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Solving trigonometric equations

Problem 13.488 · easy

Solve \( \displaystyle 2 \cos{\left(\theta \right)} = - \sqrt{3} \) for \( \displaystyle 0 \le \theta < 2\pi \).

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. 2 \cos{\left(\theta \right)} + \sqrt{3} \right|_{\substack{ \theta=\frac{5 \pi}{6} }} = 0 \]
θ = 5*pi/6 works.✓ Proved
\[ \left. 2 \cos{\left(\theta \right)} + \sqrt{3} \right|_{\substack{ \theta=\frac{7 \pi}{6} }} = 0 \]
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: pass
qwen3.6:27b-mlx: fail (error) — The solution is incomplete because it only verifies the proposed answers without deriving them or proving they are the only solutions in the given interval. A student would not learn how to solve the equation, only how to check specific values.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-09 — The solution is incomplete because it only verifies the proposed answers without deriving them or proving they are the only solutions in the given interval. A student would not learn how to solve the equation, only how to check specific values.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution claims to isolate cos(theta) but fails to perform the algebraic step (cos(theta) = -sqrt(3)/2). Furthermore, it lists 5pi/6 and 7pi/6 as solutions, but cos(5pi/6) is negative while cos(7pi/6) is also negative; however, the reference angle for sqrt(3)/2 is pi/6. The solutions for cos(theta) = -sqrt(3)/2 are indeed 5pi/6 and 7pi/6. Wait, let me re-evaluate. cos(5pi/6) = -sqrt(3)/2. cos(7pi/6) = -sqrt(3)/2. The algebraic check in lines 2 and 3 confirms they are roots. The sentence says 'Isolate cos theta...'. It skips the isolation step but the logic holds. Is it an error? The prompt asks to find what is wrong. The sentence 'Isolate cos theta and find the angles...' is a plan. The equations verify the answers. The answers are correct. Is there a subtle error? Let's check the values again. 2 cos(5pi/6) = 2 * (-sqrt(3)/2) = -sqrt(3). Correct. 2 cos(7pi/6) = 2 * (-sqrt(3)/2) = -sqrt(3). Correct. The solution is mathematically correct. The sentence is a bit brief but not false. Wait, look at the problem statement: `2 cos(theta) = -sqrt(3)`. Look at the substitution in line 2: `Subs(2*cos(theta) + sqrt(3), theta, 5*pi/6) = 0`. This implies `2*cos(theta) = -sqrt(3)`. Is there any other solution? Cosine is negative in Q2 and Q3. Reference angle for |cos| = sqrt(3)/2 is pi/6. Q2: pi - pi/6 = 5pi/6. Q3: pi + pi/6 = 7pi/6. These are the only solutions in [0, 2pi). Why would this fail? Maybe the sentence 'Isolate cos theta' is considered 'style' if it doesn't show the isolation? Or is it 'misleading' because it doesn't explicitly state the isolated value? Let's look closer. 'Isolate cos theta and find the angles...'. This is a valid strategy description. The subsequent lines verify the specific angles. However, often in these adversarial reviews, if a step is skipped that is crucial for *finding* the answer (not just verifying), it might be flagged. But the prompt says 'check that... each conclusion follows from the lines before it'. The conclusion is the stated answer. The lines before are the sentence and the verifications. The verifications prove the answers are correct. Do they prove they are the *only* answers? No. The sentence says 'find the angles'. It doesn't claim uniqueness explicitly, but 'Solve' implies finding all solutions. The verification steps only show that these two are solutions, not that there aren't others. A rigorous solution should argue why these are the only ones (e.g., by analyzing the unit circle or the range of cosine). Simply checking two values does not constitute a complete solution to 'Solve for theta' unless one also argues completeness. Therefore, the solution is incomplete/misleading because it verifies candidates but does not demonstrate that no other solutions exist in the interval.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution claims the answers are 5π/6 and 7π/6, but cos(5π/6) = -√3/2, so 2cos(5π/6) = -√3, which is correct. However, cos(7π/6) = -√3/2, so 2cos(7π/6) = -√3, which is also correct. Wait, let me re-evaluate. cos(5π/6) is -√3/2. 2 * (-√3/2) = -√3. Correct. cos(7π/6) is -√3/2. 2 * (-√3/2) = -√3. Correct. Why did I think it was wrong? Let me check the quadrants. Cosine is negative in Q2 and Q3. Reference angle for √3/2 is π/6. Q2: π - π/6 = 5π/6. Q3: π + π/6 = 7π/6. The solutions are actually correct. The verification steps show they work. The sentence is vague but not wrong. Is there a subtle error? No. The solution is correct.
gpt-oss:20b: pass 2026-11-08

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.