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

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ 3 \sin A = 4 \cos A \]

### Step 2: Divide both sides by \(\cos A\)
To isolate \(\tan A\), we can divide both sides by \(\cos A\):
\[ \frac{3 \sin A}{\cos A} = 4 \]
This simplifies to:
\[ 3 \tan A = 4 \]

### Step 3: Solve for \(\tan A\)
Now, we can solve for \(\tan A\):
\[ \tan A = \frac{4}{3} \]

### Step 4: Draw a right triangle
To find \(\sin A\), we can visualize this as a right triangle where the opposite side (perpendicular) is 4 and the adjacent side (base) is 3.

### Step 5: Use the Pythagorean theorem to find the hypotenuse
We apply the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{(\text{perpendicular})^2 + (\text{base})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 6: Calculate \(\sin A\)
Now we can find \(\sin A\) using the definition of sine:
\[ \sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \]

### Final Answer
Thus, the value of \(\sin A\) is:
\[ \sin A = \frac{4}{5} \]

---