If `sintheta=(5)/(13)andtheta` is acute , what u is valueof `sqrt((cottheta+tantheta))`?

Options

A. `(2)/(sqrt(5))`
B. `(13)/(2sqrt(15))`
C. `(-2)/(sqrt(5))`
D. Cannot be determined

Answer

To solve the problem, we need to find the value of \( \sqrt{\cot \theta + \tan \theta} \) given that \( \sin \theta = \frac{5}{13} \) and \( \theta \) is acute.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- Since \( \sin \theta = \frac{5}{13} \), we can interpret this in the context of a right triangle. Here, the opposite side (perpendicular) to angle \( \theta \) is 5, and the hypotenuse is 13.

2. **Finding the Length of the Base**:
- We can use the Pythagorean theorem to find the length of the adjacent side (base). The relationship is given by:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + b^2 \]
\[ 169 = 25 + b^2 \]
\[ b^2 = 169 - 25 = 144 \]
\[ b = \sqrt{144} = 12 \]

3. **Calculating \( \tan \theta \) and \( \cot \theta \)**:
- Now we can find \( \tan \theta \) and \( \cot \theta \):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{12} \]
\[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{12}{5} \]

4. **Finding \( \cot \theta + \tan \theta \)**:
- Now we can add \( \cot \theta \) and \( \tan \theta \):
\[ \cot \theta + \tan \theta = \frac{12}{5} + \frac{5}{12} \]
- To add these fractions, we need a common denominator:
\[ \text{LCM of } 5 \text{ and } 12 = 60 \]
\[ \cot \theta + \tan \theta = \frac{12 \times 12}{60} + \frac{5 \times 5}{60} = \frac{144 + 25}{60} = \frac{169}{60} \]

5. **Calculating \( \sqrt{\cot \theta + \tan \theta} \)**:
- Now we take the square root:
\[ \sqrt{\cot \theta + \tan \theta} = \sqrt{\frac{169}{60}} = \frac{\sqrt{169}}{\sqrt{60}} = \frac{13}{\sqrt{60}} = \frac{13}{\sqrt{4 \times 15}} = \frac{13}{2\sqrt{15}} \]

### Final Answer:
Thus, the value of \( \sqrt{\cot \theta + \tan \theta} \) is \( \frac{13}{2\sqrt{15}} \).