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Solving trigonometric equations

Problem 13.405 · easy

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

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

Answer \( \theta = \frac{2 \pi}{3},\ \frac{5 \pi}{3} \)

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 fails to isolate tan(theta) or determine the correct quadrants. tan(2pi/3) is -sqrt(3), which satisfies the equation, but tan(5pi/3) is -sqrt(3) as well? No, tan(5pi/3) = -sqrt(3). Wait. tan(2pi/3) = -sqrt(3). tan(5pi/3) = -sqrt(3). Both are solutions. However, the sentence says 'find the angles... with that value' but does not show the work of determining the reference angle or quadrants. More critically, simply checking two values is not a 'solution' method; it's verification. A student needs to see how to derive the answers. But is it 'wrong'? The prompt asks to review the solution. The solution claims to solve it but only verifies two specific answers without showing how they were found. This is misleading as a method. However, looking closer: tan(theta) = -sqrt(3). Reference angle pi/3. Tangent is negative in Q2 and Q4. Q2: pi - pi/3 = 2pi/3. Q4: 2pi - pi/3 = 5pi/3. The answers are correct. The sentence is just a summary. Is it an error? It's a 'style' or 'misleading' issue if it implies checking is solving. But usually, 'error' is reserved for mathematical mistakes. Let's look at the severity definitions. 'Misleading' (a student would learn something false). Learning that you solve equations by guessing and checking is bad practice. But is it 'false'? No. 'Style' (correct, but a sentence could be clearer). The sentence is vague. Let's look for a harder error. Did I miss a solution? No. Are the bounds correct? Yes. Is the algebra correct? Yes. The main issue is that the 'solution' is just verification. It doesn't teach how to solve it. This is 'misleading' regarding the process of solving trig equations.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution fails to isolate tan(theta) or determine the correct quadrants. tan(2pi/3) is -sqrt(3), which satisfies the equation, but tan(5pi/3) is -sqrt(3) as well? No, tan(5pi/3) = -sqrt(3). Wait. tan(2pi/3) = -sqrt(3). tan(5pi/3) = -sqrt(3). Both are solutions. However, the sentence says 'find the angles... with that value' but does not show the work of determining the reference angle or quadrants. More critically, simply checking two values is not a 'solution' method; it's verification. A student needs to see how to derive the answers. But is it 'wrong'? The prompt asks to review the solution. The solution claims to solve it but only verifies two specific answers without showing how they were found. This is misleading as a method. However, looking closer: tan(theta) = -sqrt(3). Reference angle pi/3. Tangent is negative in Q2 and Q4. Q2: pi - pi/3 = 2pi/3. Q4: 2pi - pi/3 = 5pi/3. The answers are correct. The sentence is just a summary. Is it an error? It's a 'style' or 'misleading' issue if it implies checking is solving. But usually, 'error' is reserved for mathematical mistakes. Let's look at the severity definitions. 'Misleading' (a student would learn something false). Learning that you solve equations by guessing and checking is bad practice. But is it 'false'? No. 'Style' (correct, but a sentence could be clearer). The sentence is vague. Let's look for a harder error. Did I miss a solution? No. Are the bounds correct? Yes. Is the algebra correct? Yes. The main issue is that the 'solution' is just verification. It doesn't teach how to solve it. This is 'misleading' regarding the process of solving trig equations.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution is incomplete because it only verifies the stated answers rather than deriving them. It fails to show the isolation of tan(theta) to -sqrt(3) or the identification of the reference angle and quadrants, which are the core steps of solving the problem.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly isolates tan(theta) = -sqrt(3) and identifies the two solutions in the specified interval [0, 2pi). The verification steps confirm the answers are correct.
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.