If `7 sin theta +24 cos theta =25`, then what is the value of `(sin theta +cos theta)`?

Options

A. 1
B. `(26)/(25)`
C. `(6)/(5)`
D. `(31)/(25)`

Answer

To solve the equation \( 7 \sin \theta + 24 \cos \theta = 25 \) and find the value of \( \sin \theta + \cos \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ 7 \sin \theta + 24 \cos \theta = 25 \]

2. **Divide the entire equation by 25**:
\[ \frac{7}{25} \sin \theta + \frac{24}{25} \cos \theta = 1 \]

3. **Let \( a = \frac{7}{25} \) and \( b = \frac{24}{25} \)**:
Now we can rewrite the equation as:
\[ a \sin \theta + b \cos \theta = 1 \]

4. **Use the identity for \( a \sin \theta + b \cos \theta \)**:
We know that \( a \sin \theta + b \cos \theta \) can be expressed as:
\[ R \sin(\theta + \phi) \]
where \( R = \sqrt{a^2 + b^2} \) and \( \tan \phi = \frac{b}{a} \).

5. **Calculate \( R \)**:
\[ R = \sqrt{\left(\frac{7}{25}\right)^2 + \left(\frac{24}{25}\right)^2} = \sqrt{\frac{49}{625} + \frac{576}{625}} = \sqrt{\frac{625}{625}} = 1 \]

6. **Since \( R = 1 \)**, we have:
\[ \sin(\theta + \phi) = 1 \]
This implies:
\[ \theta + \phi = \frac{\pi}{2} + 2n\pi \quad \text{for some integer } n \]

7. **Now find \( \sin \theta \) and \( \cos \theta \)**:
From the values of \( a \) and \( b \):
\[ \sin \theta = \frac{7}{25}, \quad \cos \theta = \frac{24}{25} \]

8. **Calculate \( \sin \theta + \cos \theta \)**:
\[ \sin \theta + \cos \theta = \frac{7}{25} + \frac{24}{25} = \frac{31}{25} \]

### Final Answer:
\[ \sin \theta + \cos \theta = \frac{31}{25} \]