If `cos A=(7)/(25)`, then the value of `tan A` is

Options

A. `(25)/(7)`
B. `(24)/(7)`
C. `(24)/(25)`
D. `(25)/(24)`

Answer

To find the value of \( \tan A \) given that \( \cos A = \frac{7}{25} \), we can follow these steps:

### Step 1: Understand the relationship between sides in a right triangle
In a right triangle, the cosine of an angle \( A \) is defined as the ratio of the length of the adjacent side (base) to the length of the hypotenuse. Therefore, we can express this as:
\[ \cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{7}{25} \]
From this, we can identify:
- Adjacent side (base) = 7
- Hypotenuse = 25

### Step 2: Use the Pythagorean theorem to find the opposite side (perpendicular)
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
Substituting the known values:
\[ 25^2 = 7^2 + \text{Opposite}^2 \]
Calculating the squares:
\[ 625 = 49 + \text{Opposite}^2 \]
Now, isolate the opposite side:
\[ \text{Opposite}^2 = 625 - 49 \]
\[ \text{Opposite}^2 = 576 \]
Taking the square root:
\[ \text{Opposite} = \sqrt{576} = 24 \]

### Step 3: Calculate \( \tan A \)
The tangent of angle \( A \) is defined as the ratio of the opposite side to the adjacent side:
\[ \tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{24}{7} \]

### Final Answer
Thus, the value of \( \tan A \) is:
\[ \tan A = \frac{24}{7} \]

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