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Solving trigonometric equations

Problem 13.257 · easy

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

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. 2 \cos{\left(\theta \right)} + \sqrt{2} \right|_{\substack{ \theta=\frac{3 \pi}{4} }} = 0 \]
θ = 3*pi/4 works.✓ Proved
\[ \left. 2 \cos{\left(\theta \right)} + \sqrt{2} \right|_{\substack{ \theta=\frac{5 \pi}{4} }} = 0 \]
θ = 5*pi/4 works.✓ Proved

Answer \( \theta = \frac{3 \pi}{4},\ \frac{5 \pi}{4} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution is incorrect because cos(3π/4) = -√2/2, so 2cos(3π/4) = -√2, which satisfies the equation. However, cos(5π/4) = -√2/2, so 2cos(5π/4) = -√2, which also satisfies the equation. Wait, let me re-read the problem. 2 cos(theta) = -sqrt(2). cos(theta) = -sqrt(2)/2. The solutions in [0, 2pi) are 3pi/4 and 5pi/4. The stated answer is correct. The verification steps show that substituting these values into 2*cos(theta) + sqrt(2) yields 0. This is correct. Why did I think it was wrong? Let me re-evaluate. 2 cos(3pi/4) = 2(-sqrt(2)/2) = -sqrt(2). Correct. 2 cos(5pi/4) = 2(-sqrt(2)/2) = -sqrt(2). Correct. The solution is actually correct. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution is incorrect because cos(3π/4) = -√2/2, so 2cos(3π/4) = -√2, which satisfies the equation. However, cos(5π/4) = -√2/2, so 2cos(5π/4) = -√2, which also satisfies the equation. Wait, let me re-read the problem. 2 cos(theta) = -sqrt(2). cos(theta) = -sqrt(2)/2. The solutions in [0, 2pi) are 3pi/4 and 5pi/4. The stated answer is correct. The verification steps show that substituting these values into 2*cos(theta) + sqrt(2) yields 0. This is correct. Why did I think it was wrong? Let me re-evaluate. 2 cos(3pi/4) = 2(-sqrt(2)/2) = -sqrt(2). Correct. 2 cos(5pi/4) = 2(-sqrt(2)/2) = -sqrt(2). Correct. The solution is actually correct. I will mark it as pass.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution is incomplete because it only verifies the proposed answers rather than deriving them. It fails to show the isolation of cos(θ) or the determination of the reference angle and quadrants, which is the core of the problem.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution is incorrect because cos(3π/4) = -√2/2, so 2cos(3π/4) = -√2, which is correct, but cos(5π/4) = -√2/2, so 2cos(5π/4) = -√2, which is also correct. Wait, let me re-evaluate. 2cos(θ) = -√2 implies cos(θ) = -√2/2. The angles in [0, 2π) with cosine -√2/2 are 3π/4 and 5π/4. The stated answer is actually correct. Why did I think it was wrong? Let me re-read the problem. 2cos(θ) = -√2. cos(θ) = -√2/2. Reference angle π/4. Cosine is negative in Q2 and Q3. Q2: π - π/4 = 3π/4. Q3: π + π/4 = 5π/4. The answer is correct. The sentences are minimal but not wrong. The equations check out. Is there a style issue? The solution just lists the answer and checks it. It doesn't show the derivation. But the prompt asks to review the solution. The solution is 'Isolate cos θ and find the angles...'. Then it checks the answers. This is a valid verification method, though not a derivation. Is it 'misleading'? It doesn't teach how to find them, just how to check. But usually 'Solve' implies finding. However, the prompt says 'review worked calculus solutions'. This is more pre-calc/trig. The solution is technically correct in its claims. The sentences are vague but not false. Let's look closer. 'Isolate cos θ and find the angles...'. This is a plan. Then it verifies. It doesn't actually *find* them in the text, it just asserts the answer and verifies. Is this an error? It's a very poor solution, but is it 'wrong'? The verdict options are pass/fail. If it's correct, it passes. If it's incomplete, is it a fail? The instructions say 'find what is wrong'. If nothing is wrong, say pass. The answer is correct. The verification is correct. The setup sentence is a bit hand-wavy but not false. I will mark it as pass, but maybe style? No, the instructions say 'pass with empty notes if it is sound'. It is sound. Wait, looking at the provided answer in the problem statement: \theta = \frac{3 \pi}{4},\ \frac{5 \pi}{4}. The solution checks these. It is correct. I will output pass.
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.