If `omega` is a cube root of unity then find the value of `sin((omega^(10)+omega^(23))pi -pi/4)`

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 find the value of \( \sin(\omega^{10} + \omega^{23})\pi - \frac{\pi}{4} \), where \( \omega \) is a cube root of unity.

### Step-by-Step Solution:

1. **Understanding 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 \( \omega^{10} \)**:
Since \( \omega^3 = 1 \), we can reduce the exponent modulo 3:
\[ 10 \mod 3 = 1 \quad \Rightarrow \quad \omega^{10} = \omega^1 = \omega \]

3. **Finding \( \omega^{23} \)**:
Similarly, for \( \omega^{23} \):
\[ 23 \mod 3 = 2 \quad \Rightarrow \quad \omega^{23} = \omega^2 \]

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

5. **Using the Property of Cube Roots**:
From the property \( 1 + \omega + \omega^2 = 0 \), we can express \( \omega + \omega^2 \):
\[ \omega + \omega^2 = -1 \]

6. **Substituting into the Sine Function**:
Now we substitute back into the sine function:
\[ \sin((\omega + \omega^2)\pi - \frac{\pi}{4}) = \sin(-\pi - \frac{\pi}{4}) \]

7. **Simplifying the Argument**:
We can simplify the argument of the sine function:
\[ -\pi - \frac{\pi}{4} = -\frac{4\pi}{4} - \frac{\pi}{4} = -\frac{5\pi}{4} \]

8. **Using the Sine Function Properties**:
We know that \( \sin(-\theta) = -\sin(\theta) \):
\[ \sin(-\frac{5\pi}{4}) = -\sin(\frac{5\pi}{4}) \]

9. **Finding \( \sin(\frac{5\pi}{4}) \)**:
The angle \( \frac{5\pi}{4} \) is in the third quadrant where sine is negative:
\[ \sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}} \quad \Rightarrow \quad -\sin(\frac{5\pi}{4}) = \frac{1}{\sqrt{2}} \]

10. **Final Result**:
Therefore, the final value is:
\[ \sin(\omega^{10} + \omega^{23})\pi - \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]

### Conclusion:
The value of \( \sin(\omega^{10} + \omega^{23})\pi - \frac{\pi}{4} \) is \( \frac{1}{\sqrt{2}} \).