If `4cos^(2)x=3 and x` is an acute angle, find the value of :
`sin2x`

Answer

To solve the problem step by step, we start with the given equation:

**Step 1: Start with the equation.**
Given:
\[ 4 \cos^2 x = 3 \]

**Step 2: Isolate \(\cos^2 x\).**
Divide both sides by 4:
\[ \cos^2 x = \frac{3}{4} \]

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

**Step 4: Identify the angle \(x\).**
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus, we can conclude:
\[ x = 30^\circ \]

**Step 5: Find \(2x\).**
Now, we need to find \(2x\):
\[ 2x = 2 \times 30^\circ = 60^\circ \]

**Step 6: Calculate \(\sin 2x\).**
Now we calculate:
\[ \sin 2x = \sin 60^\circ \]
We know that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

**Final Answer:**
Thus, the value of \(\sin 2x\) is:
\[ \sin 2x = \frac{\sqrt{3}}{2} \]

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