If sin B = `(12)/(13)`, then find cot B.

Answer

To find cot B given that sin B = \( \frac{12}{13} \), we can follow these steps:

### Step 1: Understand the definition of sine
We know that:
\[ \sin B = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]
From the given information, we can identify:
- Perpendicular (opposite side) = 12
- Hypotenuse = 13

### Step 2: Use the Pythagorean theorem to find the base
In a right triangle, the relationship between the sides is given by the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]
Let the base be denoted as \( b \). Then we can write:
\[ 13^2 = 12^2 + b^2 \]
Calculating the squares:
\[ 169 = 144 + b^2 \]

### Step 3: Solve for the base
Now, we can isolate \( b^2 \):
\[ b^2 = 169 - 144 \]
\[ b^2 = 25 \]
Taking the square root of both sides:
\[ b = 5 \]

### Step 4: Find cotangent
The cotangent of an angle is defined as:
\[ \cot B = \frac{\text{Base}}{\text{Perpendicular}} \]
Substituting the values we found:
\[ \cot B = \frac{5}{12} \]

### Final Answer
Thus, the value of \( \cot B \) is:
\[ \cot B = \frac{5}{12} \]

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