In right triangle RST, which is right-angled at S, the cosine of acute angle T is 48/73. Triangle RST is...

1. TRANSLATE the problem information

• Given information:
  • Right triangle RST with right angle at S
  • \(\cos \mathrm{T} = \frac{48}{73}\) (T is acute)
  • Triangle RST ~ triangle UVW
  • S corresponds to V, T corresponds to W
• Need to find: tan W

2. INFER the key relationship

• Since the triangles are similar with \(\mathrm{T} \leftrightarrow \mathrm{W}\), corresponding angles are equal
• This means angle T = angle W
• Therefore: \(\tan \mathrm{W} = \tan \mathrm{T}\)
• Strategy: Find tan T to get our answer

3. INFER what's needed to find tan T

• We have \(\cos \mathrm{T} = \frac{48}{73}\)
• To find tan T, we need: \(\tan \mathrm{T} = \frac{\sin \mathrm{T}}{\cos \mathrm{T}}\)
• So we need to find sin T first

4. SIMPLIFY using the Pythagorean identity

• Use: \(\sin^2\mathrm{T} + \cos^2\mathrm{T} = 1\)
• \(\sin^2\mathrm{T} = 1 - \cos^2\mathrm{T} = 1 - \left(\frac{48}{73}\right)^2\)
• \(\sin^2\mathrm{T} = 1 - \frac{2304}{5329} = \frac{5329 - 2304}{5329} = \frac{3025}{5329}\)
• \(\sin \mathrm{T} = \frac{\sqrt{3025}}{73} = \frac{55}{73}\) (positive since T is acute)

5. SIMPLIFY to find the final answer

• \(\tan \mathrm{T} = \frac{\sin \mathrm{T}}{\cos \mathrm{T}} = \frac{\frac{55}{73}}{\frac{48}{73}} = \frac{55}{48}\)
• Since \(\tan \mathrm{W} = \tan \mathrm{T}\), we have \(\tan \mathrm{W} = \frac{55}{48}\)

Answer: D) 55/48

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that similar triangles have equal corresponding angles

Students might think they need to use similarity ratios or set up proportions with side lengths. They may get confused about how similarity relates to trigonometric ratios and attempt to use the given cosine value directly without realizing tan W = tan T.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Missing conceptual knowledge: Forgetting the Pythagorean identity or the tangent definition

Students might remember that tan = opposite/adjacent but not know how to connect this to the given cosine value. Without the Pythagorean identity, they can't find sin T and get stuck trying to work directly with \(\cos \mathrm{T} = \frac{48}{73}\).

This may lead them to select Choice A (48/73) by incorrectly thinking tan W = cos T.

The Bottom Line:

This problem tests whether students can connect similarity properties with trigonometric identities. The key insight is recognizing that corresponding angles in similar triangles are equal, then applying fundamental trig relationships.