For two acute angles, angleA and angleB, \(\mathrm{sin(A) = cos(B)}\). The measures, in degrees, of angleA and angleB are 2x...

1. TRANSLATE the problem information

• Given information:
  • Two acute angles: \(\angle\mathrm{A}\) and \(\angle\mathrm{B}\)
  • \(\sin(\mathrm{A}) = \cos(\mathrm{B})\)
  • \(\angle\mathrm{A} = 2\mathrm{x} + 5\) degrees
  • \(\angle\mathrm{B} = 3\mathrm{x} + 10\) degrees
  • Find: value of x

2. INFER the key relationship

• The critical insight: When \(\sin(\mathrm{A}) = \cos(\mathrm{B})\) for acute angles, this means A and B are complementary angles
• Why? The cofunction identity tells us \(\sin(\theta) = \cos(90° - \theta)\)
• So if \(\sin(\mathrm{A}) = \cos(\mathrm{B})\), then \(\mathrm{A} + \mathrm{B} = 90°\)

3. TRANSLATE this relationship into an equation

• Since A and B are complementary: \(\mathrm{A} + \mathrm{B} = 90°\)
• Substituting our expressions: \((2\mathrm{x} + 5) + (3\mathrm{x} + 10) = 90\)

4. SIMPLIFY to solve for x

• Combine like terms: \(5\mathrm{x} + 15 = 90\)
• Subtract 15 from both sides: \(5\mathrm{x} = 75\)
• Divide by 5: \(\mathrm{x} = 15\)

5. Verify the answer

• \(\angle\mathrm{A} = 2(15) + 5 = 35°\)
• \(\angle\mathrm{B} = 3(15) + 10 = 55°\)
• Check: \(35° + 55° = 90°\) ✓

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the cofunction relationship between sine and cosine.

Students might see \(\sin(\mathrm{A}) = \cos(\mathrm{B})\) and think they need to find specific angle values or use more complex trigonometric identities. Without recognizing that this equation means the angles are complementary, they get stuck trying to work with the trigonometric functions directly rather than using the simple angle relationship.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when solving the linear equation.

Even if students correctly set up \(\mathrm{A} + \mathrm{B} = 90°\), they might make mistakes like:

• Incorrectly combining terms: getting \(5\mathrm{x} + 5 = 90\) instead of \(5\mathrm{x} + 15 = 90\)
• Arithmetic errors when solving: getting \(\mathrm{x} = 17\) instead of \(\mathrm{x} = 15\)

This may lead them to select Choice D (17).

The Bottom Line:

This problem tests whether students recognize that trigonometric cofunction relationships create complementary angle pairs. The actual algebra is straightforward once you realize that \(\sin(\mathrm{A}) = \cos(\mathrm{B})\) simply means \(\mathrm{A} + \mathrm{B} = 90°\).