What is the value of \(\tan(2820°)\)?

1. INFER the key strategy

• Large angles like \(2820°\) are unwieldy, but trigonometric functions are periodic
• Key insight: Tangent repeats every \(180°\) (unlike sine and cosine which repeat every \(360°\))
• Strategy: Reduce \(2820°\) to an equivalent angle between \(0°\) and \(180°\)

2. SIMPLIFY using modular arithmetic

• Divide the given angle by the period: \(2820 ÷ 180 = 15.666...\)
• The integer part tells us how many complete periods: 15 complete periods
• Calculate: \(15 × 180 = 2700°\)
• Find remainder: \(2820 - 2700 = 120°\)
• Therefore: \(\mathrm{tan}(2820°) = \mathrm{tan}(120°)\)

3. INFER the quadrant and sign

• \(120°\) is between \(90°\) and \(180°\), so it's in quadrant II
• In quadrant II, tangent values are negative
• Reference angle: \(180° - 120° = 60°\)

4. INFER the final value

• \(\mathrm{tan}(120°) = -\mathrm{tan}(60°)\)
• From unit circle: \(\mathrm{tan}(60°) = \sqrt{3}\)
• Therefore: \(\mathrm{tan}(120°) = -\sqrt{3}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about periods: Many students incorrectly use \(360°\) as the period for all trig functions because they're more familiar with sine and cosine.

If they use \(360°\): \(2820 ÷ 360 = 7.833...\), so \(7 × 360 = 2520\), remainder \(= 300°\)
Then \(\mathrm{tan}(300°) = \mathrm{tan}(-60°) = -\frac{\sqrt{3}}{3}\), leading them to select Choice B \(-\frac{\sqrt{3}}{3}\)

Second Most Common Error:

Weak INFER skill regarding signs in quadrants: Students correctly find that \(\mathrm{tan}(2820°) = \mathrm{tan}(120°)\) but forget that tangent is negative in quadrant II.

They calculate \(\mathrm{tan}(120°) = \mathrm{tan}(60°) = \sqrt{3}\) (forgetting the negative sign), leading them to select Choice D \(\sqrt{3}\)

The Bottom Line:

This problem tests whether students truly understand the period of tangent \(180°\) vs \(360°\)) and can correctly apply quadrant sign rules. The large angle is just a distraction - the real challenge is applying fundamental periodic properties accurately.