Choose the correct answer in each of the following questions:
If `cos theta =(4)/(5) " then " tan theta=?`

Options

A. `(3)/(4)`
B. `(4)/(3)`
C. `(3)/(5)`
D. `(5)/(3)`

Explanation

To find the value of \( \tan \theta \) given that \( \cos \theta = \frac{4}{5} \), we can follow these steps:

### Step 1: Understand the relationship between the sides of a right triangle
In a right triangle, the cosine of an angle \( \theta \) is defined as the ratio of the adjacent side (base) to the hypotenuse. Here, we have:
\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5} \]
This means the length of the adjacent side is 4 and the length of the hypotenuse is 5.

### Step 2: Use the Pythagorean theorem to find the opposite side
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
We can rearrange this to find the opposite side:
\[ \text{Opposite}^2 = \text{Hypotenuse}^2 - \text{Adjacent}^2 \]
Substituting the known values:
\[ \text{Opposite}^2 = 5^2 - 4^2 = 25 - 16 = 9 \]
Now, taking the square root:
\[ \text{Opposite} = \sqrt{9} = 3 \]

### Step 3: Calculate \( \tan \theta \)
The tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4} \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{3}{4} \]