Determine the sign of tan(7π/6) and the quadrant in which it lies.(1 point) Responses tan(7π6) is positive and lies in Quadrant I. tangent of open paren 7 pi over 6 close paren is positive and lies in Quadrant I. tan(7π6) is positive and lies in Quadrant III. tangent of open paren 7 pi over 6 close paren is positive and lies in Quadrant III. tan(7π6) is negative and lies in Quadrant IV. tangent of open paren 7 pi over 6 close paren is negative and lies in Quadrant IV. tan(7π6) is negative and lies in Quadrant III. Which option is correct

Answer 1

To determine the sign of \(\tan\left(\frac{7\pi}{6}\right)\) and the quadrant in which it lies, we can analyze the angle.

1. 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).

2. 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.