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

Answer

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

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

### Step 2: Divide both sides by \(\sin A\)
To isolate \(\cos A\) and \(\sin A\), we can divide both sides by \(\sin A\) (assuming \(\sin A \neq 0\)):
\[ \frac{3 \cos A}{\sin A} = 4 \]

### Step 3: Express \(\frac{\cos A}{\sin A}\)
Now, we can express \(\frac{\cos A}{\sin A}\) as:
\[ \frac{\cos A}{\sin A} = \frac{4}{3} \]

### Step 4: Recognize the trigonometric identity
The ratio \(\frac{\cos A}{\sin A}\) is equal to \(\cot A\):
\[ \cot A = \frac{4}{3} \]

### Step 5: Draw a right triangle
To visualize this, we can draw a right triangle where:
- The adjacent side (base) is 4
- The opposite side (perpendicular) is 3

### Step 6: Calculate the hypotenuse
Using the Pythagorean theorem, we can find the hypotenuse \(h\):
\[ h = \sqrt{(4^2) + (3^2)} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 7: Find \(\cos A\)
Now we can find \(\cos A\) using the definition of cosine:
\[ \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \]

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

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