If 5sin cos 4AA+= , find sin 2A.

Question

If 5sin cos 4AA+= , find sin 2A.

1. A. 1 4
2. B. 5 18
3. C. 7 8
4. D. 9 16 (Correct answer)

Correct Answer

Option D — 9 16

Detailed Solution & Explanation

The correct answer is 9 16.

Key Points

• Square the given equation: (sin A + cos A)² = (5/4)².
• The left side expands to sin²A + cos²A + 2 sin A cos A = 1 + sin 2A.
• So 1 + sin 2A = 25/16 → sin 2A = 25/16 − 1 = 9/16.
• The identity (sin A + cos A)² = 1 + sin 2A (since sin²+cos² = 1 and 2 sinA cosA = sin 2A) is the standard bridge from a sum of sin+cos to sin 2A.