If `0lt x lt 90^(@)` and `sin x=0.525`, what is the value of `cos((x)/(3))`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the angle \( x \)
Given that \( \sin x = 0.525 \), we need to find the angle \( x \) using the inverse sine function.

\[ x = \sin^{-1}(0.525) \]

### Step 2: Calculate \( x \)
Using a calculator, we find:

\[ x \approx 31.66^\circ \]

### Step 3: Find \( \frac{x}{3} \)
Now, we need to calculate \( \frac{x}{3} \):

\[ \frac{x}{3} = \frac{31.66^\circ}{3} \approx 10.555^\circ \]

### Step 4: Calculate \( \cos\left(\frac{x}{3}\right) \)
Now we will find the cosine of \( \frac{x}{3} \):

\[ \cos\left(\frac{x}{3}\right) = \cos(10.555^\circ) \]

Using a calculator, we find:

\[ \cos(10.555^\circ) \approx 0.983 \]

### Final Answer
Thus, the value of \( \cos\left(\frac{x}{3}\right) \) is approximately:

\[ \cos\left(\frac{x}{3}\right) \approx 0.983 \]

---