If `x sintheta=ysin(theta+(2pi)/(3))=z sin(theta+(4pi)/(3)),` then

Options

A. `x+y+z=0`
B. `xy+yz+zx=0`
C. `xyz+x+y+z=1`
D. none of these

Explanation

To solve the equation \( x \sin \theta = y \sin\left(\theta + \frac{2\pi}{3}\right) = z \sin\left(\theta + \frac{4\pi}{3}\right) \), we will break it down step by step.

### Step 1: Express \( y \sin\left(\theta + \frac{2\pi}{3}\right) \)
Using the sine addition formula, we have:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
For \( y \sin\left(\theta + \frac{2\pi}{3}\right) \):
\[ y \sin\left(\theta + \frac{2\pi}{3}\right) = y \left(\sin \theta \cos\left(\frac{2\pi}{3}\right) + \cos \theta \sin\left(\frac{2\pi}{3}\right)\right) \]
We know that:
\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus,
\[ y \sin\left(\theta + \frac{2\pi}{3}\right) = y \left(\sin \theta \left(-\frac{1}{2}\right) + \cos \theta \left(\frac{\sqrt{3}}{2}\right)\right) \]
This simplifies to:
\[ y \sin\left(\theta + \frac{2\pi}{3}\right) = -\frac{y}{2} \sin \theta + \frac{y \sqrt{3}}{2} \cos \theta \]

### Step 2: Set up the equation with \( x \)
Now, equate \( x \sin \theta \) to \( y \sin\left(\theta + \frac{2\pi}{3}\right) \):
\[ x \sin \theta = -\frac{y}{2} \sin \theta + \frac{y \sqrt{3}}{2} \cos \theta \]
Rearranging gives:
\[ x \sin \theta + \frac{y}{2} \sin \theta = \frac{y \sqrt{3}}{2} \cos \theta \]
Factoring out \( \sin \theta \):
\[ \left(x + \frac{y}{2}\right) \sin \theta = \frac{y \sqrt{3}}{2} \cos \theta \]
Dividing both sides by \( \sin \theta \) (assuming \( \sin \theta \neq 0 \)):
\[ x + \frac{y}{2} = \frac{y \sqrt{3}}{2} \frac{\cos \theta}{\sin \theta} \]
Thus,
\[ \frac{x}{y} = -\frac{1}{2} + \frac{\sqrt{3}}{2} \cot \theta \tag{1} \]

### Step 3: Express \( z \sin\left(\theta + \frac{4\pi}{3}\right) \)
Using the sine addition formula again for \( z \sin\left(\theta + \frac{4\pi}{3}\right) \):
\[ z \sin\left(\theta + \frac{4\pi}{3}\right) = z \left(\sin \theta \cos\left(\frac{4\pi}{3}\right) + \cos \theta \sin\left(\frac{4\pi}{3}\right)\right) \]
With:
\[ \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]
We get:
\[ z \sin\left(\theta + \frac{4\pi}{3}\right) = z \left(\sin \theta \left(-\frac{1}{2}\right) + \cos \theta \left(-\frac{\sqrt{3}}{2}\right)\right) \]
This simplifies to:
\[ z \sin\left(\theta + \frac{4\pi}{3}\right) = -\frac{z}{2} \sin \theta - \frac{z \sqrt{3}}{2} \cos \theta \]

### Step 4: Set up the equation with \( x \)
Equate \( x \sin \theta \) to \( z \sin\left(\theta + \frac{4\pi}{3}\right) \):
\[ x \sin \theta = -\frac{z}{2} \sin \theta - \frac{z \sqrt{3}}{2} \cos \theta \]
Rearranging gives:
\[ x \sin \theta + \frac{z}{2} \sin \theta = -\frac{z \sqrt{3}}{2} \cos \theta \]
Factoring out \( \sin \theta \):
\[ \left(x + \frac{z}{2}\right) \sin \theta = -\frac{z \sqrt{3}}{2} \cos \theta \]
Dividing both sides by \( \sin \theta \):
\[ x + \frac{z}{2} = -\frac{z \sqrt{3}}{2} \frac{\cos \theta}{\sin \theta} \]
Thus,
\[ \frac{x}{z} = -\frac{1}{2} - \frac{\sqrt{3}}{2} \cot \theta \tag{2} \]

### Step 5: Combine equations (1) and (2)
Now we add equations (1) and (2):
\[ \frac{x}{y} + \frac{x}{z} = \left(-\frac{1}{2} + \frac{\sqrt{3}}{2} \cot \theta\right) + \left(-\frac{1}{2} - \frac{\sqrt{3}}{2} \cot \theta\right) \]
This simplifies to:
\[ \frac{x}{y} + \frac{x}{z} = -1 \]
Multiplying through by \( yz \):
\[ xz + xy = -yz \]
Rearranging gives:
\[ xy + yz + zx = 0 \]

### Final Result
Thus, the final result is:
\[ xy + yz + zx = 0 \]