In a right triangle, the tangent of one of the two acute angles is (sqrt(3))/3. What is the tangent of...

1. TRANSLATE the problem information

• Given information:
  • We have a right triangle
  • tan of one acute angle = \(\frac{\sqrt{3}}{3}\)
  • Need to find tan of the other acute angle

2. INFER the key relationship

• In any right triangle, the two acute angles are complementary (they add up to \(90°\))
• The tangent of each angle equals opposite/adjacent for that specific angle
• If we call the legs a and b, then:
  • tan(first angle) = \(\frac{a}{b}\)
  • tan(second angle) = \(\frac{b}{a}\)
• This means the tangents are reciprocals of each other!

3. SIMPLIFY to find the answer

• If tan(first angle) = \(\frac{\sqrt{3}}{3}\)
• Then tan(second angle) = \(\frac{1}{\frac{\sqrt{3}}{3}}\)
• To compute this reciprocal:
  \(\frac{1}{\frac{\sqrt{3}}{3}}\)
  \(= 1 \times \frac{3}{\sqrt{3}}\)
  \(= \frac{3}{\sqrt{3}}\)

Answer: D. \(\frac{3}{\sqrt{3}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the reciprocal relationship between tangents of complementary angles in a right triangle. Instead, they might assume the two angles have the same tangent value, leading them to select Choice C (\(\frac{\sqrt{3}}{3}\)).

Second Most Common Error:

Conceptual confusion about angle relationships: Students might incorrectly think that if one angle has tangent \(\frac{\sqrt{3}}{3}\), the other must have the "opposite" value with a negative sign, leading them to select Choice A (\(-\frac{\sqrt{3}}{3}\)).

The Bottom Line:

This problem tests whether students understand the geometric relationship between the two acute angles in a right triangle and can translate that into the algebraic relationship between their tangents. The key insight is recognizing that these tangents are reciprocals, not equals or opposites.