If alpha and beta are complementary angles and \(\mathrm{tan}(\alpha) = \frac{3}{4}\), what is \(\mathrm{tan}(\beta)\)?

1. TRANSLATE the problem information

• Given information:
  • \(\alpha\) and \(\beta\) are complementary angles
  • \(\mathrm{tan}(\alpha) = \frac{3}{4}\)
  • Need to find \(\mathrm{tan}(\beta)\)

2. INFER the complementary angle relationship

• Since \(\alpha\) and \(\beta\) are complementary angles, they sum to 90°: \(\alpha + \beta = 90°\)
• This means \(\beta = 90° - \alpha\)
• For complementary angles, we can use cofunction identities

3. INFER the cofunction connection

• From the cofunction identity: \(\mathrm{tan}(\beta) = \mathrm{tan}(90° - \alpha) = \mathrm{cot}(\alpha)\)
• So we need to find \(\mathrm{cot}(\alpha)\) instead of \(\mathrm{tan}(\beta)\) directly

4. INFER the reciprocal relationship

• Since cotangent is the reciprocal of tangent: \(\mathrm{cot}(\alpha) = \frac{1}{\mathrm{tan}(\alpha)}\)
• Therefore: \(\mathrm{tan}(\beta) = \mathrm{cot}(\alpha) = \frac{1}{\mathrm{tan}(\alpha)}\)

5. SIMPLIFY the calculation

• Substitute the given value: \(\mathrm{tan}(\beta) = \frac{1}{\frac{3}{4}}\)
• To divide by a fraction, multiply by its reciprocal: \(\frac{1}{\frac{3}{4}} = 1 \times \frac{4}{3} = \frac{4}{3}\)

Answer: B. \(\frac{4}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the cofunction relationship for complementary angles. They might think that if the angles are complementary, their tangent values should also be complementary in some way, leading them to guess that \(\mathrm{tan}(\beta) = \mathrm{tan}(\alpha) = \frac{3}{4}\).

This may lead them to select Choice A (\(\frac{3}{4}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{tan}(\beta) = \frac{1}{\mathrm{tan}(\alpha)}\) but make an arithmetic error when calculating \(\frac{1}{\frac{3}{4}}\). They might incorrectly compute this as \(\frac{3}{4} \div 1 = \frac{3}{4}\), or get confused about fraction division rules.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

Success on this problem requires recognizing that complementary angles create cofunction relationships in trigonometry. The key insight is that \(\mathrm{tan}(\beta) = \mathrm{cot}(\alpha)\), which transforms the problem from finding a new trigonometric value to simply taking the reciprocal of the given tangent value.