If ` omega ` imaginary cube root of unity , then ` sin {(omega ^(13) + omega ^(20)) pi +(pi)/(4)}` is equal to

Options

A. `- (sqrt(3))/(2)`
B. `-(1)/(sqrt(2))`
C. `(1)/( sqrt(2))`
D. `(sqrt(3))/( 2)`

Answer

To solve the problem, we need to evaluate the expression \( \sin \left( \omega^{13} + \omega^{20} \right) \pi + \frac{\pi}{4} \), where \( \omega \) is the imaginary cube root of unity.

### Step-by-step Solution:

1. **Understanding the cube roots of unity:**
The cube roots of unity are the solutions to the equation \( x^3 = 1 \). They are given by:
\[ 1, \quad \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{4\pi i / 3} \]
where \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

2. **Finding powers of \( \omega \):**
Since \( \omega^3 = 1 \), we can reduce higher powers of \( \omega \) modulo 3:
- \( \omega^{13} = \omega^{13 \mod 3} = \omega^{1} = \omega \)
- \( \omega^{20} = \omega^{20 \mod 3} = \omega^{2} \)

3. **Combining the results:**
Now we can add these results:
\[ \omega^{13} + \omega^{20} = \omega + \omega^2 \]
From the property of cube roots of unity, we know:
\[ \omega + \omega^2 = -1 \]

4. **Substituting back into the sine function:**
We substitute this back into our sine function:
\[ \sin \left( (-1) \pi + \frac{\pi}{4} \right) \]
This simplifies to:
\[ \sin \left( -\pi + \frac{\pi}{4} \right) = \sin \left( -\left( \pi - \frac{\pi}{4} \right) \right) = \sin \left( -\frac{3\pi}{4} \right) \]

5. **Using the sine function properties:**
We know that:
\[ \sin(-x) = -\sin(x) \]
Therefore:
\[ \sin \left( -\frac{3\pi}{4} \right) = -\sin \left( \frac{3\pi}{4} \right) \]

6. **Calculating \( \sin \left( \frac{3\pi}{4} \right) \):**
The value of \( \sin \left( \frac{3\pi}{4} \right) \) is:
\[ \sin \left( \frac{3\pi}{4} \right) = \frac{1}{\sqrt{2}} \]
Thus:
\[ \sin \left( -\frac{3\pi}{4} \right) = -\frac{1}{\sqrt{2}} \]

### Final Answer:
Therefore, the value of \( \sin \left( \omega^{13} + \omega^{20} \right) \pi + \frac{\pi}{4} \) is:
\[ \boxed{-\frac{1}{\sqrt{2}}} \]