The set of values of x in `(-pi, pi)` satisfying the inequation `|4"sin" x-1| lt sqrt(5)`, is

Options

A. `(-pi//10, 3pi//10)`
B. `(-pi//10, pi)`
C. `(-pi, pi)`
D. `(-pi, 3pi//10)`

Answer

To solve the inequality \( |4 \sin x - 1| < \sqrt{5} \) for \( x \) in the interval \( (-\pi, \pi) \), we will follow these steps:

### Step 1: Remove the absolute value
The inequality \( |4 \sin x - 1| < \sqrt{5} \) can be rewritten as two separate inequalities:
\[ -\sqrt{5} < 4 \sin x - 1 < \sqrt{5} \]

### Step 2: Solve the left inequality
Starting with the left part of the inequality:
\[ 4 \sin x - 1 > -\sqrt{5} \]
Adding 1 to both sides gives:
\[ 4 \sin x > 1 - \sqrt{5} \]
Dividing by 4:
\[ \sin x > \frac{1 - \sqrt{5}}{4} \]

### Step 3: Solve the right inequality
Now, for the right part of the inequality:
\[ 4 \sin x - 1 < \sqrt{5} \]
Adding 1 to both sides gives:
\[ 4 \sin x < 1 + \sqrt{5} \]
Dividing by 4:
\[ \sin x < \frac{1 + \sqrt{5}}{4} \]

### Step 4: Find the numerical values
Now we need to calculate the numerical values of \( \frac{1 - \sqrt{5}}{4} \) and \( \frac{1 + \sqrt{5}}{4} \).

Calculating \( \sqrt{5} \) approximately gives \( 2.236 \):
\[ \frac{1 - \sqrt{5}}{4} \approx \frac{1 - 2.236}{4} \approx \frac{-1.236}{4} \approx -0.309 \]
\[ \frac{1 + \sqrt{5}}{4} \approx \frac{1 + 2.236}{4} \approx \frac{3.236}{4} \approx 0.809 \]

### Step 5: Set up the sine inequality
Now we have:
\[ -0.309 < \sin x < 0.809 \]

### Step 6: Find the angles corresponding to the sine values
To find the angles \( x \) that satisfy these sine values, we can use the inverse sine function:
1. For \( \sin x = -0.309 \):
- The reference angle is \( \theta = \arcsin(-0.309) \approx -0.314 \) radians (approximately).
- The angles in the interval \( (-\pi, \pi) \) are:
- \( x \approx -0.314 \)
- \( x \approx \pi - (-0.314) \approx 3.456 \) (but this is outside the interval, so we discard it).

2. For \( \sin x = 0.809 \):
- The reference angle is \( \theta = \arcsin(0.809) \approx 0.927 \) radians.
- The angles in the interval \( (-\pi, \pi) \) are:
- \( x \approx 0.927 \)
- \( x \approx \pi - 0.927 \approx 2.214 \) (this is also within the interval).

### Step 7: Combine the results
The solution set for \( x \) in the interval \( (-\pi, \pi) \) is:
\[ x \in (-\frac{\pi}{10}, \frac{3\pi}{10}) \]

### Final Answer
The set of values of \( x \) satisfying the inequality \( |4 \sin x - 1| < \sqrt{5} \) in the interval \( (-\pi, \pi) \) is:
\[ \left(-\frac{\pi}{10}, \frac{3\pi}{10}\right) \]