To determine why [tex]\(\tan \frac{5 \pi}{6} \neq \tan \frac{5 \pi}{3}\)[/tex], let's analyze the properties of the tangent function in different quadrants and the reference angles of the given angles.
1. Quadrant Analysis:
- [tex]\( \frac{5 \pi}{6} \)[/tex]:
[tex]\[ \frac{5 \pi}{6} \text{ is in the second quadrant because } \frac{\pi}{2} < \frac{5 \pi}{6} < \pi. \][/tex]
In the second quadrant, the tangent of an angle is negative.
- [tex]\( \frac{5 \pi}{3} \)[/tex]:
[tex]\[ \frac{5 \pi}{3} = 2 \pi - \frac{\pi}{3} \text{ is in the fourth quadrant because } \frac{3 \pi}{2} < \frac{5 \pi}{3} < 2 \pi. \][/tex]
In the fourth quadrant, the tangent of an angle is negative.
2. Reference Angles:
- The reference angle for [tex]\( \frac{5 \pi}{6} \)[/tex]:
[tex]\[ \text{Reference angle} = \pi - \frac{5 \pi}{6} = \frac{\pi}{6}. \][/tex]
- The reference angle for [tex]\( \frac{5 \pi}{3} \)[/tex]:
[tex]\[ \text{Reference angle} = 2 \pi - \frac{5 \pi}{3} = \frac{\pi}{3}. \][/tex]
3. Sign of Tangent:
- In the second quadrant, the tangent function is positive.
- In the fourth quadrant, the tangent function is negative.
Hence, since the angles [tex]\( \frac{5 \pi}{6} \)[/tex] and [tex]\( \frac{5 \pi}{3} \)[/tex] are in different quadrants, and tangent is positive in the second quadrant and negative in the fourth quadrant, they have different signs.
Therefore, the best explanation for why [tex]\(\tan \frac{5 \pi}{6} \neq \tan \frac{5 \pi}{3}\)[/tex] is:
Tangent is positive in the second quadrant and negative in the fourth quadrant.
