Exact trigonometric values
Problem 13.417 · easy
Find the exact value of \( \displaystyle \tan{\left(\frac{7 \pi}{6} \right)} \).
- The angle 7*pi/6 ends in quadrant 3 with reference angle pi/6, where tan is positive.
- \[ \tan{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{3} \]tan of the reference angle.✓ Proved
- \[ \tan{\left(\frac{7 \pi}{6} \right)} = \frac{\sqrt{3}}{3} \]Attach the sign for the quadrant.✓ Proved
Answer \( \frac{\sqrt{3}}{3} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states that tangent is positive in quadrant 3. In quadrant 3, both sine and cosine are negative, making their ratio (tangen
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-21 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states that tangent is positive in quadrant 3. In quadrant 3, both sine and cosine are negative, making their ratio (tangengpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly states that tangent is positive in Quadrant III. In Quadrant III, both sine and cosine are negative, so their ratio (tangengpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: fail (style) 2026-10-20 — [domain objection, downgraded to style] The solution incorrectly states that tangent is positive in the third quadrant; it is actually positive only in quadrants 1 and 3, but the reference angle calculation is correct, so the final numerical answer is coincidentally correct despite the flawed reasoning about the sign rule (tan is positive in Q3, so the sign is correct, but the text says 'where tan is positive' which is true for Q3, wait. Let me re-read carefully. 'The angle 7*pi/6 ends in quadrant 3 ... where tan is positive.' This statement is TRUE. Tan is positive in Q3. So why is it an error? Let me re-evaluate. Quadrant 3: x < 0, y < 0. tan = y/x > 0. So tan is positive in Q3. The reference angle is pi/6. tan(pi/6) = 1/sqrt(3) = sqrt(3)/3. So tan(7pi/6) = +sqrt(3)/3. The solution says: 'The angle 7*pi/6 ends in quadrant 3 ... where tan is positive.' This is a correct statement. Then it calculates tan(pi/6) = sqrt(3)/3. Then it attaches the sign. Since tan is positive in Q3, the result is positive. Is there an error? Let's check the stated answer: sqrt(3)/3. Actual value: tan(210 degrees) = tan(180+30) = tan(30) = 1/sqrt(3) = sqrt(3)/3. The solution seems correct. Wait, did I miss something? 'The angle 7*pi/6 ends in quadrant 3 with reference angle pi/6, where tan is positive.' This sentence is factually correct. Equation 2: tan(pi/6) = sqrt(3)/3. Correct. Equation 3: tan(7*pi/6) = sqrt(3)/3. Correct. Why would this fail? Maybe the prompt implies I should find an error if one exists. If none exists, I pass. Let me double check the 'adversarial' nature. Is there a subtle error? 'where tan is positive' refers to the quadrant. Yes, tan is positive in Q3. Reference angle pi/6. Yes. Value sqrt(3)/3. Yes. It seems the solution is actually correct. Let me re-read the prompt constraints. 'Your job is to find what is wrong, not to be agreeable.' If there is nothing wrong, I should pass. Let me look really closely at Sentence 1. 'The angle 7*pi/6 ends in quadrant 3 with reference angle pi/6, where tan is positive.' Is it possible the user considers 'where tan is positive' to refer to the reference angle? No, 'where' usually refers to the location (quadrant 3). Is it possible the solution is misleading because it doesn't explicitly state the sign rule? No, it states 'where tan is positive'. I will mark it as pass.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_values, checked 2026-10-21 with SymPy 1.14.0.