In a right triangle, the legs have lengths 5 and 5sqrt(3). If theta is the smaller of the two acute...

1. INFER which angle is smaller

• Given information:
  • Right triangle with legs 5 and \(5\sqrt{3}\)
  • \(\theta\) is the smaller acute angle
  • Need to find \(\tan(90° - \theta)\)

• Since \(5 \lt 5\sqrt{3}\), the smaller angle \(\theta\) is opposite the shorter leg (5) and adjacent to the longer leg \((5\sqrt{3})\)

2. SIMPLIFY to find tan θ

• \(\tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} = \frac{5}{5\sqrt{3}} = \frac{1}{\sqrt{3}}\)

• Rationalize the denominator:
  \(\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)

3. INFER the solution approach

• I can use the cofunction identity: \(\tan(90° - \theta) = \cot \theta = \frac{1}{\tan \theta}\)
• Alternatively, \((90° - \theta)\) is the other acute angle in the triangle

4. SIMPLIFY using the cofunction identity

• \(\tan(90° - \theta) = \frac{1}{\tan \theta} = \frac{1}{\frac{\sqrt{3}}{3}} = \frac{3}{\sqrt{3}}\)

• Rationalize:
  \(\frac{3}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}\)

Answer: \(\sqrt{3}\) (Choice D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly identify which angle is smaller, thinking \(\theta\) is opposite the longer leg \((5\sqrt{3})\) instead of the shorter leg (5).

This leads them to calculate \(\tan \theta = \frac{5\sqrt{3}}{5} = \sqrt{3}\), and then \(\tan(90° - \theta) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\). This may lead them to select Choice B \((\frac{\sqrt{3}}{3})\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find \(\tan \theta = \frac{\sqrt{3}}{3}\) but make algebraic errors when computing \(\frac{1}{\tan \theta}\), perhaps getting \(\frac{1}{\frac{\sqrt{3}}{3}} = \frac{\sqrt{3}}{3}\) instead of \(\frac{3}{\sqrt{3}} = \sqrt{3}\).

This calculation error causes confusion about which trigonometric values correspond to which angles, leading to guessing among the radical answer choices.

The Bottom Line:

This problem tests whether students can correctly identify angle relationships in right triangles and accurately manipulate expressions involving radicals. The key insight is recognizing that the smaller acute angle is always opposite the shorter leg, and then applying cofunction identities with careful algebraic manipulation.