If `omega (ne 1)` is a cube root of unity, then the value of `tan[(omega^(2017) + omega^(2225)) pi - pi//3]`

Options

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

Answer

To solve the problem, we need to find the value of \( \tan\left[\left(\omega^{2017} + \omega^{2225}\right) \pi - \frac{\pi}{3}\right] \), where \( \omega \) is a cube root of unity (specifically, \( \omega \neq 1 \)).

### Step-by-Step Solution:

1. **Understanding Cube Roots of Unity**:
The cube roots of unity are \( 1, \omega, \omega^2 \) where \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{4\pi i / 3} \). They satisfy the equation \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \).

2. **Reducing the Exponents**:
Since \( \omega^3 = 1 \), we can reduce the exponents \( 2017 \) and \( 2225 \) modulo \( 3 \):
- For \( 2017 \):
\[ 2017 \mod 3 = 1 \quad (\text{since } 2017 = 3 \times 672 + 1) \]
Thus, \( \omega^{2017} = \omega^1 = \omega \).
- For \( 2225 \):
\[ 2225 \mod 3 = 1 \quad (\text{since } 2225 = 3 \times 741 + 2) \]
Thus, \( \omega^{2225} = \omega^2 \).

3. **Calculating \( \omega^{2017} + \omega^{2225} \)**:
Now we can add these results:
\[ \omega^{2017} + \omega^{2225} = \omega + \omega^2 \]
From the property of cube roots of unity, we know:
\[ \omega + \omega^2 = -1 \]

4. **Substituting into the Tangent Function**:
We substitute back into the tangent function:
\[ \tan\left[\left(\omega^{2017} + \omega^{2225}\right) \pi - \frac{\pi}{3}\right] = \tan\left[-\pi - \frac{\pi}{3}\right] \]
Simplifying the angle:
\[ -\pi - \frac{\pi}{3} = -\frac{3\pi}{3} - \frac{\pi}{3} = -\frac{4\pi}{3} \]

5. **Using Tangent Properties**:
We know that:
\[ \tan(-\theta) = -\tan(\theta) \]
Therefore:
\[ \tan\left(-\frac{4\pi}{3}\right) = -\tan\left(\frac{4\pi}{3}\right) \]
The angle \( \frac{4\pi}{3} \) is in the third quadrant, where tangent is positive:
\[ \tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus:
\[ \tan\left(-\frac{4\pi}{3}\right) = -\sqrt{3} \]

6. **Final Result**:
Therefore, the value of \( \tan\left[\left(\omega^{2017} + \omega^{2225}\right) \pi - \frac{\pi}{3}\right] \) is:
\[ \boxed{-\sqrt{3}} \]