If `theta={(3^(2n))/(8)}`, where {x} = Fractional part of x then the value of `sec^(-1)(8theta)` is

Options

A. `pi`
B. `2pi`
C. `3pi`
D. 0

Answer

To solve the problem, we need to find the value of \( \sec^{-1}(8\theta) \) given that \( \theta = \left\{ \frac{3^{2n}}{8} \right\} \), where \( \{x\} \) denotes the fractional part of \( x \).

### Step-by-Step Solution:

1. **Understanding the Fractional Part**:
The fractional part \( \{x\} \) is defined as \( x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \). Thus, we can express:
\[ \theta = \left\{ \frac{3^{2n}}{8} \right\} = \frac{3^{2n}}{8} - \left\lfloor \frac{3^{2n}}{8} \right\rfloor \]

2. **Rewriting \( 3^{2n} \)**:
We can express \( 3^{2n} \) as \( 9^n \):
\[ \theta = \left\{ \frac{9^n}{8} \right\} \]

3. **Using the Binomial Theorem**:
We can rewrite \( \frac{9^n}{8} \) as:
\[ \frac{9^n}{8} = \frac{1 + 8^n}{8} \]
This can be expanded using the binomial theorem:
\[ \frac{1 + 8^n}{8} = \frac{1}{8} + \frac{8^n}{8} \]

4. **Identifying the Integer Part**:
The term \( \frac{8^n}{8} = 8^{n-1} \) is an integer. Therefore, we can express:
\[ \theta = \frac{1}{8} + \text{integer} - \lfloor \frac{1}{8} + \text{integer} \rfloor \]
Since \( \frac{1}{8} \) is less than 1, the integer part will not affect the fractional part. Thus:
\[ \theta = \frac{1}{8} \]

5. **Calculating \( 8\theta \)**:
Now we can find \( 8\theta \):
\[ 8\theta = 8 \times \frac{1}{8} = 1 \]

6. **Finding \( \sec^{-1}(8\theta) \)**:
We need to find \( \sec^{-1}(1) \). The secant function is defined as:
\[ \sec(x) = \frac{1}{\cos(x)} \]
Therefore, \( \sec^{-1}(1) \) corresponds to the angle where the secant is equal to 1, which is:
\[ \sec^{-1}(1) = 0 \]

### Final Answer:
Thus, the value of \( \sec^{-1}(8\theta) \) is:
\[ \boxed{0} \]