A 170-centimeter ladder leans against a vertical wall, forming a right triangle with the ground. The sine of the angle...

1. TRANSLATE the problem information

• Given information:
  • 170-cm ladder leans against wall
  • Forms right triangle with ground
  • \(\sin \theta = \frac{3}{5}\) (where \(\theta\) is angle between ladder and ground)
  • Need: distance from foot of ladder to wall
• What this tells us: The ladder is the hypotenuse, and we need the horizontal leg (adjacent to angle \(\theta\))

2. INFER the approach

• We have \(\sin \theta\) but need the adjacent side
• Since \(\cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\), we need to find \(\cos \theta\) first
• We can use the Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\)

3. SIMPLIFY to find cos \(\theta\)

• \(\sin^2\theta + \cos^2\theta = 1\)
• \((\frac{3}{5})^2 + \cos^2\theta = 1\)
• \(\frac{9}{25} + \cos^2\theta = 1\)
• \(\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}\)
• \(\cos \theta = \frac{4}{5}\) (positive since this is an acute angle)

4. SIMPLIFY to find the horizontal distance

• \(\cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\)
• \(\frac{4}{5} = \frac{\mathrm{adjacent}}{170}\)
• \(\mathrm{adjacent} = 170 \times \frac{4}{5} = 136\)

Answer: C (136)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which side they're looking for or mix up sin and cos relationships.

Many students see \(\sin \theta = \frac{3}{5}\) and think they can directly calculate \(170 \times \frac{3}{5} = 102\), not realizing this gives them the vertical height (opposite side), not the horizontal distance (adjacent side) that the question asks for. This leads them to select Choice B (102).

Second Most Common Error:

Missing conceptual knowledge of Pythagorean identity: Students know they need \(\cos \theta\) but don't remember how to find it from \(\sin \theta\).

Without the identity \(\sin^2\theta + \cos^2\theta = 1\), students get stuck trying to find \(\cos \theta\) and may resort to guessing or using incorrect relationships. This leads to confusion and random answer selection.

The Bottom Line:

This problem requires careful reading to identify what distance is being asked for, then connecting trigonometric ratios through the Pythagorean identity. The key insight is recognizing that you need a different trig ratio than what's given, and knowing how to derive it.