if x and y are acute angles such that `cosx + cosy = 3/2` and `sinx + siny = 3/4` then `sin(x+y)=`

Answer

To solve the problem, we need to find the value of \( \sin(x+y) \) given the equations:

1. \( \cos x + \cos y = \frac{3}{2} \)
2. \( \sin x + \sin y = \frac{3}{4} \)

### Step 1: Use the sum-to-product identities
We can express \( \cos x + \cos y \) and \( \sin x + \sin y \) using sum-to-product identities:

\[ \cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

\[ \sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

### Step 2: Set up the equations
From the first equation, we have:

\[ 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) = \frac{3}{2} \]

Dividing both sides by 2 gives:

\[ \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) = \frac{3}{4} \quad \text{(Equation 1)} \]

From the second equation, we have:

\[ 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) = \frac{3}{4} \]

Dividing both sides by 2 gives:

\[ \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) = \frac{3}{8} \quad \text{(Equation 2)} \]

### Step 3: Divide Equation 2 by Equation 1
Now, we can divide Equation 2 by Equation 1:

\[ \frac{\sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)}{\cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)} = \frac{\frac{3}{8}}{\frac{3}{4}} \]

This simplifies to:

\[ \tan\left(\frac{x+y}{2}\right) = \frac{3/8}{3/4} = \frac{3}{8} \cdot \frac{4}{3} = \frac{1}{2} \]

### Step 4: Find \( \sin(x+y) \)
Using the identity for \( \sin(2\theta) \):

\[ \sin(2\theta) = \frac{2 \tan \theta}{1 + \tan^2 \theta} \]

Let \( \theta = \frac{x+y}{2} \). Then:

\[ \tan\left(\frac{x+y}{2}\right) = \frac{1}{2} \]

Now substituting this into the formula:

\[ \sin(x+y) = \sin(2\theta) = \frac{2 \cdot \frac{1}{2}}{1 + \left(\frac{1}{2}\right)^2} = \frac{1}{1 + \frac{1}{4}} = \frac{1}{\frac{5}{4}} = \frac{4}{5} \]

### Final Answer
Thus, the value of \( \sin(x+y) \) is:

\[ \sin(x+y) = \frac{4}{5} \]