To determine the correct order of the given angles from least to greatest, we need to convert all angles to a common unit, typically radians, since angles in radians are often easier to compare, especially when dealing with a mix of degrees and radian measures.
First, let's list the given angles and convert the ones given in degrees to radians:
1. [tex]\(2\pi\)[/tex] radians
2. [tex]\(\frac{7\pi}{6}\)[/tex] radians
3. [tex]\(80^{\circ}\)[/tex]
4. [tex]\(\frac{\pi}{4}\)[/tex] radians
5. [tex]\(38^{\circ}\)[/tex]
Now, converting [tex]\(80^{\circ}\)[/tex] and [tex]\(38^{\circ}\)[/tex] to radians:
[tex]\[
80^{\circ} = 80 \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \text{ radians}
\][/tex]
[tex]\[
38^{\circ} = 38 \times \frac{\pi}{180} = \frac{38\pi}{180} = \frac{19\pi}{90} \text{ radians}
\][/tex]
With all angles converted to radians, we have:
1. [tex]\(2\pi\)[/tex]
2. [tex]\(\frac{7\pi}{6}\)[/tex]
3. [tex]\(\frac{4\pi}{9}\)[/tex]
4. [tex]\(\frac{\pi}{4}\)[/tex]
5. [tex]\(\frac{19\pi}{90}\)[/tex]
Next, we compare these angles to arrange them from least to greatest. We can use approximations or direct comparison of the fractional values:
1. [tex]\(\frac{19\pi}{90} \approx 0.6632\)[/tex]
2. [tex]\(\frac{\pi}{4} = 0.7854\)[/tex]
3. [tex]\(\frac{4\pi}{9} \approx 1.3963\)[/tex]
4. [tex]\(\frac{7\pi}{6} \approx 3.6652\)[/tex]
5. [tex]\(2\pi \approx 6.2832\)[/tex]
Arranging these angles from smallest to largest based on their approximate values:
[tex]\[
\frac{19\pi}{90}, \frac{\pi}{4}, \frac{4\pi}{9}, \frac{7\pi}{6}, 2\pi
\][/tex]
Translating back to the original representations:
1. [tex]\(38^{\circ}\)[/tex]
2. [tex]\(\frac{\pi}{4}\)[/tex]
3. [tex]\(80^{\circ}\)[/tex]
4. [tex]\(\frac{7\pi}{6}\)[/tex]
5. [tex]\(2\pi\)[/tex]
Thus, the angles in the correct order from least to greatest are:
[tex]\[
38^{\circ}, \frac{\pi}{4}, 80^{\circ}, \frac{7\pi}{6}, 2\pi
\][/tex]
Checking the given options, the correct order matches the second option:
[tex]\[
38^{\circ}, \frac{\pi}{4}, 80^{\circ}, \frac{7\pi}{6}, 2\pi
\][/tex]
