If `cos A = 4/5` than tan A = ?

Options

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

Answer

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

### Step 1: Understand the relationship between the sides of a right triangle
In a right triangle, if we denote:
- The hypotenuse as \( h \)
- The base (adjacent side) as \( b \)
- The perpendicular (opposite side) as \( p \)

We know that:
\[ \cos A = \frac{\text{base}}{\text{hypotenuse}} = \frac{b}{h} \]

### Step 2: Assign values based on the given cosine
From the problem, we have:
\[ \cos A = \frac{4}{5} \]
This means:
- Base \( b = 4 \)
- Hypotenuse \( h = 5 \)

### Step 3: Use the Pythagorean theorem to find the perpendicular
According to the Pythagorean theorem:
\[ h^2 = p^2 + b^2 \]
Substituting the known values:
\[ 5^2 = p^2 + 4^2 \]
This simplifies to:
\[ 25 = p^2 + 16 \]

### Step 4: Solve for the perpendicular
Rearranging the equation gives:
\[ p^2 = 25 - 16 \]
\[ p^2 = 9 \]
Taking the square root of both sides:
\[ p = \sqrt{9} = 3 \]

### Step 5: Calculate \( \tan A \)
Now that we have the lengths of all sides, we can find \( \tan A \):
\[ \tan A = \frac{\text{perpendicular}}{\text{base}} = \frac{p}{b} = \frac{3}{4} \]

### Final Answer
Thus, \( \tan A = \frac{3}{4} \).

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