If `2cos3theta=sqrt(3)(0^(@)lethetale90^(@))`,then find the value of `theta`.

Answer

To solve the equation \(2 \cos(3\theta) = \sqrt{3}\) for \(\theta\) where \(\theta\) lies between \(0^\circ\) and \(90^\circ\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2 \cos(3\theta) = \sqrt{3} \]

### Step 2: Divide both sides by 2
To isolate \(\cos(3\theta)\), we divide both sides by 2:
\[ \cos(3\theta) = \frac{\sqrt{3}}{2} \]

### Step 3: Identify the angle for cosine
We know that \(\cos(x) = \frac{\sqrt{3}}{2}\) corresponds to angles where:
\[ x = 30^\circ + 360^\circ n \quad \text{or} \quad x = 330^\circ + 360^\circ n \quad (n \in \mathbb{Z}) \]
However, since we are dealing with \(3\theta\), we can set:
\[ 3\theta = 30^\circ \quad \text{or} \quad 3\theta = 330^\circ \]

### Step 4: Solve for \(\theta\)
Now we solve for \(\theta\) in both cases.

**Case 1:**
\[ 3\theta = 30^\circ \]
Dividing both sides by 3:
\[ \theta = \frac{30^\circ}{3} = 10^\circ \]

**Case 2:**
\[ 3\theta = 330^\circ \]
Dividing both sides by 3:
\[ \theta = \frac{330^\circ}{3} = 110^\circ \]

### Step 5: Determine valid solutions
Since \(\theta\) must lie between \(0^\circ\) and \(90^\circ\), we discard \(110^\circ\) as it is outside the specified range.

### Final Answer
Thus, the only valid solution is:
\[ \theta = 10^\circ \]

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