Evaluate: `sin(1/4sin^(- 1)(sqrt 63/8)i s`

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(2sqrt(2))`
D. `(1)/(5)`

Explanation

To evaluate the expression \( \sin\left(\frac{1}{4} \sin^{-1}\left(\frac{\sqrt{63}}{8}\right)\right) \), we will follow these steps:

### Step 1: Let \( \theta = \sin^{-1}\left(\frac{\sqrt{63}}{8}\right) \)
This implies that \( \sin \theta = \frac{\sqrt{63}}{8} \).

### Step 2: Construct a right triangle
We can represent \( \sin \theta \) in terms of a right triangle. In this triangle:
- The opposite side to angle \( \theta \) is \( \sqrt{63} \).
- The hypotenuse is \( 8 \).

### Step 3: Find the adjacent side using the Pythagorean theorem
Using the Pythagorean theorem:
\[ \text{adjacent}^2 + \text{opposite}^2 = \text{hypotenuse}^2 \]
Let the length of the adjacent side be \( x \):
\[ x^2 + (\sqrt{63})^2 = 8^2 \]
\[ x^2 + 63 = 64 \]
\[ x^2 = 64 - 63 = 1 \]
\[ x = 1 \]

### Step 4: Calculate \( \cos \theta \)
Now that we have the lengths of all sides, we can find \( \cos \theta \):
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{8} \]

### Step 5: Use the half-angle formula for \( \cos\left(\frac{\theta}{2}\right) \)
We can use the half-angle formula:
\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} = \sqrt{\frac{1 + \frac{1}{8}}{2}} = \sqrt{\frac{\frac{9}{8}}{2}} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Step 6: Use the half-angle formula for \( \sin\left(\frac{\theta}{2}\right) \)
Now, we can find \( \sin\left(\frac{\theta}{2}\right) \) using:
\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} = \sqrt{\frac{1 - \frac{1}{8}}{2}} = \sqrt{\frac{\frac{7}{8}}{2}} = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \]

### Step 7: Find \( \sin\left(\frac{1}{4} \sin^{-1}\left(\frac{\sqrt{63}}{8}\right)\right) \)
Now, we need to find \( \sin\left(\frac{\theta}{4}\right) \). We can use the half-angle formula again:
\[ \sin\left(\frac{\theta}{4}\right) = \sqrt{\frac{1 - \cos\left(\frac{\theta}{2}\right)}{2}} = \sqrt{\frac{1 - \frac{3}{4}}{2}} = \sqrt{\frac{\frac{1}{4}}{2}} = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}} \]

### Final Answer
Thus, the value of \( \sin\left(\frac{1}{4} \sin^{-1}\left(\frac{\sqrt{63}}{8}\right)\right) \) is:
\[ \frac{1}{2\sqrt{2}} \]