Solving trigonometric equations
Problem 13.264 · easy
Solve \( \displaystyle \tan{\left(\theta \right)} = \sqrt{3} \) for \( \displaystyle 0 \le \theta < 2\pi \).
- Isolate tan θ and find the angles in [0, 2π) with that value.
- \[ \left. \tan{\left(\theta \right)} - \sqrt{3} \right|_{\substack{ \theta=\frac{\pi}{3} }} = 0 \]θ = pi/3 works.✓ Proved
- \[ \left. \tan{\left(\theta \right)} - \sqrt{3} \right|_{\substack{ \theta=\frac{4 \pi}{3} }} = 0 \]θ = 4*pi/3 works.✓ Proved
Answer \( \theta = \frac{\pi}{3},\ \frac{4 \pi}{3} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution verifies that the given answers are solutions but fails to prove they are the only solutions in the interval [0, 2π). It does not use the periodicity of the tangent function or the unit circle to derive the complete set of solutions.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution verifies that the given answers are solutions but fails to prove they are the only solutions in the interval [0, 2π). It does not use the periodicity of the tangent function or the unit circle to derive the complete set of solutions.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to demonstrate that the listed values are the *only* solutions in the given interval. It merely verifies that two specific values are solutions, omitting the necessary reasoning about the periodicity of the tangent function and the uniqueness of solutions within one period.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The solution merely verifies the given answers rather than deriving them, failing to demonstrate the method for solving trigonometric equations (e.g., using reference angles and quadrant signs). This teaches a student that verification is equivalent to solving.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.