If `x sin theta = y sin ( theta + ( 2pi )/( 3)) = z sin( ( theta + 4pi )/( 3))`, then `Sigma xy =`

Options

A. 1
B. `1//2`
C. 0
D. none of these

Answer

To solve the equation \( x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right) \) and find \( \Sigma xy \), we will follow these steps:

### Step 1: Set up the equations

From the given condition, we can write:

1. \( x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) \)
2. \( x \sin \theta = z \sin \left( \theta + \frac{4\pi}{3} \right) \)

### Step 2: Express \( y \) in terms of \( x \)

From the first equation, we can express \( y \):

\[ y = \frac{x \sin \theta}{\sin \left( \theta + \frac{2\pi}{3} \right)} \]

### Step 3: Use the sine addition formula

Using the sine addition formula, we have:

\[ \sin \left( \theta + \frac{2\pi}{3} \right) = \sin \theta \cos \frac{2\pi}{3} + \cos \theta \sin \frac{2\pi}{3} \]

We know:

\[ \cos \frac{2\pi}{3} = -\frac{1}{2}, \quad \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \]

So,

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

### Step 4: Substitute back into the equation for \( y \)

Now substituting this back into the equation for \( y \):

\[ y = \frac{x \sin \theta}{-\frac{1}{2} \sin \theta + \frac{\sqrt{3}}{2} \cos \theta} \]

### Step 5: Express \( z \) in terms of \( x \)

Similarly, from the second equation, we can express \( z \):

\[ z = \frac{x \sin \theta}{\sin \left( \theta + \frac{4\pi}{3} \right)} \]

Using the sine addition formula again:

\[ \sin \left( \theta + \frac{4\pi}{3} \right) = \sin \theta \cos \frac{4\pi}{3} + \cos \theta \sin \frac{4\pi}{3} \]

Where:

\[ \cos \frac{4\pi}{3} = -\frac{1}{2}, \quad \sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2} \]

So,

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

### Step 6: Substitute back into the equation for \( z \)

Now substituting this back into the equation for \( z \):

\[ z = \frac{x \sin \theta}{-\frac{1}{2} \sin \theta - \frac{\sqrt{3}}{2} \cos \theta} \]

### Step 7: Form the equations for \( \frac{x}{y} \) and \( \frac{x}{z} \)

Now we can form the equations:

1. \( \frac{x}{y} = -\frac{1}{2} + \frac{\sqrt{3}}{2} \cot \theta \)
2. \( \frac{x}{z} = -\frac{1}{2} - \frac{\sqrt{3}}{2} \cot \theta \)

### Step 8: Add the two equations

Adding these two equations:

\[ \frac{x}{y} + \frac{x}{z} = -1 \]

### Step 9: Rearranging gives us the final equation

This leads us to:

\[ xz + xy + yz = 0 \]

### Step 10: Conclusion

Thus, we find that:

\[ \Sigma xy = xy + yz + zx = 0 \]

### Final Answer

\[ \Sigma xy = 0 \]

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