What is the value of 3/2 × (cos 39° / sin 51°) − √(sin² 39° + sin² 51°) ?

Question

What is the value of 3/2 × (cos 39° / sin 51°) − √(sin² 39° + sin² 51°) ?

1. A. 3/2
2. B. 7/8
3. C. 5/6
4. D. 1/2 (Correct answer)

Correct Answer

Option D — 1/2

Detailed Solution & Explanation

The correct answer is 1/2.

Key Points

• Use the complementary relationship: since 39° + 51° = 90°, cos 39° = sin 51°.
• So the first term collapses: (3/2) × (cos 39° / sin 51°) = (3/2) × 1 = 3/2.
• For the radical, sin 51° = cos 39°, so:
  • sin²39° + sin²51° = sin²39° + cos²39° = 1, and √1 = 1
• Therefore the expression is 3/2 − 1 = 1/2.

Additional Information

• The complementary identities: sin(90° − θ) = cosθ, cos(90° − θ) = sinθ, tan(90° − θ) = cotθ, sec(90° − θ) = cosecθ.
• Angle pairs summing to 90° are placed deliberately in these questions — 39/51, 20/70, 35/55, 25/65. Spotting the pair is the whole solution.
• Without that observation you would need actual values of sin 39° and sin 51°, which are not standard angles — a strong hint that a simplification must exist.
• The same trick handles expressions such as sin²10° + sin²20° + … + sin²80°, where terms pair off to give 1 each.