What is the value of \(\mathrm{tan}\left(\frac{92\pi}{3}\right)\)?

1. INFER the strategy for large angles

• The angle \(\frac{92\pi}{3}\) is much larger than \(2\pi\) (one full revolution)
• Since trigonometric functions repeat every \(2\pi\) radians, I need to find a coterminal angle
• Strategy: Find how many complete revolutions, then work with the remainder

2. SIMPLIFY to find the coterminal angle

• Divide the given angle by \(2\pi\):
  \(\frac{92\pi}{3} \div 2\pi = \frac{92\pi}{3} \times \frac{1}{2\pi} = \frac{92}{6} = \frac{46}{3}\)
• Convert to mixed number: \(\frac{46}{3} = 15\frac{1}{3}\)
• This means 15 complete revolutions plus \(\frac{1}{3}\) revolution
• The coterminal angle is: \(\frac{1}{3} \times 2\pi = \frac{2\pi}{3}\)

3. INFER the quadrant and sign

• \(\frac{2\pi}{3}\) radians = 120°, which is in quadrant II
• In quadrant II, tangent is negative (sine positive, cosine negative)

4. INFER the reference angle approach

• Reference angle = \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\)
• \(\mathrm{tan}(\frac{\pi}{3}) = \sqrt{3}\)
• Since we're in quadrant II: \(\mathrm{tan}(\frac{2\pi}{3}) = -\sqrt{3}\)

Answer: A. \(-\sqrt{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the need to reduce the large angle to a coterminal angle

Students might try to work directly with \(\frac{92\pi}{3}\), leading to confusion about which quadrant this massive angle ends up in. Without systematic angle reduction, they may randomly guess among the answer choices or try to apply trigonometric identities incorrectly.

Second Most Common Error:

Conceptual confusion about quadrant signs: Correctly finding that the coterminal angle is \(\frac{2\pi}{3}\) but forgetting that tangent is negative in quadrant II

This leads them to calculate \(\mathrm{tan}(\frac{2\pi}{3}) = \sqrt{3}\) (positive), causing them to select Choice D (\(\sqrt{3}\)) instead of the correct negative value.

The Bottom Line:

This problem tests whether students can systematically handle angles larger than \(2\pi\) and remember that the signs of trigonometric functions depend on which quadrant the terminal side lands in. The key insight is that no matter how many times you spin around, what matters is where you end up.