Given A is an acute angle and
13 sin A = 5 , evaluate :
` ( 5 sin A - 2 cos A)/( tan A) `

Answer

To solve the problem step by step, we will start from the given equation and work through the necessary trigonometric identities and relationships.

### Step 1: Given Information
We are given that \( 13 \sin A = 5 \).

### Step 2: Solve for \( \sin A \)
To find \( \sin A \), we can rearrange the equation:
\[ \sin A = \frac{5}{13} \]

### Step 3: Construct a Right Triangle
Since \( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} \), we can visualize a right triangle where:
- The opposite side (to angle A) is 5 (let's denote it as \( BC = 5x \)).
- The hypotenuse is 13 (denote it as \( AC = 13x \)).

### Step 4: Use Pythagorean Theorem to Find the Adjacent Side
Using the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ (13x)^2 = AB^2 + (5x)^2 \]
\[ 169x^2 = AB^2 + 25x^2 \]
Rearranging gives:
\[ AB^2 = 169x^2 - 25x^2 = 144x^2 \]
Taking the square root:
\[ AB = 12x \]

### Step 5: Find \( \cos A \) and \( \tan A \)
Now we can find \( \cos A \) and \( \tan A \):
- \( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{12x}{13x} = \frac{12}{13} \)
- \( \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} = \frac{5x}{12x} = \frac{5}{12} \)

### Step 6: Substitute Values into the Expression
We need to evaluate:
\[ \frac{5 \sin A - 2 \cos A}{\tan A} \]
Substituting the values we found:
\[ = \frac{5 \left(\frac{5}{13}\right) - 2 \left(\frac{12}{13}\right)}{\frac{5}{12}} \]
Calculating the numerator:
\[ = \frac{\frac{25}{13} - \frac{24}{13}}{\frac{5}{12}} = \frac{\frac{1}{13}}{\frac{5}{12}} \]

### Step 7: Simplify the Expression
To simplify:
\[ = \frac{1}{13} \cdot \frac{12}{5} = \frac{12}{65} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{12}{65} \]