If sin `theta = (5)/(13)` then the value of tan `theta` is ……… .

Answer

To find the value of \( \tan \theta \) given that \( \sin \theta = \frac{5}{13} \), we can follow these steps:

### Step 1: Understand the relationship of sine in a right triangle
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side (perpendicular) to the length of the hypotenuse.

### Step 2: Set up the triangle
Given \( \sin \theta = \frac{5}{13} \), we can denote:
- Opposite side (perpendicular) = 5
- Hypotenuse = 13

### Step 3: Use the Pythagorean theorem to find the base
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Base}^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + \text{Base}^2 \]
Calculating the squares:
\[ 169 = 25 + \text{Base}^2 \]

### Step 4: Solve for the base
Rearranging the equation to isolate the base:
\[ \text{Base}^2 = 169 - 25 \]
Calculating the right side:
\[ \text{Base}^2 = 144 \]
Taking the square root:
\[ \text{Base} = 12 \]

### Step 5: Calculate \( \tan \theta \)
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side (base):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Base}} = \frac{5}{12} \]

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

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