If cos 3x= 0 and x is acute find the value of :
` cos 2x `

Answer

To solve the problem step by step, we start with the given equation and follow through the necessary calculations.

### Step 1: Understand the given equation
We are given that \( \cos 3x = 0 \). This means that the angle \( 3x \) must be equal to an odd multiple of \( \frac{\pi}{2} \) since cosine is zero at those angles.

### Step 2: Set up the equation
The general solution for \( \cos \theta = 0 \) is:
\[ \theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
For our case, we can take the first solution:
\[ 3x = \frac{\pi}{2} \]

### Step 3: Solve for \( x \)
Now, we solve for \( x \):
\[ x = \frac{\pi}{2} \cdot \frac{1}{3} = \frac{\pi}{6} \]

### Step 4: Find \( \cos 2x \)
Next, we need to find \( \cos 2x \):
\[ 2x = 2 \cdot \frac{\pi}{6} = \frac{\pi}{3} \]
Now we calculate \( \cos 2x \):
\[ \cos 2x = \cos\left(\frac{\pi}{3}\right) \]

### Step 5: Use the known value of cosine
We know from trigonometric values that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \cos 2x \) is:
\[ \cos 2x = \frac{1}{2} \]

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