 0 \]θ = 5*pi/6 works.✓ Proved
- \[ \left. \cos{\left(\theta \right)} + \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{7 \pi}{6} }} = 0 \]θ = 7*pi/6 works.✓ Proved
Answer \( \theta = \frac{5 \pi}{6},\ \frac{7 \pi}{6} \)
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 is incomplete because it only verifies two specific values without deriving them or proving they are the only solutions in the interval. A student would not learn how to find the solutions, only how to check them.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution is incomplete because it only verifies two specific values without deriving them or proving they are the only solutions in the interval. A student would not learn how to find the solutions, only how to check them.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer is incorrect. The solutions to cos(θ) = -√3/2 in [0, 2π) are 5π/6 and 7π/6? No, cos(5π/6) = -√3/2 is correct, but cos(7π/6) = -√3/2 is also correct. Wait, let me re-evaluate. cos(5π/6) = -√3/2. cos(7π/6) = -√3/2. The stated answer IS correct. Why did I think it was wrong? Ah, usually students confuse 5π/6 and 7π/6 with 2π/3 and 4π/3. Let's check the values again. cos(5π/6) = -√3/2. cos(7π/6) = -√3/2. The answer provided in the problem statement is actually correct. However, the solution only verifies two specific points without deriving them or explaining why these are the *only* solutions. It fails to establish uniqueness or the method of finding them, merely checking two guesses. This is misleading as a solution method.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer is incorrect; cos(5π/6) and cos(7π/6) equal -√3/2, but the problem asks to solve cos(θ) = -√3/2, which has solutions 5π/6 and 7π/6. Wait, cos(5π/6) = -√3/2 is correct. Let me re-read carefully. cos(θ) = -√3/2. Reference angle π/6. Cosine is negative in Q2 and Q3. Q2: π - π/6 = 5π/6. Q3: π + π/6 = 7π/6. The stated answer IS correct. Why did I think it was wrong? Ah, I misread the sign in my head. Let me double check. cos(5pi/6) = -sqrt(3)/2. Yes. cos(7pi/6) = -sqrt(3)/2. Yes. The solution is actually correct. However, the 'solution' provided is just checking the answers, not deriving them. The prompt asks to 'Solve'. The sentences say 'Isolate cos theta and find the angles'. This is a valid method description. The equations verify the answers. Is it a 'solution'? It's a verification. But usually 'solve' implies finding. However, the prompt says 'review worked calculus solutions'. If the student just guessed and checked, is it a valid solution? In many contexts, yes, if the search space is small. But here, the 'sentence' claims to 'find' the angles, but the subsequent lines only 'check' them. It doesn't show the finding process (reference angle, quadrants). Is this an error? It's misleading because it presents verification as the method of finding. But technically, if you know the answers, checking them is sufficient to confirm they are solutions. Does it prove they are the ONLY solutions? No. The problem asks to solve, implying finding all solutions. The solution does not demonstrate that there are no other solutions. Therefore, it is incomplete/misleading.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.