To determine the sign of \(\tan\left(\frac{7\pi}{6}\right)\) and the quadrant in which it lies, we can analyze the angle.
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Identify the quadrant:
- The angle \(\frac{7\pi}{6}\) is equal to \(210^\circ\) (since \(\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ\)).
- The angle \(210^\circ\) lies in the third quadrant (Quadrant III).
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Determine the sign of tangent in that quadrant:
- In the third quadrant, both sine and cosine values are negative, hence the tangent, which is the ratio of sine to cosine, will be positive:
\[ \tan\left(\frac{7\pi}{6}\right) = \frac{\sin\left(\frac{7\pi}{6}\right)}{\cos\left(\frac{7\pi}{6}\right)} \]
Since both sine and cosine are negative in the third quadrant, their ratio (tangent) is positive.
So the correct option is: tan(7π/6) is positive and lies in Quadrant III.