If `2cos2A=sqrt(3)` and A is acute, find :
sin 3A

Answer

To solve the problem step by step, we will follow the given information and apply trigonometric identities.

### Step 1: Start with the given equation
We are given:
\[ 2 \cos 2A = \sqrt{3} \]

### Step 2: Isolate \(\cos 2A\)
To isolate \(\cos 2A\), divide both sides of the equation by 2:
\[ \cos 2A = \frac{\sqrt{3}}{2} \]

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

### Step 4: Solve for \(A\)
Now, divide both sides by 2 to find \(A\):
\[ A = \frac{30^\circ}{2} = 15^\circ \]

### Step 5: Find \(\sin 3A\)
Now we need to find \(\sin 3A\):
\[ 3A = 3 \times 15^\circ = 45^\circ \]
So, we need to calculate \(\sin 45^\circ\).

### Step 6: Use the known value of \(\sin 45^\circ\)
We know that:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the value of \(\sin 3A\) is:
\[ \sin 3A = \frac{1}{\sqrt{2}} \]

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