If x and y are acute angles such that `cosx+cosy=9/4` and `sinx+siny=9/8` then `sin(x+y)=`

Options

A. `2/5`
B. `3/4`
C. `3/5`
D. `4/5`

Explanation

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

1. \( \cos x + \cos y = \frac{9}{4} \)
2. \( \sin x + \sin y = \frac{9}{8} \)

### Step 1: Rewrite the equations using sum-to-product identities

Using the sum-to-product identities, we can rewrite the equations:

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

### Step 2: Set up the equations

From the first equation, we have:

\[ 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) = \frac{9}{4} \]

From the second equation, we have:

\[ 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) = \frac{9}{8} \]

### Step 3: Divide the two equations

Dividing the second equation by the first equation gives:

\[ \frac{2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)}{2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)} = \frac{\frac{9}{8}}{\frac{9}{4}} \]

This simplifies to:

\[ \tan\left(\frac{x+y}{2}\right) = \frac{9/8}{9/4} = \frac{4}{8} = \frac{1}{2} \]

### Step 4: Find \( \sin(x+y) \)

Now we know:

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

Using the double angle formula for sine:

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

We can express \( \sin\left(\frac{x+y}{2}\right) \) and \( \cos\left(\frac{x+y}{2}\right) \) in terms of \( \tan\left(\frac{x+y}{2}\right) \):

Let \( \tan\left(\frac{x+y}{2}\right) = \frac{1}{2} \). Then:

\[ \sin\left(\frac{x+y}{2}\right) = \frac{1}{\sqrt{1^2 + 2^2}} = \frac{1}{\sqrt{5}} \]
\[ \cos\left(\frac{x+y}{2}\right) = \frac{2}{\sqrt{1^2 + 2^2}} = \frac{2}{\sqrt{5}} \]

Now substituting these values into the sine double angle formula:

\[ \sin(x+y) = 2 \cdot \frac{1}{\sqrt{5}} \cdot \frac{2}{\sqrt{5}} = \frac{4}{5} \]

### Final Answer

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

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