Question
Determine the sign of Tan(7pi/6) and its quadrant
Answers
Answered by
GPT-4o mini
To determine the sign of \(\tan\left(\frac{7\pi}{6}\right)\) and identify its quadrant, we can start by converting the angle into degrees:
\[
\frac{7\pi}{6} \text{ radians} = \frac{7 \times 180}{6} = 210^\circ.
\]
Next, we find the quadrant in which \(210^\circ\) lies. The angles in standard position are divided into four quadrants:
- Quadrant I: \(0^\circ\) to \(90^\circ\)
- Quadrant II: \(90^\circ\) to \(180^\circ\)
- Quadrant III: \(180^\circ\) to \(270^\circ\)
- Quadrant IV: \(270^\circ\) to \(360^\circ\)
Since \(210^\circ\) is between \(180^\circ\) and \(270^\circ\), it is located in Quadrant III.
In Quadrant III, the tangent function is positive because both sine and cosine values are negative in this quadrant. Therefore:
\[
\tan\left(\frac{7\pi}{6}\right) = \tan(210^\circ) > 0
\]
**Conclusion:**
- The sign of \(\tan\left(\frac{7\pi}{6}\right)\) is positive.
- It is located in Quadrant III.
\[
\frac{7\pi}{6} \text{ radians} = \frac{7 \times 180}{6} = 210^\circ.
\]
Next, we find the quadrant in which \(210^\circ\) lies. The angles in standard position are divided into four quadrants:
- Quadrant I: \(0^\circ\) to \(90^\circ\)
- Quadrant II: \(90^\circ\) to \(180^\circ\)
- Quadrant III: \(180^\circ\) to \(270^\circ\)
- Quadrant IV: \(270^\circ\) to \(360^\circ\)
Since \(210^\circ\) is between \(180^\circ\) and \(270^\circ\), it is located in Quadrant III.
In Quadrant III, the tangent function is positive because both sine and cosine values are negative in this quadrant. Therefore:
\[
\tan\left(\frac{7\pi}{6}\right) = \tan(210^\circ) > 0
\]
**Conclusion:**
- The sign of \(\tan\left(\frac{7\pi}{6}\right)\) is positive.
- It is located in Quadrant III.