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

Answer

To solve the problem, we need to find the value of \( \sin x \) given that \( \cos 3x = 0 \) and \( x \) is an acute angle. Here’s the step-by-step solution:

### Step 1: Set up the equation
We start with the equation given in the problem:
\[ \cos 3x = 0 \]

### Step 2: Determine the angle where cosine is zero
The cosine function is equal to zero at odd multiples of \( \frac{\pi}{2} \). Therefore, we can write:
\[ 3x = \frac{\pi}{2} + n\pi \quad \text{(where \( n \) is an integer)} \]
Since \( x \) is acute, we will consider only the first solution where \( n = 0 \):
\[ 3x = \frac{\pi}{2} \]

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

### Step 4: Find \( \sin x \)
Now that we have \( x = \frac{\pi}{6} \), we need to find \( \sin x \):
\[ \sin x = \sin \left( \frac{\pi}{6} \right) \]

### Step 5: Use the known value of sine
From trigonometric values, we know:
\[ \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \]

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

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