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Exact trigonometric values

Problem 13.245 · easy

Find the exact value of \( \displaystyle \tan{\left(- \frac{4 \pi}{3} \right)} \).

The angle -4*pi/3 ends in quadrant 2 with reference angle pi/3, where tan is negative.
\[ \tan{\left(\frac{\pi}{3} \right)} = \sqrt{3} \]
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: pass
qwen3.6:27b-mlx: fail (error) — The angle -4pi/3 is coterminal with 2pi/3, which is in Quadrant II where tangent is negative. However, the reference angle is pi/3, and tan(pi/3) is positive sqrt(3). Therefore, tan(-4pi/3) should be -sqrt(3). Wait, let me re-evaluate. -4pi/3 + 2pi = 2pi/3. 2pi/3 is in Q2. Tan is negative in Q2. Reference angle is pi - 2pi/3 = pi/3. Tan(pi/3) = sqrt(3). So Tan(2pi/3) = -sqrt(3). The final answer -sqrt(3) is correct. Let me re-read the sentence. 'The angle -4*pi/3 ends in quadrant 2 with reference angle pi/3, where tan is negative.' This sentence is factually correct. The logic holds. Why did I think it was wrong? Let me check the sign again. Tan(x) = sin(x)/cos(x). In Q2, sin is +, cos is -, so tan is -. Correct. The solution is actually correct. Let me look closer. Is there a subtle error? No. The steps are correct. The answer is correct. The reasoning is correct. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The angle -4pi/3 is coterminal with 2pi/3, which is in Quadrant II where tangent is negative. However, the reference angle is pi/3, and tan(pi/3) is positive sqrt(3). Therefore, tan(-4pi/3) should be -sqrt(3). Wait, let me re-evaluate. -4pi/3 + 2pi = 2pi/3. 2pi/3 is in Q2. Tan is negative in Q2. Reference angle is pi - 2pi/3 = pi/3. Tan(pi/3) = sqrt(3). So Tan(2pi/3) = -sqrt(3). The final answer -sqrt(3) is correct. Let me re-read the sentence. 'The angle -4*pi/3 ends in quadrant 2 with reference angle pi/3, where tan is negative.' This sentence is factually correct. The logic holds. Why did I think it was wrong? Let me check the sign again. Tan(x) = sin(x)/cos(x). In Q2, sin is +, cos is -, so tan is -. Correct. The solution is actually correct. Let me look closer. Is there a subtle error? No. The steps are correct. The answer is correct. The reasoning is correct. I will mark it as pass.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The angle -4pi/3 is coterminal with 2pi/3, which is in Quadrant II, not Quadrant I as implied by the reference angle logic, but more importantly, ta
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The angle -4pi/3 is coterminal with 2pi/3, which lies in Quadrant II where tangent is negative. However, the reference angle is pi/3, and tan(pi/3) is sqrt(3). Therefore, tan(-4pi/3) should be -sqrt(3). Wait, let me re-evaluate. -4pi/3 = -240 degrees. Adding 360 gives 120 degrees. 120 degrees is in Q2. Reference angle is 60 degrees (pi/3). Tan(60) is sqrt(3). In Q2, tan is negative. So the answer -sqrt(3) is actually correct. Let me re-read the solution text. 'The angle -4*pi/3 ends in quadrant 2 with reference angle pi/3, where tan is negative.' This sentence is correct. 'tan(pi/3) = sqrt(3)' is correct. 'tan(-4*pi/3) = -sqrt(3)' is correct. Why did I think it was wrong? Let me check the sign again. Tan is sin/cos. Sin(120) is positive. Cos(120) is negative. Positive/Negative is Negative. So the answer is indeed -sqrt(3). The solution is correct. Wait, is there a subtle error? Let's check the quadrant again. -4pi/3. Rotate clockwise 4pi/3. 4pi/3 is 240 degrees. Clockwise 240 degrees lands you at 120 degrees (360-240=120). 120 degrees is in Q2. Correct. Reference angle is 180-120=60 degrees = pi/3. Correct. Tan is negative in Q2. Correct. Value is -tan(pi/3) = -sqrt(3). Correct. The solution seems perfectly sound. Is there a style issue? No. Is there a misleading issue? No. It passes.
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.