The general solution of the system of equation `sin^(3) x + sin^(3) ((2pi )/( 3) + x) +sin^(3) ((4pi)/( 3) +x ) ` `+(3)/( 4) cos 2x= 0` `cos x cancel(=) 0` is

Options

A. `x = ((2k +1 )pi)/(10) , k in Z`
B. `x = ((2k +1 )pi)/(5) , k in Z`
C. `x = ((4k +1 )pi)/(10) , k in Z`
D. `x = ((4k +1 )/(5))pi , k in Z`

Answer

To solve the equation

\[ \sin^3 x + \sin^3 \left(\frac{2\pi}{3} + x\right) + \sin^3 \left(\frac{4\pi}{3} + x\right) + \frac{3}{4} \cos 2x = 0 \]

with the condition that \(\cos x \neq 0\), we will follow these steps:

### Step 1: Use the identity for \(\sin^3 \theta\)

Recall the identity:

\[ \sin^3 \theta = \frac{3 \sin \theta - \sin 3\theta}{4} \]

We will apply this identity to each term in the equation.

### Step 2: Apply the identity

Using the identity, we rewrite each sine term:

\[ \sin^3 x = \frac{3 \sin x - \sin 3x}{4} \]
\[ \sin^3 \left(\frac{2\pi}{3} + x\right) = \frac{3 \sin \left(\frac{2\pi}{3} + x\right) - \sin \left(3\left(\frac{2\pi}{3} + x\right)\right)}{4} \]
\[ \sin^3 \left(\frac{4\pi}{3} + x\right) = \frac{3 \sin \left(\frac{4\pi}{3} + x\right) - \sin \left(3\left(\frac{4\pi}{3} + x\right)\right)}{4} \]

### Step 3: Substitute back into the equation

Substituting these back into the original equation gives:

\[ \frac{3 \sin x - \sin 3x}{4} + \frac{3 \sin \left(\frac{2\pi}{3} + x\right) - \sin \left(3\left(\frac{2\pi}{3} + x\right)\right)}{4} + \frac{3 \sin \left(\frac{4\pi}{3} + x\right) - \sin \left(3\left(\frac{4\pi}{3} + x\right)\right)}{4} + \frac{3}{4} \cos 2x = 0 \]

### Step 4: Factor out \(\frac{1}{4}\)

Factor out \(\frac{1}{4}\):

\[ \frac{1}{4} \left( 3 \sin x + 3 \sin \left(\frac{2\pi}{3} + x\right) + 3 \sin \left(\frac{4\pi}{3} + x\right) - \sin 3x - \sin \left(3\left(\frac{2\pi}{3} + x\right)\right) - \sin \left(3\left(\frac{4\pi}{3} + x\right)\right) + 3 \cos 2x \right) = 0 \]

### Step 5: Simplify the equation

This simplifies to:

\[ 3 \sin x + 3 \sin \left(\frac{2\pi}{3} + x\right) + 3 \sin \left(\frac{4\pi}{3} + x\right) - \sin 3x - \sin \left(3\left(\frac{2\pi}{3} + x\right)\right) - \sin \left(3\left(\frac{4\pi}{3} + x\right)\right) + 3 \cos 2x = 0 \]

### Step 6: Use sine addition formulas

Using the sine addition formulas, we can express \(\sin \left(\frac{2\pi}{3} + x\right)\) and \(\sin \left(\frac{4\pi}{3} + x\right)\):

\[ \sin \left(\frac{2\pi}{3} + x\right) = \frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x \]
\[ \sin \left(\frac{4\pi}{3} + x\right) = -\frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x \]

### Step 7: Substitute and simplify

Substituting these values back into the equation and simplifying will lead to a more manageable form of the equation.

### Step 8: Solve for \(x\)

After simplifying, we will isolate \(x\) and solve for the general solution.