For two acute angles, angleQ and angleR, \(\cos(\mathrm{Q}) = \sin(\mathrm{R})\). The measures, in degrees, of angleQ and angleR are x...

1. TRANSLATE the problem information

• Given information:
  • Two acute angles: \(\angle\mathrm{Q}\) and \(\angle\mathrm{R}\)
  • \(\cos(\mathrm{Q}) = \sin(\mathrm{R})\)
  • \(\angle\mathrm{Q} = \mathrm{x} + 61\) degrees
  • \(\angle\mathrm{R} = 4\mathrm{x} + 4\) degrees
  • Need to find: value of x

2. INFER the key relationship

• The critical insight: When \(\cos(\mathrm{Q}) = \sin(\mathrm{R})\) for acute angles, this means Q and R are complementary angles
• Why? Because \(\cos(\theta) = \sin(90° - \theta)\), so if \(\cos(\mathrm{Q}) = \sin(\mathrm{R})\), then \(\mathrm{Q} = 90° - \mathrm{R}\), which means \(\mathrm{Q} + \mathrm{R} = 90°\)
• This gives us our strategy: set up an equation where the angles sum to 90°

3. TRANSLATE into an equation

• Since \(\mathrm{Q} + \mathrm{R} = 90°\):
  \((\mathrm{x} + 61) + (4\mathrm{x} + 4) = 90\)

4. SIMPLIFY to solve for x

• Combine like terms on the left side:
  \(\mathrm{x} + 4\mathrm{x} + 61 + 4 = 90\)
  \(5\mathrm{x} + 65 = 90\)
• Subtract 65 from both sides:
  \(5\mathrm{x} = 25\)
• Divide by 5:
  \(\mathrm{x} = 5\)

Answer: A. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge about complementary angles: Students may not recognize that \(\cos(\mathrm{Q}) = \sin(\mathrm{R})\) means the angles are complementary. Instead, they might try to use more complex trigonometric identities or get confused about what relationship this equation represents. Without this key insight, they cannot set up the correct equation and may end up guessing.

Second Most Common Error:

Weak INFER reasoning about trigonometric relationships: Some students might recognize complementary angles but confuse the setup. For example, they might think \(\cos(\mathrm{Q}) = \cos(\mathrm{R})\) instead of recognizing the sine-cosine relationship, leading them to set up \(\mathrm{Q} = \mathrm{R}\) instead of \(\mathrm{Q} + \mathrm{R} = 90°\). This could lead them to select Choice B (19), which the solution notes would result from treating this as \(\cos(\mathrm{Q}) = \cos(\mathrm{R})\).

The Bottom Line:

This problem tests whether students can connect trigonometric function relationships to geometric angle relationships. The key breakthrough is recognizing that the sine and cosine relationship implies complementary angles - without this insight, the algebraic setup becomes impossible.