Exact trigonometric values
Problem 13.248 · easy
Find the exact value of \( \displaystyle \tan{\left(\frac{4 \pi}{3} \right)} \).
- The angle 4*pi/3 ends in quadrant 3 with reference angle pi/3, where tan is positive.
- \[ \tan{\left(\frac{\pi}{3} \right)} = \sqrt{3} \]tan of the reference angle.✓ Proved
- \[ \tan{\left(\frac{4 \pi}{3} \right)} = \sqrt{3} \]Attach the sign for the quadrant.✓ Proved
Answer \( \sqrt{3} \)
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 | mpmath evaluates the function at the angle |
Reviewers
gpt-oss:20b: fail (error) — The solution incorrectly states that tan is positive in quadrant III; in fact tan is negative there, so the final value should be -√3, not √3.qwen3.6:27b-mlx: pass — The solution correctly identifies the quadrant, reference angle, and sign of the tangent function, leading to the correct exact value.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the quadrant, reference angle, and sign of the tangent function, leading to the correct exact value.gpt-oss:20b: fail (error) 2026-10-18 — The solution incorrectly states that tan is positive in quadrant III; in fact tan is negative there, so the final value should be -√3, not √3.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the quadrant, reference angle, and sign of the tangent function, leading to the correct result.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the quadrant, reference angle, and sign of the tangent function, leading to the correct result.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_values, checked 2026-10-18 with SymPy 1.14.0.