If 3 sin A = 4 cos A , find the value of :
cos A

Answer

To find the value of \( \cos A \) given that \( 3 \sin A = 4 \cos A \), we can follow these steps:

### Step 1: Rearranging the equation
We start with the equation:
\[ 3 \sin A = 4 \cos A \]
We can rearrange this to express the ratio of \( \sin A \) to \( \cos A \):
\[ \frac{\sin A}{\cos A} = \frac{4}{3} \]

### Step 2: Using the definition of tangent
From the definition of tangent, we know that:
\[ \tan A = \frac{\sin A}{\cos A} \]
Thus, we can write:
\[ \tan A = \frac{4}{3} \]

### Step 3: Drawing a right triangle
To visualize this, we can draw a right triangle where:
- The opposite side (perpendicular) to angle \( A \) is \( 4 \) (the value of \( \sin A \)).
- The adjacent side (base) to angle \( A \) is \( 3 \) (the value of \( \cos A \)).

### Step 4: Finding the hypotenuse
Using the Pythagorean theorem, we can find the hypotenuse \( h \):
\[ h^2 = \text{(perpendicular)}^2 + \text{(base)}^2 \]
Substituting the values:
\[ h^2 = 4^2 + 3^2 = 16 + 9 = 25 \]
Taking the square root:
\[ h = \sqrt{25} = 5 \]

### Step 5: Finding \( \cos A \)
Now, we can find \( \cos A \):
\[ \cos A = \frac{\text{base}}{\text{hypotenuse}} = \frac{3}{5} \]

Thus, the value of \( \cos A \) is:
\[ \cos A = \frac{3}{5} \]

### Summary
The final answer is:
\[ \cos A = \frac{3}{5} \]

---