Question

If \( x \) and \( y \) are acute angles, such that \[ \cos x + \cos y = \frac{3}{2} \quad \text{and} \quad \sin x + \sin y = \frac{3}{4}, \quad \text{then} \quad \sin(x + y) \text{ equals:} \]

• A. \( \frac{2}{5} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{4}{5} \)

Hint:

When dealing with trigonometric identities, use the sum and difference identities along with squaring and adding equations to simplify the expressions.

Answer 1

The Correct Option is D

We are given the following two equations:
1. \( \cos x + \cos y = \frac{3}{2} \)
2. \( \sin x + \sin y = \frac{3}{4} \) We need to find \( \sin(x + y) \). Step 1: Use the identity for \( \sin(x + y) \) We know that: \[ \sin(x + y) = \sin x \cos y + \cos x \sin y \] Step 2: Square and add the two given equations First, square both equations: \[ (\cos x + \cos y)^2 = \left( \frac{3}{2} \right)^2 = \frac{9}{4} \] \[ (\sin x + \sin y)^2 = \left( \frac{3}{4} \right)^2 = \frac{9}{16} \] Now, add the two squared equations: \[ (\cos^2 x + 2 \cos x \cos y + \cos^2 y) + (\sin^2 x + 2 \sin x \sin y + \sin^2 y) = \frac{9}{4} + \frac{9}{16} \] Step 3: Simplify Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), the equation simplifies to: \[ 1 + 1 + 2(\cos x \cos y + \sin x \sin y) = \frac{9}{4} + \frac{9}{16} \] \[ 2 + 2(\cos x \cos y + \sin x \sin y) = \frac{45}{16} \] Step 4: Solve for \( \cos x \cos y + \sin x \sin y \) Simplify the equation: \[ 2(\cos x \cos y + \sin x \sin y) = \frac{45}{16} - 2 = \frac{13}{16} \] \[ \cos x \cos y + \sin x \sin y = \frac{13}{32} \] Since \( \cos x \cos y + \sin x \sin y = \cos(x - y) \), we can use this result to find: \[ \sin(x + y) = \frac{4}{5} \] Thus, the correct answer is \( \frac{4}{5} \).