If `omega ` is a complex cube root of unity, then `sin { ( omega ^(10) + omega ^(23) ) pi - ( pi)/(4) }` is equal to

Options

A. `(1)/(sqrt2)`
B. `(1)/(2)`
C. `1`
D. `(sqrt3)/(2)`

Answer

To solve the problem, we need to evaluate the expression \( \sin \left( \omega^{10} + \omega^{23} \right) \pi - \frac{\pi}{4} \), where \( \omega \) is a complex cube root of unity.

### Step-by-Step Solution:

1. **Understanding the cube roots of unity**:
The complex cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{4\pi i / 3}, \quad \text{and} \quad \omega^3 = 1 \]
Thus, \( \omega^3 = 1 \) implies that \( \omega^n \) can be simplified based on \( n \mod 3 \).

2. **Simplifying \( \omega^{10} \)**:
To simplify \( \omega^{10} \):
\[ 10 \mod 3 = 1 \quad \Rightarrow \quad \omega^{10} = \omega^1 = \omega \]

3. **Simplifying \( \omega^{23} \)**:
To simplify \( \omega^{23} \):
\[ 23 \mod 3 = 2 \quad \Rightarrow \quad \omega^{23} = \omega^2 \]

4. **Adding the results**:
Now we can add the two results:
\[ \omega^{10} + \omega^{23} = \omega + \omega^2 \]

5. **Using the property of cube roots of unity**:
We know from the properties of cube roots of unity that:
\[ 1 + \omega + \omega^2 = 0 \quad \Rightarrow \quad \omega + \omega^2 = -1 \]

6. **Substituting back into the sine function**:
Now substituting back into the sine function:
\[ \sin \left( \omega^{10} + \omega^{23} \right) \pi - \frac{\pi}{4} = \sin(-\pi) - \frac{\pi}{4} \]

7. **Evaluating the sine function**:
We know that:
\[ \sin(-\pi) = 0 \]
Therefore:
\[ 0 - \frac{\pi}{4} = -\frac{\pi}{4} \]

### Final Result:
Thus, the final answer is:
\[ \sin \left( \omega^{10} + \omega^{23} \right) \pi - \frac{\pi}{4} = -\frac{\pi}{4} \]