In triangle ABC, angle C measures `90^(@)`. If `cosB=(12)/(13)`, what is the value of sinB?

Options

A. `(5)/(12)`
B. `(5)/(13)`
C. `(12)/(13)`
D. `(13)/(12)`

Answer

To find the value of \( \sin B \) in triangle ABC where angle C is \( 90^\circ \) and \( \cos B = \frac{12}{13} \), we can use the Pythagorean identity:

### Step-by-step Solution:

1. **Use the Pythagorean Identity**:
The Pythagorean identity states that:
\[ \cos^2 B + \sin^2 B = 1 \]

2. **Substitute the value of \( \cos B \)**:
Given \( \cos B = \frac{12}{13} \), we can find \( \cos^2 B \):
\[ \cos^2 B = \left(\frac{12}{13}\right)^2 = \frac{144}{169} \]

3. **Plug \( \cos^2 B \) into the identity**:
Substitute \( \cos^2 B \) into the Pythagorean identity:
\[ \frac{144}{169} + \sin^2 B = 1 \]

4. **Isolate \( \sin^2 B \)**:
To find \( \sin^2 B \), rearrange the equation:
\[ \sin^2 B = 1 - \frac{144}{169} \]

5. **Convert 1 to a fraction with the same denominator**:
Convert 1 to a fraction:
\[ 1 = \frac{169}{169} \]
Now substitute:
\[ \sin^2 B = \frac{169}{169} - \frac{144}{169} = \frac{25}{169} \]

6. **Take the square root**:
To find \( \sin B \), take the square root of both sides:
\[ \sin B = \sqrt{\frac{25}{169}} = \frac{5}{13} \]

7. **Consider the positive value**:
Since \( B \) is an angle in a triangle, we take the positive value:
\[ \sin B = \frac{5}{13} \]

### Final Answer:
\[ \sin B = \frac{5}{13} \]