If `0 lt x lt (pi)/(2)` and `cos x = 0.34`, what is the value of sin `((x)/(2))` ?

Options

A. `0.574`
B. `0.733`
C. `0.819`
D. `0.917`

Explanation

To solve the problem, we will use the trigonometric identity for sine of half an angle. The identity states:

\[ \sin\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos x}{2}} \]

Given that \( \cos x = 0.34 \) and \( 0 < x < \frac{\pi}{2} \), we can proceed with the following steps:

### Step 1: Substitute the value of \( \cos x \) into the identity
We start by substituting \( \cos x = 0.34 \) into the half-angle formula:

\[ \sin\left(\frac{x}{2}\right) = \sqrt{\frac{1 - 0.34}{2}} \]

### Step 2: Simplify the expression inside the square root
Now, we simplify the expression inside the square root:

\[ 1 - 0.34 = 0.66 \]

So we have:

\[ \sin\left(\frac{x}{2}\right) = \sqrt{\frac{0.66}{2}} \]

### Step 3: Calculate \( \frac{0.66}{2} \)
Next, we calculate \( \frac{0.66}{2} \):

\[ \frac{0.66}{2} = 0.33 \]

### Step 4: Take the square root
Now we take the square root of \( 0.33 \):

\[ \sin\left(\frac{x}{2}\right) = \sqrt{0.33} \]

### Step 5: Use a calculator to find the value
Using a calculator, we find:

\[ \sqrt{0.33} \approx 0.574 \]

### Final Answer
Thus, the value of \( \sin\left(\frac{x}{2}\right) \) is approximately \( 0.574 \).