If sin `theta = (5)/(13)` then the valut 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
We know that:
\[ \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]
From the given information, we have:
\[ \sin \theta = \frac{5}{13} \]
This means that the length of the perpendicular side (opposite to the angle \( \theta \)) is 5 and the length of the hypotenuse is 13.

### Step 2: Use the Pythagorean theorem to find the base
In a right triangle, we can use the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + \text{Base}^2 \]
Calculating the squares:
\[ 169 = 25 + \text{Base}^2 \]
Now, isolate the base:
\[ \text{Base}^2 = 169 - 25 = 144 \]
Taking the square root gives:
\[ \text{Base} = \sqrt{144} = 12 \]

### Step 3: Calculate \( \tan \theta \)
Now that we have both the perpendicular and the base, we can find \( \tan \theta \):
\[ \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{5}{12} \]

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

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