Given : `4 sintheta = 3 cos theta` , find the value of:
`sin theta`

Answer

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

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

### Step 2: Isolate \(\tan \theta\)
We can divide both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)):
\[ \frac{4 \sin \theta}{\cos \theta} = 3 \]
This simplifies to:
\[ 4 \tan \theta = 3 \]
Now, we can isolate \(\tan \theta\):
\[ \tan \theta = \frac{3}{4} \]

### Step 3: Use the Pythagorean identity
We know that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Substituting \(\tan \theta = \frac{3}{4}\):
\[ \sec^2 \theta = 1 + \left(\frac{3}{4}\right)^2 \]
Calculating \(\left(\frac{3}{4}\right)^2\):
\[ \sec^2 \theta = 1 + \frac{9}{16} = \frac{16}{16} + \frac{9}{16} = \frac{25}{16} \]

### Step 4: Find \(\sec \theta\)
Taking the square root:
\[ \sec \theta = \sqrt{\frac{25}{16}} = \frac{5}{4} \]

### Step 5: Find \(\cos \theta\)
Since \(\sec \theta = \frac{1}{\cos \theta}\), we can find \(\cos \theta\):
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{5}{4}} = \frac{4}{5} \]

### Step 6: Substitute back to find \(\sin \theta\)
Now we can substitute \(\cos \theta\) back into the original equation:
\[ 4 \sin \theta = 3 \cos \theta \]
Substituting \(\cos \theta = \frac{4}{5}\):
\[ 4 \sin \theta = 3 \times \frac{4}{5} \]
Calculating the right side:
\[ 4 \sin \theta = \frac{12}{5} \]
Now, divide both sides by 4:
\[ \sin \theta = \frac{12}{5} \times \frac{1}{4} = \frac{12}{20} = \frac{3}{5} \]

### Final Answer:
Thus, the value of \(\sin \theta\) is:
\[ \sin \theta = \frac{3}{5} \]