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

Options

A. `1/sqrt(2)`
B. `sqrt(3)/2`
C. `-1/sqrt(3)`
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:
\[ 1, \quad \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{-2\pi i / 3} \]
where \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

2. **Calculating \( \omega^{10} \) and \( \omega^{23} \)**:
Since \( \omega^3 = 1 \), we can reduce the exponents modulo 3:
- For \( \omega^{10} \):
\[ 10 \mod 3 = 1 \quad \Rightarrow \quad \omega^{10} = \omega^1 = \omega \]
- For \( \omega^{23} \):
\[ 23 \mod 3 = 2 \quad \Rightarrow \quad \omega^{23} = \omega^2 \]

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

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

5. **Substituting into the Sine Function**:
Now we substitute this back into the sine function:
\[ \sin((\omega^{10} + \omega^{23})\pi - \frac{\pi}{4}) = \sin((-1)\pi - \frac{\pi}{4}) \]
Simplifying this gives:
\[ \sin(-\pi - \frac{\pi}{4}) = \sin(-(\pi + \frac{\pi}{4})) = \sin(-\frac{5\pi}{4}) \]

6. **Using the Sine Function Properties**:
We know that \( \sin(-\theta) = -\sin(\theta) \):
\[ \sin(-\frac{5\pi}{4}) = -\sin(\frac{5\pi}{4}) \]
The angle \( \frac{5\pi}{4} \) is in the third quadrant where sine is negative:
\[ \sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}} \quad \Rightarrow \quad -\sin(\frac{5\pi}{4}) = -(-\frac{1}{\sqrt{2}}) = \frac{1}{\sqrt{2}} \]

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