To determine which of the given sequences correctly orders the values \(2 \pi, \frac{7 \pi}{6}, 80^{\circ}, \frac{\pi}{4}, 38^{\circ}\) from least to greatest, we firstly convert every value to the same units, typically radians for this type of problem. Here are the conversions:
1. \(2 \pi\) already in radians.
2. \(\frac{7 \pi}{6}\) already in radians.
3. \(80^{\circ}\) converted to radians is \(\frac{80 \pi}{180} = \frac{4 \pi}{9}\).
4. \(\frac{\pi}{4}\) already in radians.
5. \(38^{\circ}\) converted to radians is \(\frac{38 \pi}{180} = \frac{19 \pi}{90}\).
Now list these values:
1. \(2 \pi \approx 6.2832\)
2. \(\frac{7 \pi}{6} \approx 3.6652\)
3. \( \frac{80 \pi}{180} = \frac{4 \pi}{9} \approx 1.3963\)
4. \(\frac{\pi}{4} \approx 0.7854\)
5. \(\frac{38 \pi}{180} = \frac{19 \pi}{90} \approx 0.6635\)
Next, we order them from least to greatest:
- \(\frac{38 \pi}{180} = \frac{19 \pi}{90} \approx 0.6635\)
- \(\frac{\pi}{4} \approx 0.7854\)
- \(\frac{80 \pi}{180} = \frac{4 \pi}{9} \approx 1.3963\)
- \(\frac{7 \pi}{6} \approx 3.6652\)
- \(2 \pi \approx 6.2832\)
This sequence in degrees and radians becomes:
\(38^{\circ}, \frac{\pi}{4}, 80^{\circ}, \frac{7 \pi}{6}, 2 \pi\).
Hence, the correct order from least to greatest is:
[tex]\[ 38^{\circ}, \frac{\pi}{4}, 80^{\circ}, \frac{7 \pi}{6}, 2 \pi \][/tex]
Thus the correct choice is:
[tex]\[ 38^{\circ}, \frac{\pi}{4}, 80^{\circ}, \frac{7 \pi}{6}, 2 \pi \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{3} \][/tex]
