In a right - angled triangle , it is given that A is an acute angle and `tan A=5/12`
cos A

Answer

To find the value of cos A given that tan A = 5/12, we can follow these steps:

### Step 1: Understand the relationship between tan, sec, and cos
We know that:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} \]
Given that \(\tan A = \frac{5}{12}\), we can assume the opposite side is 5 and the adjacent side is 12.

### Step 2: Use the Pythagorean theorem to find the hypotenuse
In a right-angled triangle, the relationship between the sides is given by:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
Substituting the values we have:
\[ \text{hypotenuse}^2 = 5^2 + 12^2 = 25 + 144 = 169 \]
Thus, the hypotenuse is:
\[ \text{hypotenuse} = \sqrt{169} = 13 \]

### Step 3: Find cos A using the definition of cosine
The cosine of an angle in a right triangle is given by:
\[ \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Substituting the values we found:
\[ \cos A = \frac{12}{13} \]

### Conclusion
Thus, the value of \(\cos A\) is:
\[ \cos A = \frac{12}{13} \]

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