Given that `"sin"^(-1)("sin""(3pi)/(4))=(2pi)/(k)`,then k=

Options

A. 3
B. 8
C. 6
D. 9

Answer

To solve the equation given in the problem, we will follow these steps:

### Step 1: Understand the expression
We start with the expression \( \sin^{-1}(\sin(3\pi/4)) \).

### Step 2: Simplify the sine function
The sine function has a periodicity of \( 2\pi \). Therefore, we can express \( 3\pi/4 \) in terms of a related angle within the range of \( \sin^{-1} \), which is \( [-\pi/2, \pi/2] \).

Since \( 3\pi/4 \) is greater than \( \pi/2 \), we can use the identity:
\[ \sin(x) = \sin(\pi - x) \]
Thus, we can rewrite:
\[ \sin(3\pi/4) = \sin(\pi - 3\pi/4) = \sin(\pi/4) \]

### Step 3: Substitute back into the inverse sine function
Now we substitute back into the inverse sine function:
\[ \sin^{-1}(\sin(3\pi/4)) = \sin^{-1}(\sin(\pi/4)) \]

### Step 4: Evaluate the inverse sine
Since \( \pi/4 \) is within the range of \( [-\pi/2, \pi/2] \), we have:
\[ \sin^{-1}(\sin(\pi/4)) = \pi/4 \]

### Step 5: Set up the equation
Now we set up the equation given in the problem:
\[ \frac{\pi}{4} = \frac{2\pi}{k} \]

### Step 6: Solve for \( k \)
To find \( k \), we can cross-multiply:
\[ \pi \cdot k = 8\pi \]
Dividing both sides by \( \pi \):
\[ k = 8 \]

### Final Answer
Thus, the value of \( k \) is \( 8 \).

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