Which of the following expressions is equivalent to \((\sin 24°)(\cos 66°) + (\cos 24°)(\sin 66°)\)?
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Which of the following expressions is equivalent to \((\sin 24°)(\cos 66°) + (\cos 24°)(\sin 66°)\)?
\(2(\cos 66°)(\sin 24°)\)
\(2(\cos 66°) + 2(\cos 24°)\)
\((\cos 66°)^2 + (\cos 24°)^2\)
\((\cos 66°)^2 + (\sin 24°)^2\)
1. TRANSLATE the problem information
- Given expression: \(\sin 24° \cos 66° + \cos 24° \sin 66°\)
- Need to find: An equivalent expression from the given choices
2. INFER the key relationship
- Notice that \(24° + 66° = 90°\)
- This means 24° and 66° are complementary angles
- Key insight: Use complementary angle relationships rather than trying complex trigonometric identities
3. INFER and apply complementary angle relationships
- For complementary angles: \(\sin \theta = \cos(90° - \theta)\)
- Therefore: \(\sin 24° = \cos(90° - 24°) = \cos 66°\)
- Similarly: \(\sin 66° = \cos(90° - 66°) = \cos 24°\)
4. SIMPLIFY by substitution
- Original: \(\sin 24° \cos 66° + \cos 24° \sin 66°\)
- Substitute: \(\cos 66° \cdot \cos 66° + \cos 24° \cdot \cos 24°\)
- Final form: \(\cos^2 66° + \cos^2 24°\)
Answer: C. \(\cos^2 66° + \cos^2 24°\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that 24° and 66° are complementary angles, so they miss the key relationship that makes this problem simple.
Instead, they might try to apply the sine addition formula backwards: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\), thinking the given expression equals \(\sin(24° + 66°) = \sin 90° = 1\). While this is mathematically correct, it doesn't help them match any of the answer choices. This leads to confusion and guessing.
Second Most Common Error:
Conceptual confusion about complementary angles: Students might know complementary angle relationships exist but apply them incorrectly, such as thinking \(\sin 24° = \sin 66°\) instead of \(\sin 24° = \cos 66°\).
This incorrect substitution could lead them to select Choice D (\(\cos^2 66° + \sin^2 24°\)) because they substitute incorrectly but still follow the right general approach.
The Bottom Line:
The key insight is recognizing the complementary angle relationship immediately. Once you see that \(24° + 66° = 90°\), the complementary angle identities make this problem straightforward. Students who miss this relationship often overcomplicate the solution or make substitution errors.
\(2(\cos 66°)(\sin 24°)\)
\(2(\cos 66°) + 2(\cos 24°)\)
\((\cos 66°)^2 + (\cos 24°)^2\)
\((\cos 66°)^2 + (\sin 24°)^2\)