If ` x+y=(4pi)/(3) and sin x = 2 sin y` , then

Options

A. (a)` x=npi +(pi)/(2), n in I `
B. (b)`y=(5pi)/(6)-npi, n in I `
C. (c)Both (a) and (b)
D. (d)None of the above

Answer

To solve the problem, we are given two equations:

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

We need to find the values of \( x \) and \( y \).

### Step 1: Express \( y \) in terms of \( x \)
From the first equation, we can express \( y \) as:
\[ y = \frac{4\pi}{3} - x \]

### Step 2: Substitute \( y \) into the second equation
Now, we substitute \( y \) into the second equation:
\[ \sin x = 2 \sin\left(\frac{4\pi}{3} - x\right) \]

### Step 3: Use the sine subtraction formula
Using the sine subtraction formula, \( \sin(a - b) = \sin a \cos b - \cos a \sin b \), we can rewrite the equation:
\[ \sin\left(\frac{4\pi}{3} - x\right) = \sin\left(\frac{4\pi}{3}\right) \cos x - \cos\left(\frac{4\pi}{3}\right) \sin x \]
We know:
\[ \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}, \quad \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \]
Substituting these values gives:
\[ \sin\left(\frac{4\pi}{3} - x\right) = -\frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x \]

### Step 4: Substitute back into the equation
Now substituting this back into our equation:
\[ \sin x = 2\left(-\frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x\right) \]
This simplifies to:
\[ \sin x = -\sqrt{3} \cos x + \sin x \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ \sin x + \sqrt{3} \cos x = \sin x \]
This simplifies to:
\[ \sqrt{3} \cos x = 0 \]

### Step 6: Solve for \( x \)
The equation \( \sqrt{3} \cos x = 0 \) implies:
\[ \cos x = 0 \]
The solutions for \( x \) are:
\[ x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]

### Step 7: Find \( y \)
Substituting \( x \) back into the equation for \( y \):
\[ y = \frac{4\pi}{3} - \left(\frac{\pi}{2} + n\pi\right) \]
This simplifies to:
\[ y = \frac{4\pi}{3} - \frac{\pi}{2} - n\pi \]
Finding a common denominator (which is 6):
\[ y = \frac{8\pi}{6} - \frac{3\pi}{6} - n\pi = \frac{5\pi}{6} - n\pi \]

### Final Solutions
Thus, we have:
\[ x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
\[ y = \frac{5\pi}{6} - n\pi \quad (n \in \mathbb{Z}) \]