Which of the following expressions is equivalent to \(6[(\sin 10°)(\cos 80°) + (\cos 10°)(\sin 80°)]\)?

1. INFER the key pattern recognition

• Given expression: \(\mathrm{6[(sin\ 10°)(cos\ 80°) + (cos\ 10°)(sin\ 80°)]}\)
• Key insight: The expression inside the brackets follows the pattern \(\mathrm{sin\ A\ cos\ B + cos\ A\ sin\ B}\)
• This matches the right side of the sine addition formula

2. INFER the trigonometric identity to apply

• Sine addition formula: \(\mathrm{sin(A + B) = sin\ A\ cos\ B + cos\ A\ sin\ B}\)
• Here: \(\mathrm{A = 10°}\) and \(\mathrm{B = 80°}\)
• Therefore: \(\mathrm{(sin\ 10°)(cos\ 80°) + (cos\ 10°)(sin\ 80°) = sin(10° + 80°)}\)

3. SIMPLIFY using the addition formula

• \(\mathrm{sin(10° + 80°) = sin(90°)}\)
• Since \(\mathrm{sin(90°) = 1}\)
• The expression becomes: \(\mathrm{6[sin(90°)] = 6(1) = 6}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the sine addition formula pattern

Students see the four separate trigonometric terms and think they need to calculate each one individually: sin 10°, cos 80°, cos 10°, and sin 80°. This approach requires either a calculator or memorized values for these specific angles, making the problem unnecessarily complicated. Without recognizing the pattern, they become overwhelmed by the complexity and may guess randomly.

This leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Not knowing that \(\mathrm{sin(90°) = 1}\)

Even if students correctly apply the sine addition formula to get \(\mathrm{6[sin(90°)]}\), they may not remember that \(\mathrm{sin(90°) = 1}\). This is a fundamental trigonometric value that should be memorized. Without this knowledge, they cannot complete the final step.

This may lead them to select Choice A (1) if they think \(\mathrm{sin(90°) = 1/6}\), or other incorrect choices based on incorrect values for \(\mathrm{sin(90°)}\).

The Bottom Line:

Success depends on pattern recognition rather than calculation. The problem tests whether students can identify when to apply trigonometric identities instead of computing individual trigonometric values.