Let a, b,c are respectively the sines and p, q, r are respectively the consines of `alpha, alpha+(2pi)/(3) and alpha+(4pi)/(3)`, then :
Q. The value of `(qc-rb)` is :

Options

A. 0
B. `-(sqrt(3))/(2)`
C. `(sqrt(3))/(2)`
D. depends on `alpha`

Answer

To solve the problem, we need to find the value of \( qc - rb \) where:

- \( a = \sin(\alpha) \)
- \( b = \sin\left(\alpha + \frac{2\pi}{3}\right) \)
- \( c = \sin\left(\alpha + \frac{4\pi}{3}\right) \)
- \( p = \cos(\alpha) \)
- \( q = \cos\left(\alpha + \frac{2\pi}{3}\right) \)
- \( r = \cos\left(\alpha + \frac{4\pi}{3}\right) \)

### Step 1: Write down the expressions for \( qc \) and \( rb \)

We need to calculate:
\[ qc - rb = q \cdot c - r \cdot b \]

### Step 2: Substitute the values of \( q, c, r, \) and \( b \)

Substituting the values:
\[ qc = \cos\left(\alpha + \frac{2\pi}{3}\right) \cdot \sin\left(\alpha + \frac{4\pi}{3}\right) \]
\[ rb = \cos\left(\alpha + \frac{4\pi}{3}\right) \cdot \sin\left(\alpha + \frac{2\pi}{3}\right) \]

### Step 3: Use the sine subtraction formula

Notice that:
\[ qc - rb = \cos\left(\alpha + \frac{2\pi}{3}\right) \cdot \sin\left(\alpha + \frac{4\pi}{3}\right) - \cos\left(\alpha + \frac{4\pi}{3}\right) \cdot \sin\left(\alpha + \frac{2\pi}{3}\right) \]

This expression can be simplified using the sine subtraction formula:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
Here, let \( A = \alpha + \frac{4\pi}{3} \) and \( B = \alpha + \frac{2\pi}{3} \).

Thus, we have:
\[ qc - rb = \sin\left(\left(\alpha + \frac{4\pi}{3}\right) - \left(\alpha + \frac{2\pi}{3}\right)\right) \]

### Step 4: Simplify the angle

Now, simplifying the angle:
\[ \left(\alpha + \frac{4\pi}{3}\right) - \left(\alpha + \frac{2\pi}{3}\right) = \frac{4\pi}{3} - \frac{2\pi}{3} = \frac{2\pi}{3} \]

### Step 5: Calculate \( \sin\left(\frac{2\pi}{3}\right) \)

Now we find:
\[ qc - rb = \sin\left(\frac{2\pi}{3}\right) \]

Using the sine value:
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Final Answer

Thus, the value of \( qc - rb \) is:
\[ \frac{\sqrt{3}}{2} \]

---