If `4cos^(2)x=3 and x` is an acute angle, find the value of :
x

Answer

To solve the equation \(4 \cos^2 x = 3\) for \(x\) where \(x\) is an acute angle, we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 4 \cos^2 x = 3 \]

### Step 2: Isolate \(\cos^2 x\)
Divide both sides of the equation by 4:
\[ \cos^2 x = \frac{3}{4} \]

### Step 3: Take the square root
Now, take the square root of both sides. Since \(x\) is an acute angle, we only consider the positive root:
\[ \cos x = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 4: Identify the angle
We know from trigonometric ratios that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus, we can equate:
\[ x = 30^\circ \]

### Step 5: Verify the angle
Since \(30^\circ\) is indeed an acute angle (between 0 and 90 degrees), we confirm that our solution is valid.

### Final Answer
\[ x = 30^\circ \]

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