If `alpha, beta ` are solutin of `6 cos theta + 8 sin theta=9,` then `sin (alpha + beta)=`

Options

A. `3/5`
B. `4/5`
C. `24/25`
D. `12/13`

Answer

To solve the problem, we need to find the value of \( \sin(\alpha + \beta) \) given that \( \alpha \) and \( \beta \) are solutions to the equation \( 6 \cos \theta + 8 \sin \theta = 9 \).

### Step-by-Step Solution:

1. **Rewrite the equation**:
\[ 6 \cos \theta + 8 \sin \theta = 9 \]
We can divide the entire equation by 10 to simplify it:
\[ \frac{6}{10} \cos \theta + \frac{8}{10} \sin \theta = \frac{9}{10} \]
This simplifies to:
\[ 0.6 \cos \theta + 0.8 \sin \theta = 0.9 \]

2. **Identify sine and cosine values**:
We can express \( 0.6 \) and \( 0.8 \) in terms of sine and cosine of an angle \( m \):
\[ \sin m = \frac{3}{5}, \quad \cos m = \frac{4}{5} \]
This is because \( \sin^2 m + \cos^2 m = 1 \) holds true:
\[ \left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 = \frac{9}{25} + \frac{16}{25} = 1 \]

3. **Substitute into the equation**:
Now we can rewrite the equation using the sine and cosine values:
\[ \frac{3}{5} \cos \theta + \frac{4}{5} \sin \theta = \frac{9}{10} \]
This can be rewritten as:
\[ \sin m \cos \theta + \cos m \sin \theta = \frac{9}{10} \]
This is of the form \( \sin(m + \theta) = \frac{9}{10} \).

4. **Find \( \alpha + \beta \)**:
Since \( \alpha \) and \( \beta \) are the solutions, we have:
\[ \sin(m + \alpha) = \sin(m + \beta) = \frac{9}{10} \]
From the sine function, we know:
\[ m + \alpha = n\pi + (-1)^k (m + \beta) \]
This gives us two cases:
- \( m + \alpha = m + \beta \) (which is trivial)
- \( m + \alpha = \pi - (m + \beta) \)

5. **Solving for \( \alpha + \beta \)**:
From the second case, we can rearrange:
\[ \alpha + \beta = \pi - 2m \]

6. **Calculate \( \sin(\alpha + \beta) \)**:
Now we need to find:
\[ \sin(\alpha + \beta) = \sin(\pi - 2m) \]
Using the sine identity \( \sin(\pi - x) = \sin x \):
\[ \sin(\alpha + \beta) = \sin(2m) \]

7. **Use the double angle formula**:
We can use the double angle formula:
\[ \sin(2m) = 2 \sin m \cos m \]
Substituting the values of \( \sin m \) and \( \cos m \):
\[ \sin(2m) = 2 \left(\frac{3}{5}\right) \left(\frac{4}{5}\right) = 2 \cdot \frac{12}{25} = \frac{24}{25} \]

### Final Answer:
Thus, the value of \( \sin(\alpha + \beta) \) is:
\[ \sin(\alpha + \beta) = \frac{24}{25} \]