The equation `x^3=3/4x=-(sqrt(3))/8` is satisfied by `x=cos((5pi)/(18))` (b) `x=cos((7pi)/(18))` `x=cos((23pi)/(18))` (d) `x=cos((17pi)/(18))`

Options

A. `x=cos((5pi)/(18))`
B. `x =cos((7pi)/(18))`
C. `x=cos((23pi)/(18))`
D. `x=-sin((7pi)/(9))`

Explanation

To solve the equation \( x^3 - \frac{3}{4}x = -\frac{\sqrt{3}}{8} \) and determine which values of \( x \) satisfy it, we will follow these steps:

### Step 1: Rearrange the Equation
Start with the given equation:
\[ x^3 - \frac{3}{4}x + \frac{\sqrt{3}}{8} = 0 \]
To eliminate the fractions, we can multiply through by 8 (the least common multiple of the denominators):
\[ 8x^3 - 6x + \sqrt{3} = 0 \]

### Step 2: Substitute \( x \) with \( \cos(\theta) \)
Let \( x = \cos(\theta) \). Then, substituting this into the equation gives:
\[ 8\cos^3(\theta) - 6\cos(\theta) + \sqrt{3} = 0 \]

### Step 3: Use the Identity for Cosine
We can use the identity \( 4\cos^3(\theta) - 3\cos(\theta) = \cos(3\theta) \). Thus, we rewrite the equation:
\[ 2\cos(3\theta) + \sqrt{3} = 0 \]
This simplifies to:
\[ \cos(3\theta) = -\frac{\sqrt{3}}{2} \]

### Step 4: Solve for \( 3\theta \)
The cosine function equals \(-\frac{\sqrt{3}}{2}\) at specific angles:
\[ 3\theta = \frac{5\pi}{6} + 2n\pi \quad \text{and} \quad 3\theta = \frac{7\pi}{6} + 2n\pi \quad (n \in \mathbb{Z}) \]

### Step 5: Solve for \( \theta \)
Now, we divide by 3 to find \( \theta \):
1. From \( 3\theta = \frac{5\pi}{6} + 2n\pi \):
\[ \theta = \frac{5\pi}{18} + \frac{2n\pi}{3} \]
2. From \( 3\theta = \frac{7\pi}{6} + 2n\pi \):
\[ \theta = \frac{7\pi}{18} + \frac{2n\pi}{3} \]

### Step 6: Find Specific Values of \( x \)
Now, we can find specific values by substituting \( n = 0 \) and \( n = 1 \):
- For \( n = 0 \):
- From \( \theta = \frac{5\pi}{18} \)
- From \( \theta = \frac{7\pi}{18} \)

- For \( n = 1 \):
- From \( \theta = \frac{5\pi}{18} + \frac{2\pi}{3} = \frac{5\pi}{18} + \frac{12\pi}{18} = \frac{17\pi}{18} \)
- From \( \theta = \frac{7\pi}{18} + \frac{2\pi}{3} = \frac{7\pi}{18} + \frac{12\pi}{18} = \frac{19\pi}{18} \) (but this is not in the range of \( \cos \))

### Conclusion
The valid solutions for \( x \) are:
- \( x = \cos\left(\frac{5\pi}{18}\right) \)
- \( x = \cos\left(\frac{7\pi}{18}\right) \)
- \( x = \cos\left(\frac{17\pi}{18}\right) \)

Thus, the values of \( x \) that satisfy the equation are:
- \( \cos\left(\frac{5\pi}{18}\right) \)
- \( \cos\left(\frac{7\pi}{18}\right) \)
- \( \cos\left(\frac{17\pi}{18}\right) \)